###### Acknowledgements.

This research was supported by the German Research Foundation (DFG) under
grant number ME 3281/9-1 within the priority program Non-smooth and
Complementarity-based Distributed Parameter Systems: Simulation and
Hierarchical Optimization (SPP 1962). 2019-10-01 2020-03-19 1 2020 5800
10.46298/jnsao-2020-5800

# Optimal control of an abstract evolution variational inequality with
application to homogenized plasticity

Hannes Meinlschmidt Johann Radon Institute for Computational and Applied
Mathematics (RICAM), Alternberger Straße 66a, 4040 Linz, Austria (,
https://people.ricam.oeaw.ac.at/h.meinlschmidt/)
<EMAIL_ADDRESS>Christian Meyer TU Dortmund, Faculty of
Mathematics, Vogelpothsweg 87, 44227 Dortmund, Germany (,
http://www.mathematik.tu-dortmund.de/lsx/cms/de/mitarbeiter/cmeyer.html, ,
http://www.mathematik.tu-dortmund.de/lsx/cms/de/mitarbeiter/swalther.html).
<EMAIL_ADDRESS><EMAIL_ADDRESS>Stephan
Walther22footnotemark: 2

###### Abstract

The paper is concerned with an optimal control problem governed by a state
equation in form of a generalized abstract operator differential equation
involving a maximal monotone operator. The state equation is uniquely
solvable, but the associated solution operator is in general not Gâteaux
differentiable. In order to derive optimality conditions, we therefore
regularize the state equation and its solution operator, respectively, by
means of a (smoothed) Yosida approximation. We show convergence of global
minimizers for regularization parameter tending to zero and derive necessary
and sufficient optimality conditions for the regularized problems. The paper
ends with an application of the abstract theory to optimal control of
homogenized quasi-static elastoplasticity.

##  1 Introduction

This paper is concerned with an optimal control problem of the following form,
governed by an operator differential equation:

(P) $\left\\{\quad\begin{aligned} \min\text{ \ \
}&J(z,\ell)\coloneqq\Psi(z,\ell)+\Phi(\ell),\\\ \text{ s.t. }\text{ \ \
}&\overset{\boldsymbol{.}}{z}\in A(R\ell-Qz),\text{ \ \ \ \ }z(0)=z_{0},\\\
&(z,\ell)\in
H^{1}(0,T;\mathcal{H})\times\big{(}H^{1}(0,T;\mathcal{X}_{c})\cap\mathcal{U}(z_{0};M)\big{)}.\end{aligned}\right.$

Herein, $A$ is a maximal monotone operator, while $R$ and $Q$ are linear and
bounded operators in a Hilbert space $\mathcal{H}$. The control variable is
denoted by $\ell$, whereas $z$ is the state of the system. The precise
assumptions on the data are given in Section 2 below.

The particular feature of the problem under consideration is the set-valued
mapping $A$. Due to its maximal monotony, one can show that there is a well-
defined single-valued control-to-state mapping $\ell\mapsto z$ (in suitable
function spaces), but this mapping is in general _not Gâteaux differentiable_.
We are thus faced with a non-smooth optimal control problem, for which the
derivation of necessary and sufficient optimality conditions is a particular
challenge.

Depending on the precise choice of $A$, $R$, and $Q$, problem (P) covers
various application problems. For instance, quasi-static elastoplasticity is
frequently modeled in this way. Here, $R$ is the solution operator associated
with the equations of linear elasticity for given load distribution $\ell$.
Moreover, $Q$ is the sum of the solution operator of linear elasticity for
given stress distribution, the elasticity tensor, and a coercive operator
modeling hardening effects. Finally, $A$ is the convex subdifferential of the
indicator functional of the closed and convex set of feasible stresses defined
by a suitable yield condition. Details on models in elastoplasticity can be
found in [25]. Another model which is covered by the state equation in (P) is
the system of homogenized elastoplasticity, which we will study in detail in
Section 7 below.

Let us put our work into perspective. Assume for a moment that $A$ is the
convex subdifferential of a proper, convex, and lower semicontinuous
functional $\phi$ and that $Q$ is self adjoint. Then, by convex duality, the
state equation is equivalent to

(1)
$0\in\partial\phi^{*}(\overset{\boldsymbol{.}}{z})+\mathcal{E}^{\prime}(z),\quad
z(0)=z_{0},$

where $\mathcal{E}$ is the quadratic energy functional given by

$\mathcal{E}(z)\coloneqq\tfrac{1}{2}(Qz,z)_{\mathcal{H}}-\langle
R\ell,z\rangle.$

Systems of this type have been intensively studied concerning existence of
solutions and their numerical approximation, and we only refer to [29] and the
references therein. In contrast to this, the literature on optimization
problems governed by (1) is rather scarce. The research on optimal control of
equations of type (1) probably started with the sweeping process, where
$\phi=I_{-C(t)}$ is the indicator functional of a moving convex set $C(t)$,
see [30]. In the optimal control setting, $C(t)$ is most frequently set to
$C(t)=\ell(t)-Z$ with a convex set $Z$ and a driving force $\ell$. This fits
into the setting of (1) by defining $\phi\coloneqq I_{Z}$ and $Q=R=\text{id}$
(identity). Optimal control problems of this type are investigated in [9, 10,
11, 1, 15, 16, 17, 2], where the underlying Hilbert space is mostly finite
dimensional. Problems in an infinite dimensional Hilbert space are
investigated in [37, 21]. To be more precise, in these contributions,
$\mathcal{H}$ is the Sobolev space $H^{1}_{0}(\Omega)$ and $\mathcal{E}$ is
the Dirichlet energy. Moreover, $\phi^{*}$ is set to
$\phi^{*}(z)=\|z\|_{L^{1}(\Omega)}$ and its viscous regularization,
respectively, i.e.,
$\phi^{*}_{\delta}(z)=\|z\|_{L^{1}(\Omega)}+\frac{\delta}{2}\|z\|_{H^{1}_{0}(\Omega)}^{2}$.
Optimal control problems governed by quasi-static elastoplasticity with linear
kinematic hardening and von Mises yield conditions are treated in [40, 42,
43]. As already indicated above, $\phi$ is the indicator functional of the
convex set of feasible stresses in this case. All mentioned problems fit into
our framework and can thus be seen as special cases of our non-smooth
evolution. Our analysis therefore represents a generalization of existing
results on optimal control of non-smooth evolution problems and can also be
applied to application problems that were not treated in the literature so far
such as optimal control of homogenized plasticity, which is investigated in
Section 7. We emphasize that problems with non-convex energies such as damage
evolution are not covered by our analysis. Optimal control problems governed
by (1) with non-convex energy are investigated in [31, 32].

Our strategy to analyze (P) is as follows: After showing well-posedness of the
state equation and the optimal control problem, we employ the Yosida-
regularization with an additional smoothing to obtain a smooth (i.e., Fréchet
differentiable) control-to-state map. We will prove that accumulation points
of global minimizers of the regularized optimal control problems for vanishing
regularization are solutions of the original non-smooth problem (P). Moreover,
first-order necessary and second-order sufficient optimality conditions for
the regularized problems are derived. The passage to the limit to establish
optimality conditions for the original problem goes beyond the scope of this
paper and is subject to future work. The results of [43, 37] indicate that the
optimality conditions obtained in this way are rather weak and we expect the
same all the more for our general setting. Let us underline that
regularization is a widely used approach to treat optimal control problems
governed by non-smooth evolutions. We only refer to [10, 42, 21] and the
references therein.

The paper is organized as follows. After stating our standing assumptions in
Section 2, we investigate the state equation and its regularization in Section
3. In Section 4, we then turn to the optimal control problem and show that it
admits an optimal solution under our standing assumptions. Moreover, we
establish the convergence result for vanishing regularization indicated above.
Section 5 is devoted to first-order necessary optimality conditions for the
regularized problems in form of a KKT-system involving an adjoint equation. In
Section 6, we address second-order sufficient conditions for the regularized
problem. The paper ends with the adaptation of our general results to a
concrete application problem, namely the optimal control of homogenized
elastoplasticity.

##  2 Notation and Standing Assumptions

We start with a short introduction in the notation used throughout the paper.

#### Notation

Given two vector spaces $\mathcal{X}$ and $\mathcal{Y}$, we denote the space
of linear and continuous functions from $\mathcal{X}$ into $\mathcal{Y}$ by
$L(\mathcal{X},\mathcal{Y})$. If $\mathcal{X}=\mathcal{Y}$, we simply write
$L(\mathcal{X})$. The dual space of $\mathcal{X}$ is denoted by
$\mathcal{X}^{*}=L(\mathcal{X},\mathbb{R})$. If $\mathcal{H}$ is a Hilbert
space, we denote its inner product by
$\left(\cdot,\cdot\right)_{\mathcal{H}}$. For the whole paper, we fix the
final time $T>0$. We denote the Bochner space of square-integrable functions
by $L^{2}(0,T;\mathcal{X})$ and the Bochner-Sobolev space by
$H^{1}(0,T;\mathcal{X})$. Given a coercive operator
$G:\mathcal{H}\rightarrow\mathcal{H}$ in a Hilbert space $\mathcal{H}$, we
denote its coercivity constant by $\gamma_{G}$, i.e.,
$\left(Gh,h\right)_{\mathcal{H}}\geq\gamma_{G}\|h\|_{\mathcal{H}}^{2}$ for all
$h\in\mathcal{H}$. Finally, $c>0$ and $C>0$ denote generic constants.

### Standing Assumptions

The following standing assumptions are tacitly assumed for the rest of the
paper without mentioning them every time.

#### Spaces

Throughout the paper,
$\mathcal{X},\mathcal{X}_{c},\mathcal{Y},\mathcal{Z},\mathcal{W}$ are real
Banach spaces. Moreover, $\mathcal{X}_{c}$ reflexive and $\mathcal{H}$ is a
separable Hilbert space. The space $\mathcal{X}_{c}$ is compactly embedded
into $\mathcal{X}$ and the embeddings
$\mathcal{Y}\hookrightarrow\mathcal{Z}\hookrightarrow\mathcal{H}\hookrightarrow\mathcal{W}$
are continuous.

#### Operators

The operator $A\colon\mathcal{H}\rightarrow 2^{\mathcal{H}}$ is maximal
monotone, its domain $D(A)$ is closed and we define

(2) $A^{0}\colon D(A)\rightarrow\mathcal{H},\quad
h\mapsto\operatorname*{arg\,min}_{v\in A(h)}\|v\|_{\mathcal{H}}.$

Furthermore, by $A_{\lambda}\colon\mathcal{H}\rightarrow\mathcal{H}$,
$\lambda>0$, we denote the Yosida-approximation of $A$ and by
$R_{\lambda}=(I+\lambda A)^{-1}$ the resolvent of $A$, so that
$A_{\lambda}=\frac{1}{\lambda}(I-R_{\lambda})$. We assume that the operator
$A^{0}\colon D(A)\rightarrow\mathcal{H}$ is bounded on bounded sets. For
further reference on maximal monotone operators, we refer to [7], [44, Ch.
32], [45, Ch. 55], and [35, Ch. 55]. Furthermore, $R\in
L(\mathcal{X};\mathcal{Y})$ and $Q\in L(\mathcal{W};\mathcal{W})$, are linear
and bounded operators, and the restriction of $Q$ to $\mathcal{H}$,
$\mathcal{Z}$, or $\mathcal{Y}$ maps into these spaces and is again linear and
bounded. To ease notation, we denote this restriction by the same symbol.
Moreover, $Q\colon\mathcal{H}\rightarrow\mathcal{H}$ is coercive and self-
adjoint.

#### Optimization Problem

By $J\colon H^{1}(0,T;\mathcal{W})\times
H^{1}(0,T;\mathcal{X}_{c})\rightarrow\mathbb{R}$ we denote the objective
function. We assume that both $\Psi\colon H^{1}(0,T;\mathcal{W})\times
H^{1}(0,T;\mathcal{X}_{c})\to\mathbb{R}$ and $\Phi\colon
H^{1}(0,T;\mathcal{X}_{c})\to\mathbb{R}$ are weakly lower semicontinuous.
Moreover, $\Psi$ is bounded from below and continuous in the first argument,
while $\Phi$ is coercive. The set $M$ is a nonempty and closed subset of
$D(A)$ and $z_{0}\in\mathcal{Y}$ is a given initial state.

######  Remark 2.1.

We emphasize that not all of the above assumptions are always needed. For
instance, in the next two sections, $Q$ and $R$ are only considered as
operators with values in $\mathcal{H}$ and the spaces $\mathcal{Y}$ and
$\mathcal{Z}$ are not needed, before the investigation of optimality
conditions starts in Sections 5 and 6. However, to keep the discussion
concise, we present the standing assumption in the present form.

##  3 State Equation

###  3.1 Existence and Uniqueness

We start the investigation of (P) with the discussion of the state equation,
i.e.,

(3) $\overset{\boldsymbol{.}}{z}\in A(R\ell-Qz),\text{ \ \ \ \ }z(0)=z_{0}.$

######  Definition 3.1.

Let $\ell\in H^{1}(0,T;\mathcal{X})$ and $z_{0}\in\mathcal{H}$. Then $z\in
H^{1}(0,T;\mathcal{H})$ is called solution of Eq. 3, if $z(0)=z_{0}$ and
$\overset{\boldsymbol{.}}{z}(t)\in A(Rl(t)-Qz(t))$ holds for almost all
$t\in[0,T]$.

In order to obtain the existence of a solution to (3), the data have to
fulfill a certain compatibility condition. For this reason, we introduce the
following

######  Definition 3.2.

For $z_{0}\in\mathcal{H}$ and $M\subset D(A)$, we define the set

$\displaystyle\mathcal{U}(z_{0},M)$

of admissible loads.

######  Theorem 3.3 (Existence result for the state equation).

Let $z_{0}\in\mathcal{H}$ and $\ell\in\mathcal{U}(z_{0},D(A))$. Then there
exists a unique solution $z\in H^{1}(0,T;\mathcal{H})$ of Eq. 3. Furthermore,
there exists a constant $C$, independent of $z_{0}$ and $\ell$, such that

(4) $\displaystyle\|z\|_{C([0,T];\mathcal{H})}$ $\displaystyle\leq
C\big{(}1+\|z_{0}\|_{\mathcal{H}}+\|\ell\|_{C([0,T];\mathcal{X})}+\|\overset{\boldsymbol{.}}{\ell}\|_{L^{1}(0,T;\mathcal{X})}\big{)},$
(5) $\displaystyle\|\overset{\boldsymbol{.}}{z}\|_{L^{2}(0,T;\mathcal{H})}$
$\displaystyle\leq
C\big{(}\|\overset{\boldsymbol{.}}{\ell}\|_{L^{2}(0,T;\mathcal{X})}+\sup_{\tau\in[0,T]}\|A^{0}(R\ell(\tau)-Qz(\tau))\|_{\mathcal{H}}\big{)},$

where $A^{0}$ is as defined in (2).

######  Proof 3.4.

The proof essentially follows the lines of [24, Theorem 4.1] and [7,
Proposition 3.4]. For convenience of the reader, we sketch the main arguments.
At first, one employs the transformation $H^{1}(0,T;\mathcal{H})\ni z\mapsto
q$ with its inverse $H^{1}(0,T;\mathcal{H})\ni q\mapsto z$ to see that Eq. 3
is equivalent to

(6) $\displaystyle\overset{\boldsymbol{.}}{q}+QA(q)\ni
R\overset{\boldsymbol{.}}{\ell},\text{ \ \ \ \ }q(0)=R\ell(0)-Qz_{0}.$

Since $Q$ is coercive the operator, $\tilde{A}\colon\mathcal{H}\rightarrow
2^{\mathcal{H}}$, $h\mapsto QA(h)$ is maximal monotone with respect to the
scalar product

$\displaystyle\left(h_{1},h_{2}\right)_{\mathcal{H},Q^{-1}}$

Therefore, [7, Proposition 3.4] yields the existence of a unique solution
$q\in H^{1}(0,T;\mathcal{H})$ of Eq. 6. To verify the estimate in (4), we
employ [7, Lemme 3.1], which gives

$\displaystyle\|q(t)-\tilde{q}(t)\|_{\mathcal{H},Q^{-1}}\leq\|R\ell(0)-Qz_{0}-a\|_{\mathcal{H},Q^{-1}}+\int_{0}^{t}\|R\overset{\boldsymbol{.}}{\ell}(\tau)\|_{\mathcal{H},Q^{-1}}\,d\tau,$

where $\tilde{q}$ is the unique solution of

$\displaystyle\overset{\boldsymbol{.}}{\tilde{q}}+QA(\tilde{q})\ni 0,\text{ \
\ \ \ }q(0)=a$

with an arbitrary element $a\in D(A)$. This gives the desired first
inequality.

To prove the second inequality, we deduce from [7, Proposition 3.4] and the
associated proof that

$\displaystyle\|q(t)-q(s)\|_{\mathcal{H},Q^{-1}}\leq\int_{s}^{t}\|R\overset{\boldsymbol{.}}{\ell}(\tau)\|_{\mathcal{H},Q^{-1}}\,d\tau+\sup_{\tau\in[0,T]}\|\tilde{A}^{0}(q(\tau))\|_{\mathcal{H},Q^{-1}}(t-s).$

Dividing this inequality by $(t-s)$ and letting $t\rightarrow s$ yields

(7)
$\|\overset{\boldsymbol{.}}{q}(s)\|_{\mathcal{H},Q^{-1}}\leq\|R\overset{\boldsymbol{.}}{\ell}(s)\|_{\mathcal{H},Q^{-1}}+\sup_{\tau\in[0,T]}\|\tilde{A}^{0}(q(\tau))\|_{\mathcal{H},Q^{-1}}$

for almost all $s\in[0,T]$. From the definition of $\tilde{A}^{0}$ (with
respect to $\left(\cdot,\cdot\right)_{\mathcal{H},Q^{-1}}$) we see that
$\|\tilde{A}^{0}(h)\|_{\mathcal{H},Q^{-1}}\leq\|Qv\|_{\mathcal{H},Q^{-1}}$ for
all $v\in A(h)$. This holds in particular for $v=A^{0}(h)$ so that
$z=Q^{-1}(R\ell-q)$ and (7) imply

$\displaystyle\|\overset{\boldsymbol{.}}{z}(s)\|_{\mathcal{H}}\leq
C\big{(}\|\overset{\boldsymbol{.}}{\ell}(s)\|_{\mathcal{X}}+\|\overset{\boldsymbol{.}}{q}(s)\|_{\mathcal{H},Q^{-1}}\big{)}\leq
C\Big{(}\|\overset{\boldsymbol{.}}{\ell}(s)\|_{\mathcal{X}}+\sup_{\tau\in[0,T]}\|A^{0}(R\ell(\tau)-Qz(\tau))\|_{\mathcal{H}}\Big{)},$

which gives the second inequality.

######  Remark 3.5.

In order to prove Theorem 3.3, it is sufficient to require that $A^{0}$ is
bounded on compact subsets (in addition to the closedness of $D(A)$), cf. [7,
Proposition 3.4]. However, the boundedness on bounded sets of $A^{0}$ is
needed to prove Theorem 3.16 below and therefore, we impose it as a standing
assumption.

Based on Theorem 3.3, we may introduce the solution operator associated with
(3) and reduce (P) to an optimization problem in the control variable only,
see Definition 4.1 below. Due to the set-valued operator $A$, this solution
operator will in general be _non-smooth_ , which complicates the derivation of
first- and second-order optimality conditions. A prominent way to overcome
this issue is to regularize the state equation in order to obtain a smooth
solution operator. This is frequently done by means of the Yosida-
approximation, see e.g. [5], and we will pursue the same approach. For this
purpose, we will investigate the Yosida-approximation and its convergence
properties in the next subsection.

###  3.2 Regularization and Convergence Results

For the rest of this section, we fix $z_{0}\in\mathcal{H}$ and
$\ell\in\mathcal{U}(z_{0},D(A))$ and denote the unique solution of (3) by $z$.
We start with a convergence result of the Yosida-approximation for fixed data
$z_{0}$ and $\ell$ and then turn to perturbation of the data.

######  Proposition 3.6 (Convergence of the Yosida-approximation for fixed
data).

Let $z_{\lambda}\in H^{1}(0,T;\mathcal{H})$ be the solution of

(8) $\displaystyle\overset{\boldsymbol{.}}{z}_{\lambda}=A_{\lambda}(R\ell-
Qz_{\lambda}),\text{ \ \ \ \ \ \ }z_{\lambda}(0)=z_{0}$

for all $\lambda>0$. Then $z_{\lambda}\rightarrow z$ in
$H^{1}(0,T;\mathcal{H})$ as $\lambda\searrow 0$ and the following inequality
holds

(9)
$\|z_{\lambda}-z\|_{C([0,T];\mathcal{H})}^{2}+\frac{\lambda}{\gamma_{Q}}\|\overset{\boldsymbol{.}}{z}_{\lambda}\|_{L^{2}(0,T;\mathcal{H})}^{2}+\frac{\lambda}{\gamma_{Q}}\|\overset{\boldsymbol{.}}{z}_{\lambda}-\overset{\boldsymbol{.}}{z}\|_{L^{2}(0,T;\mathcal{H})}^{2}\leq\frac{\lambda}{\gamma_{Q}}\|\overset{\boldsymbol{.}}{z}\|_{L^{2}(0,T;\mathcal{H})}^{2}.$

######  Proof 3.7.

The proof in principle follows the lines of [7, Proposition 3.11], since our
assumptions and assertions however are slightly different, we provide the
arguments in detail.

First of all, since $z\mapsto A_{\lambda}(R\ell-Qz)$ is Lipschitz-continuous
by [45, Proposition 55.2(b)], the existence of a unique solution of Eq. 8
follows from Banach’s contraction principle by standard arguments, cf. e.g.
[19]. Moreover, [45, Proposition 55.2(a)] and the definition of $A_{\lambda}$
give

$\displaystyle\frac{d}{dt}\left(Q(z_{\lambda}(t)-z(t)),z_{\lambda}(t)-z(t)\right)_{\mathcal{H}}=2\left(\overset{\boldsymbol{.}}{z}_{\lambda}(t)-\overset{\boldsymbol{.}}{z}(t),Q(z_{\lambda}(t)-z(t))\right)_{\mathcal{H}}$
$\displaystyle\quad=-2\left(\overset{\boldsymbol{.}}{z}_{\lambda}(t)-\overset{\boldsymbol{.}}{z}(t),R_{\lambda}\big{[}R\ell(t)-Qz_{\lambda}(t)\big{]}-\big{[}R\ell(t)-Qz(t)\big{]}\right)_{\mathcal{H}}$
$\displaystyle\qquad-2\left(\overset{\boldsymbol{.}}{z}_{\lambda}(t)-\overset{\boldsymbol{.}}{z}(t),R\ell(t)-Qz_{\lambda}(t)-R_{\lambda}\big{[}R\ell(t)-Qz_{\lambda}(t)\big{]}\right)_{\mathcal{H}}$
$\displaystyle\quad\leq-2\lambda\left(\overset{\boldsymbol{.}}{z}_{\lambda}(t)-\overset{\boldsymbol{.}}{z}(t),\overset{\boldsymbol{.}}{z}_{\lambda}(t)\right)_{\mathcal{H}}=\lambda\Big{(}\|\overset{\boldsymbol{.}}{z}(t)\|_{\mathcal{H}}^{2}-\|\overset{\boldsymbol{.}}{z}_{\lambda}(t)\|_{\mathcal{H}}^{2}-\|\overset{\boldsymbol{.}}{z}_{\lambda}(t)-\overset{\boldsymbol{.}}{z}(t)\|_{\mathcal{H}}^{2}\Big{)}.$

By integrating this inequality and using the coercivity of $Q$, we obtain the
desired inequality.

In order to prove the strong convergence of $z_{\lambda}$ to $z$ in
$H^{1}(0,T;\mathcal{H})$, we note that $z_{\lambda}\rightarrow z$ in
$C([0,T];\mathcal{H})$ and
$\|\overset{\boldsymbol{.}}{z}_{\lambda}\|_{L^{2}(0,T;\mathcal{H})}\leq\|\overset{\boldsymbol{.}}{z}\|_{L^{2}(0,T;\mathcal{H})}$
follow from the gained inequality. Hence, $z_{\lambda}\rightharpoonup z$ in
$H^{1}(0,T;\mathcal{H})$ and the desired strong convergence follows from [8,
Proposition 3.32].

######  Remark 3.8.

The above proof shows that the inequality in (9) is by no means restricted to
the specific setting in (3), i.e., whenever $\zeta\in H^{1}(0,T;\mathcal{H})$
and $\zeta_{\lambda}\in H^{1}(0,T;\mathcal{H})$ solve

$\displaystyle\overset{\boldsymbol{.}}{\zeta}$
$\displaystyle=\mathcal{A}(Rg-Q\zeta),$ $\displaystyle\zeta(0)$
$\displaystyle=\zeta_{0},$
$\displaystyle\overset{\boldsymbol{.}}{\zeta}_{\lambda}$
$\displaystyle=\mathcal{A}_{\lambda}(Rg-Q\zeta_{\lambda}),$
$\displaystyle\zeta_{\lambda}(0)$ $\displaystyle=\zeta_{0},$

where $\mathcal{A}\colon\mathcal{H}\to 2^{\mathcal{H}}$ is a maximal monotone
operator, $\mathcal{A}_{\lambda}\colon\mathcal{H}\to\mathcal{H}$ its Yosida-
approximation and $g\in L^{2}(0,T;\mathcal{X})$ and $\zeta_{0}\in\mathcal{H}$
are given, then an inequality analogue to (9) holds (with $\zeta$ and
$\zeta_{\lambda}$ instead of $z$ and $z_{\lambda}$).

Since we are concerned with an optimal control problem with the external loads
as control variable, the continuity of the solution operator of (3) and its
regularization w.r.t. variations in the external loads is of particular
interest, for instance when it comes to the existence of optimal controls, see
Section 4 below. Since we aim to have a less restrictive control space in
order to allow for as many control functions as possible, the topology for the
variations of the loads needed for our continuity results should be as weak as
possible. In particular, we aim to avoid _strong convergence_ of
(time-)derivatives of the loads. The underlying idea is similar to [7, Theorem
3.16] and leads to the following

######  Lemma 3.9.

Let $\\{z_{n,0}\\}_{n\in\mathbb{N}}\subset\mathcal{H}$ and
$\\{\ell_{n}\\}_{n\in\mathbb{N}}\subset L^{2}(0,T;\mathcal{X})$ be sequences
such that $z_{n,0}\rightarrow z_{0}$ in $\mathcal{H}$ and
$\ell_{n}\rightarrow\ell$ in $L^{1}(0,T;\mathcal{X})$. Assume further that
$\\{A_{n}\\}_{n\in\mathbb{N}}$ is a sequence of maximal monotone operators
such that

(10) $A_{n,\lambda}(h)\rightarrow A_{\lambda}(h)$

for all $\lambda>0$ and all $h\in(R\ell-Qz_{\lambda})([0,T])$, as
$n\rightarrow\infty$, where $z_{\lambda}$ is the solution of (8) and
$A_{n,\lambda}$ denotes the Yosida-approximation of $A_{n}$. Then, if a
sequence $\\{z_{n}\\}_{n\in\mathbb{N}}\subset H^{1}(0,T;\mathcal{H})$
satisfies

(11) $\overset{\boldsymbol{.}}{z}_{n}\in A_{n}(R\ell_{n}-Qz_{n}),\text{ \ \ \
\ }z_{n}(0)=z_{n,0}.$

and the derivatives $\overset{\boldsymbol{.}}{z}_{n}$ are bounded in
$L^{2}(0,T;\mathcal{H})$, then $z_{n}\rightharpoonup z$ in
$H^{1}(0,T;\mathcal{H})$ and $z_{n}\rightarrow z$ in $C([0,T];\mathcal{H})$.

######  Proof 3.10.

Let $\lambda>0$ be fixed, but arbitrary and define $z_{n,\lambda}\in
H^{1}(0,T;\mathcal{H})$ as solution of

$\overset{\boldsymbol{.}}{z}_{n,\lambda}=A_{n,\lambda}(R\ell_{n}-Qz_{n,\lambda}),\quad
z_{n,\lambda}(0)=z_{n,0},$

whose existence and uniqueness can again be shown by Banach’s contraction
principle as in case of (8). Owing to [45, Proposition 55.2(b)], we obtain

$\displaystyle\|\overset{\boldsymbol{.}}{z}_{\lambda}(t)-\overset{\boldsymbol{.}}{z}_{n,\lambda}(t)\|_{\mathcal{H}}$
$\displaystyle\leq\|A_{\lambda}(R\ell(t)-Qz_{\lambda}(t))-A_{n,\lambda}(R\ell(t)-Qz_{\lambda}(t))\|_{\mathcal{H}}$
$\displaystyle\qquad+\|A_{n,\lambda}(R\ell(t)-Qz_{\lambda}(t))-A_{n,\lambda}(R\ell_{n}(t)-Qz_{n,\lambda}(t))\|_{\mathcal{H}}$
$\displaystyle\leq\|A_{\lambda}(R\ell(t)-Qz_{\lambda}(t))-A_{n,\lambda}(R\ell(t)-Qz_{\lambda}(t))\|_{\mathcal{H}}$
$\displaystyle\qquad+\frac{\|Q\|_{L(\mathcal{H};\mathcal{H})}}{\lambda}\|z_{\lambda}(t)-z_{n,\lambda}(t)\|_{\mathcal{H}}+\frac{\|R\|_{L(\mathcal{X};\mathcal{H})}}{\lambda}\|\ell(t)-\ell_{n}(t)\|_{\mathcal{X}},$

and therefore, Gronwall’s inequality implies

$\displaystyle\|z_{\lambda}-z_{n,\lambda}\|_{C([0,T];\mathcal{H})}$
$\displaystyle\leq
C(\lambda)\Big{(}\|z_{0}-z_{n,0}\|_{\mathcal{H}}+\|\ell-\ell_{n}\|_{L^{1}(0,T;\mathcal{X})}$
$\displaystyle\qquad\qquad+\|A_{\lambda}(R\ell-
Qz_{\lambda})-A_{n,\lambda}(R\ell-
Qz_{\lambda})\|_{L^{1}(0,T;\mathcal{H})}\Big{)}.$

The operators $A_{n,\lambda}$ are uniformly Lipschitz continuous with
Lipschitz constant $\lambda^{-1}$. Thus, thanks to assumption Eq. 10, we can
apply Lemma C.1 with $\mathcal{M}$, $\mathcal{N}$, $G_{n}$ and $G$. Together
with the assumptions on $\ell_{n}$ and $z_{n,0}$ this gives that the right
side in the inequality above converges to zero as $n\rightarrow\infty$. Using
this, Proposition 3.6, and Remark 3.8 (with $\mathcal{A}=A_{n}$), we conclude

(12)
$\displaystyle\limsup_{n\rightarrow\infty}\|z-z_{n}\|_{C([0,T];\mathcal{H})}$
$\displaystyle\leq\|z-z_{\lambda}\|_{C([0,T];\mathcal{H})}+\limsup_{n\rightarrow\infty}\|z_{n,\lambda}-z_{n}\|_{C([0,T];\mathcal{H})}$
$\displaystyle\leq\sqrt{\frac{\lambda}{\gamma_{Q}}}\big{(}\|\overset{\boldsymbol{.}}{z}\|_{L^{2}(0,T;\mathcal{H})}+\sup_{n\in\mathbb{N}}\|\overset{\boldsymbol{.}}{z}_{n}\|_{L^{2}(0,T;\mathcal{H})}\big{)}.$

Now, since $\lambda$ was arbitrary, (12) holds for every $\lambda>0$.
Therefore, as $\overset{\boldsymbol{.}}{z}_{n}$ is bounded in
$L^{2}(0,T;\mathcal{H})$ by assumption, we obtain $z_{n}\rightarrow z$ in
$C([0,T];\mathcal{H})$. Moreover, again due to the boundedness assumption on
$\overset{\boldsymbol{.}}{z}_{n}$, there is a weakly converging subsequence in
$H^{1}(0,T;\mathcal{H})$. Due to $z_{n}\rightarrow z$ in
$C([0,T];\mathcal{H})$, the weak limit is unique and hence, the whole sequence
$z_{n}$ converges weakly to $z$ in $H^{1}(0,T;\mathcal{H})$.

######  Lemma 3.11.

Let $\\{\lambda_{n}\\}_{n\in\mathbb{N}}\subset(0,\infty)$ be a sequence
converging towards zero. Then the sequence $A_{n}$, $n\in\mathbb{N}$, of
maximal monotone operators fulfills Eq. 10 for all $\lambda>0$ and all
$h\in\mathcal{H}$.

######  Proof 3.12.

At first we prove that, for all $h\in\mathcal{H}$ and $2\lambda>\mu>0$, the
following inequality holds

(13)
$\displaystyle\|R_{\lambda}(h)-R_{\lambda+\mu}(h)\|_{\mathcal{H}}\leq\sqrt{\frac{\mu}{2\lambda-\mu}}\,\|h-R_{\lambda}(h)\|_{\mathcal{H}}.$

For this purpose, let $h\in\mathcal{H}$ be arbitrary and set $y_{1}$ and
$y_{2}$. Then we have $h\in y_{1}+\lambda A(y_{1})$, hence,
$\frac{h-y_{1}}{\lambda}\in A(y_{1})$ and analogously
$\frac{h-y_{2}}{\lambda+\mu}\in A(y_{2})$. The monotonicity of $A$ thus
implies

$\displaystyle
0\leq\left(\frac{\lambda+\mu}{\lambda}(h-y_{1})-(h-y_{2}),y_{1}-y_{2}\right)_{\mathcal{H}}\leq\Big{(}\frac{\mu}{2\lambda}-1\Big{)}\|y_{1}-y_{2}\|_{\mathcal{H}}^{2}+\frac{\mu}{2\lambda}\|h-y_{1}\|_{\mathcal{H}}^{2},$

hence,

$\displaystyle\|y_{1}-y_{2}\|_{\mathcal{H}}^{2}\leq\frac{\mu}{2\lambda-\mu}\|h-y_{1}\|_{\mathcal{H}}^{2},$

which yields Eq. 13. With this inequality and [45, Proposition 55.2 (d)] at
hand, we obtain

$\displaystyle(A_{\lambda_{n}})_{\lambda}(h)=A_{\lambda_{n}+\lambda}(h)=\frac{1}{\lambda_{n}+\lambda}(h-R_{\lambda_{n}+\lambda}(h))\rightarrow\frac{1}{\lambda}(h-R_{\lambda}(h))=A_{\lambda}(h),$

which completes the proof.

Now, we are in the position to state our main convergence results in Theorem
3.14 and Theorem 3.16, where the loads and initial data are no longer fixed.
The first theorem addresses the continuity properties of the solution operator
to the original equation (3), whereas Theorem 3.16 deals with the Yosida-
approximation. In order to sharpen these convergence results and prove strong
convergence in $H^{1}(0,T;\mathcal{H})$, we additionally need the following

######  Assumption 3.13.

The maximal monotone operator $A$ is given as a subdifferential of a proper,
convex and lower semicontinuous function
$\phi\colon\mathcal{H}\rightarrow(-\infty,\infty]$, that is, $A=\partial\phi$.

######  Theorem 3.14 (Continuity of the solution operator).

Let $\\{z_{n,0}\\}_{n\in\mathbb{N}}\subset\mathcal{H}$ and
$\\{\ell_{n}\\}_{n\in\mathbb{N}}\subset\mathcal{U}(z_{n,0},D(A))$ be sequences
such that $z_{n,0}\rightarrow z_{0}$ in $\mathcal{H}$,
$\ell_{n}\rightharpoonup\ell$ in $H^{1}(0,T;\mathcal{X})$ and
$\ell_{n}\rightarrow\ell$ in $L^{1}(0,T;\mathcal{X})$. Moreover, denote the
solution of

$\overset{\boldsymbol{.}}{z}_{n}\in A(R\ell_{n}-Qz_{n}),\text{ \ \ \ \
}z_{n}(0)=z_{n,0}$

by $z_{n}\in H^{1}(0,T;\mathcal{H})$ (whose existence is guaranteed by Theorem
3.3). Then $z_{n}\rightharpoonup z$ in $H^{1}(0,T;\mathcal{H})$ and
$z_{n}\rightarrow z$ in $C([0,T];\mathcal{H})$.

If additionally $\ell_{n}\rightarrow\ell$ in $H^{1}(0,T;\mathcal{X})$, $A$
fulfills 3.13, and
$\phi(R\ell_{n}(0)-Qz_{n,0})\rightarrow\phi(R\ell(0)-Qz_{0})$, then
$z_{n}\rightarrow z$ in $H^{1}(0,T;\mathcal{H})$.

######  Proof 3.15.

Thanks to Theorem 3.3, to be more precise (4), $\\{z_{n}\\}$ is bounded in
$C([0,T];\mathcal{H})$. Since $A^{0}$ is bounded on bounded sets by
assumption, (5) then gives that $\\{\overset{\boldsymbol{.}}{z}_{n}\\}$ is
bounded in $L^{2}(0,T;\mathcal{H})$. Therefore, we can apply Lemma 3.9 with
$A_{n}$ for all $n\in\mathbb{N}$ to obtain $z_{n}\rightharpoonup z$ in
$H^{1}(0,T;\mathcal{H})$ and $z_{n}\rightarrow z$ in $C([0,T];\mathcal{}H)$.

If additionally $\ell_{n}\rightarrow\ell$ in $H^{1}(0,T;\mathcal{X})$, $A$
fulfills 3.13, and
$\phi(R\ell_{n}(0)-Qz_{n,0})\rightarrow\phi(R\ell(0)-Qz_{0})$, we can follow
the lines of [24, Theorem 4.2 step 3)] to get

$\displaystyle\limsup_{n\rightarrow\infty}\int_{0}^{T}(Q\overset{\boldsymbol{.}}{z}_{n},\overset{\boldsymbol{.}}{z}_{n})_{\mathcal{H}}\,dt$
$\displaystyle=\limsup_{n\rightarrow\infty}-(R\overset{\boldsymbol{.}}{\ell}_{n}-Q\overset{\boldsymbol{.}}{z}_{n},\overset{\boldsymbol{.}}{z}_{n})_{L^{2}(0,T;\mathcal{H})}+(R\overset{\boldsymbol{.}}{\ell},\overset{\boldsymbol{.}}{z})_{L^{2}(0,T;\mathcal{H})}$
$\displaystyle=\limsup_{n\rightarrow\infty}\phi(R\ell_{n}(0)-Qz_{n,0})-\phi(R\ell_{n}(T)-Qz_{n}(T))+(R\overset{\boldsymbol{.}}{\ell},\overset{\boldsymbol{.}}{z})_{L^{2}(0,T;\mathcal{H})}$
$\displaystyle\leq\phi(R\ell(0)-Qz_{0})-\phi(R\ell(T)-Qz(T))+(R\overset{\boldsymbol{.}}{\ell},\overset{\boldsymbol{.}}{z})_{L^{2}(0,T;\mathcal{H})}$
$\displaystyle=\int_{0}^{T}(Q\overset{\boldsymbol{.}}{z},\overset{\boldsymbol{.}}{z})_{\mathcal{H}}\,dt$

where the second and last equation follows from [7, Lemme 3.3]. Hence, by
equipping $\mathcal{H}$ with the equivalent norm
$\sqrt{(Q\cdot,\cdot)_{\mathcal{H}}}$, the strong convergence
$z_{n}\rightarrow z$ in $H^{1}(0,T;\mathcal{H})$ follows from [8, Proposition
3.32].

######  Theorem 3.16 (Convergence of the Yosida-approximation).

The statement of Theorem 3.14 holds true when $z_{n}$ is, for every
$n\in\mathbb{N}$, the solution of

(14)
$\displaystyle\overset{\boldsymbol{.}}{z}_{n}=A_{\lambda_{n}}(R\ell_{n}-Qz_{n}),\text{
\ \ \ \ }z_{n}(0)=z_{n,0},$

where $\\{\lambda_{n}\\}_{n\in\mathbb{N}}\subset(0,\infty)$ is a sequence
converging to zero.

######  Proof 3.17.

According to Lemma 3.11, the sequence of maximal monotone operators $A_{n}$
fulfills Eq. 10 so that it only remains to prove that
$\overset{\boldsymbol{.}}{z}_{n}$ is bounded in $L^{2}(0,T;\mathcal{H})$ to
apply again Lemma 3.9. To this end, let $v_{n}\in H^{1}(0,T;\mathcal{H})$ be
the solution of

$\displaystyle\overset{\boldsymbol{.}}{v}_{n}\in A(R\ell_{n}-Qv_{n}),\text{ \
\ \ \ }v_{n}(0)=z_{n,0},$

whose existence is guaranteed by Theorem 3.3 (note that
$\ell_{n}\in\mathcal{U}(z_{n,0},D(A))$ by assumption). Thanks to Theorem 3.14,
it holds $v_{n}\rightharpoonup z$ in $H^{1}(0,T;\mathcal{H})$. From
Proposition 3.6, it follows
$\|\overset{\boldsymbol{.}}{z}_{n}\|_{L^{2}(0,T;\mathcal{H})}\leq\|\overset{\boldsymbol{.}}{v}_{n}\|_{L^{2}(0,T;\mathcal{H})}$
and consequently, $\overset{\boldsymbol{.}}{z}_{n}$ is bounded in
$L^{2}(0,T;\mathcal{H})$. Thus, Lemma 3.9 yields $z_{n}\rightharpoonup z$ in
$H^{1}(0,T;\mathcal{H})$ and $z_{n}\rightarrow z$ in $C([0,T];\mathcal{H})$,
as claimed.

If additionally $\ell_{n}\rightarrow\ell$ in $H^{1}(0,T;\mathcal{X})$, $A$
fulfills 3.13, and
$\phi(R\ell_{n}(0)-Qz_{n,0})\rightarrow\phi(R\ell(0)-Qz_{0})$, then Theorem
3.14 implies $v_{n}\rightarrow z$ in $H^{1}(0,T;\mathcal{H})$ so that [8,
Proposition 3.32] gives the strong convergence $z_{n}\rightarrow z$ in
$H^{1}(0,T;\mathcal{H})$ because of
$\|\overset{\boldsymbol{.}}{z}_{n}\|_{L^{2}(0,T;\mathcal{H})}\leq\|\overset{\boldsymbol{.}}{v}_{n}\|_{L^{2}(0,T;\mathcal{H})}$
as seen above.

######  Remark 3.18.

The assertions of Theorem 3.14 and Theorem 3.16 are remarkable due to the
following: As a first approach to prove the (strong) convergence of the states
in $H^{1}(0,T;\mathcal{H})$, one is tempted to follow the lines of the proofs
of [7, Lemme 3.1] and Proposition 3.6, respectively. This would however
require the strong convergence of the _derivatives_ of the given loads, which
we want to avoid in order to enable less regular controls. The detour via the
Yosida-regularization in Lemma 3.9 allows to overcome this issue.

######  Remark 3.19.

If we would allow for more regular controls, then we could weaken the
assumptions on the maximal monotone operator $A$. For instance, if $\ell\in
H^{2}(0,T;\mathcal{X})$, then we can drop the assumptions that $D(A)$ is
closed and $A^{0}$ is bounded on bounded sets. In this case, one can use [45,
Theorem 55.A] instead of [7, Proposition 3.4] in the proof of Theorem 3.3. The
proof of [45, Theorem 55.A] also gives the boundedness of
$\overset{\boldsymbol{.}}{z}_{n}$ in $L^{\infty}(0,T;\mathcal{H})$ in this
case. Thus, Lemma 3.9 is again applicable and we can argue similar as we did
in the proof of Theorem 3.16 to verify the previous convergence results. In
this setting, we would not have any restrictions on $A$ (except monotonicity),
but would need more regular loads, which is not favorable, as the latter serve
as control variables in our optimization problem. Moreover, the boundedness
assumption on $A$ is fulfilled for our concrete application problem in Section
7.4. Therefore, we decided to choose the present setting and to impose the
additional boundedness assumption on $A$.

######  Remark 3.20.

It is to be noted that most of the above results can also be shown in more
general Bochner-Sobolev spaces, that is, when loads are contained in
$W^{1,r}(0,T;\mathcal{X})$ and states in $W^{1,r}(0,T;\mathcal{H})$ for some
$r\in[1,\infty)$. However, since a Hilbert space setting is advantageous when
it comes to the derivation of optimality conditions, we focus on the case
$r=2$.

Unfortunately, the Yosida-approximation is frequently not sufficient for the
derivation of optimality conditions by means of the standard adjoint approach,
since the solution operator associated with (8) is in general still not
Gâteaux differentiable. Therefore, we apply a second regularization turning
the Yosida-approximation of $A$ into a smooth operator. The properties needed
to ensure convergence of this second regularization are investigated in the
following

######  Lemma 3.21 (Convergence of the Regularized Yosida-Approximation).

Consider a sequence $\\{\lambda_{n}\\}_{n\in\mathbb{N}}\subset(0,\infty)$ and
a sequence of Lipschitz continuous operators
$A_{n}\colon\mathcal{H}\rightarrow\mathcal{H}$, $n\in\mathbb{N}$, such that

(15) $\lambda_{n}\searrow
0\quad\text{and}\quad\frac{1}{\lambda_{n}}\exp\Big{(}\frac{T\|Q\|_{L(\mathcal{H};\mathcal{H})}}{\lambda_{n}}\Big{)}\,\sup_{h\in\mathcal{H}}\|A_{n}(h)-A_{\lambda_{n}}(h)\|_{\mathcal{H}}\,\rightarrow
0.$

Let moreover $\\{\ell_{n}\\}_{n\in\mathbb{N}}\subset C([0,T];\mathcal{X})$ be
given and denote by $z_{n},z_{\lambda_{n}}\in C^{1}([0,T];\mathcal{H})$ the
solutions of

(16) $\displaystyle\overset{\boldsymbol{.}}{z}_{n}$
$\displaystyle=A_{n}(R\ell_{n}-Qz_{n}),$ $\displaystyle\quad z_{n}(0)$
$\displaystyle=z_{0},$ (17)
$\displaystyle\text{and}\qquad\overset{\boldsymbol{.}}{z}_{\lambda_{n}}$
$\displaystyle=A_{\lambda_{n}}(R\ell_{n}-Qz_{\lambda_{n}}),$
$\displaystyle\quad z_{\lambda_{n}}(0)$ $\displaystyle=z_{0}.$

Then $\|z_{n}-z_{\lambda_{n}}\|_{C^{1}([0,T];\mathcal{H})}\rightarrow 0$.

######  Proof 3.22.

Again, thanks to the Lipschitz continuity of $A_{n}$ and $A_{\lambda_{n}}$,
the existence and uniqueness of $z_{n}$ and $z_{n,\lambda}$ follows from
Banach’s contraction principle by classical arguments. Moreover, the
continuity of $\ell_{n}$ carries over to the continuity of
$\overset{\boldsymbol{.}}{z}_{n}$ and
$\overset{\boldsymbol{.}}{z}_{n,\lambda}$. Let us abbreviate $c_{n}$. Then, in
light of (16) and (17), we find

$\displaystyle\|\overset{\boldsymbol{.}}{z}_{n}(t)-\overset{\boldsymbol{.}}{z}_{\lambda_{n}}(t)\|_{\mathcal{H}}\leq
c_{n}+\frac{\|Q\|_{L(\mathcal{H};\mathcal{H})}}{\lambda_{n}}\|z_{n}(t)-z_{\lambda_{n}}(t)\|_{\mathcal{H}}\quad\forall\,t\in[0,T]$

so that Gronwall’s inequality yields

$\displaystyle\|\overset{\boldsymbol{.}}{z}_{n}(t)-\overset{\boldsymbol{.}}{z}_{\lambda_{n}}(t)\|_{\mathcal{H}}\leq\frac{\|Q\|_{L(\mathcal{H};\mathcal{H})}}{\lambda_{n}}\Big{(}T\exp\Big{(}\frac{\|Q\|_{L(\mathcal{H};\mathcal{H})}}{\lambda_{n}}\,T\Big{)}+1\Big{)}c_{n}\quad\forall\,t\in[0,T],$

which completes the proof.

##  4 Existence and Approximation of Optimal Controls

Now we turn to the optimal control problem (P). We first address the existence
of optimal solutions and afterwards discuss the approximation of (P) in
Section 4.2.

###  4.1 Existence of Optimal Controls

Based on Theorem 3.3, we reduce the optimal control problem (P) into a problem
in the control variable only. Recall that the control space $\mathcal{X}_{c}$
embeds compactly in $\mathcal{X}$.

######  Definition 4.1.

Let $z_{0}\in\mathcal{H}$ and $M\subset D(A)$. Due to Theorem 3.3, there
exists for every $\ell\in H^{1}(0,T;\mathcal{X}_{c})\cap\mathcal{U}(z_{0};M)$
a solution $z\in H^{1}(0,T;\mathcal{H})$ of the state equation in Eq. 3.
Consequently, we may define the solution operator

$\mathcal{S}\colon
H^{1}(0,T;\mathcal{X}_{c})\cap\mathcal{U}(z_{0};M)\ni\ell\rightarrow z\in
H^{1}(0,T;\mathcal{H}).$

This operator will be frequently called control-to-state map.

With the definition above, problem Eq. P is equivalent to the _reduced
problem_ :

$~{}\eqref{eq:optprob}\quad\Longleftrightarrow\quad\left\\{\;\begin{aligned}
\min\quad&J(\mathcal{S}(\ell),\ell),\\\ \text{s.t.}\quad&\ell\in
H^{1}(0,T;\mathcal{X}_{c})\cap\mathcal{U}(z_{0};M).\end{aligned}\right.$

Recall our standing assumptions on the objective, namely that
$J(z,\ell)=\Psi(z,\ell)+\Phi(\ell)$, where $\Psi\colon
H^{1}(0,T;\mathcal{W})\times H^{1}(0,T;\mathcal{X}_{c})\to\mathbb{R}$ is
weakly lower semicontinuous and bounded from below and $\Phi\colon
H^{1}(0,T;\mathcal{X}_{c})\to\mathbb{R}$ is weakly lower semicontinuous and
coercive. These assumptions allow us to show the existence of (globally)
optimal solutions:

######  Theorem 4.2 (Existence of Optimal Solutions).

Let $z_{0}\in\mathcal{H}$ and $M$ be a closed subset of $D(A)$. Then there
exists a global solution of Eq. P.

######  Proof 4.3.

Based on Theorem 3.14, the proof follows the standard direct method of the
calculus of variations. First of all, since $\Psi$ is bounded from below and
$\Phi$ is coercive, every infimal sequence of controls is bounded in
$H^{1}(0,T;\mathcal{X}_{c})$ and thus admits a weakly converging subsequence.
Due to the compact embedding of $\mathcal{X}_{c}$ in $\mathcal{X}$, this
sequence converges strongly in $C([0,T];\mathcal{X})$ so that the weak limit
belongs to $\mathcal{U}(z_{0};M)$, due the closeness of $M$. Moreover, thanks
to weak convergence in $H^{1}(0,T;\mathcal{X})$ and strong convergence in
$C([0,T];\mathcal{X})$, Theorem 3.14 gives weak convergence of the associated
states in $H^{1}(0,T;\mathcal{H})$. The weak lower semicontinuity of $\Psi$
and $\Phi$ together with $\mathcal{H}\hookrightarrow\mathcal{W}$ then implies
the optimality of the weak limit.

Clearly, in view of the nonlinear state equation, one cannot expect the
optimal solution to be unique. Note that, since $D(A)$ is closed by our
standing assumptions, the choice $M=D(A)$ is feasible.

###  4.2 Convergence of Global Minimizers

While the existence of optimal solutions for (P) can be shown by well-
established techniques as seen above, the derivation of optimality conditions
is all but standard because of the lack of differentiability of the control-
to-state map. We therefore apply a regularization of $A$ built upon the
Yosida-approximation in order to obtain a smooth control-to-state mapping. In
view of Lemma 3.21, this regularization is assumed to satisfy the following

######  Assumption 4.4.

Let $\\{A_{n}\\}_{n\in\mathbb{N}}$ be a sequence of Lipschitz continuous
operators from $\mathcal{H}$ to $\mathcal{H}$ such that, together with a
sequence $\\{\lambda_{n}\\}_{n\in\mathbb{N}}\subset(0,\infty)$, it holds

(18) $\lambda_{n}\searrow
0\quad\text{and}\quad\frac{1}{\lambda_{n}}\exp\Big{(}\frac{T\|Q\|_{L(\mathcal{H};\mathcal{H})}}{\lambda_{n}}\Big{)}\,\sup_{h\in\mathcal{H}}\|A_{n}(h)-A_{\lambda_{n}}(h)\|_{\mathcal{H}}\,\rightarrow
0,$

i.e., the requirements in Lemma 3.21 are fulfilled.

In Section 7.4, we show how to construct such a regularization for a concrete
application problem. Given the regularization of $A$, we define the
corresponding optimal control problem:

(Pn) $\left\\{\quad\begin{aligned} \min\text{ \ \ }&J(z,\ell),\\\ \text{ s.t.
}\text{ \ \ }&\overset{\boldsymbol{.}}{z}=A_{n}(R\ell-Qz),\text{ \ \ \ \
}z(0)=z_{0},\\\ &(z,\ell)\in
H^{1}(0,T;\mathcal{H})\times\big{(}H^{1}(0,T;\mathcal{X}_{c})\cap\mathcal{U}(z_{0};M)\big{)}.\end{aligned}\right.$

Since $A_{n}$ is Lipschitz continuous, the equation

$\overset{\boldsymbol{.}}{z}=A_{n}(R\ell-Qz),\quad z(0)=z_{0}$

admits a unique solution for every $z_{0}\in\mathcal{H}$ and every $\ell\in
L^{2}(0,T;\mathcal{X})$. Similar to Definition 4.1, we denote the associated
solution operator by

$\mathcal{S}_{n}\colon L^{2}(0,T;\mathcal{X})\to H^{1}(0,T;\mathcal{H}).$

Moreover, the solution operator associated with the Yosida-approximation,
i.e., the solution operator of
$\overset{\boldsymbol{.}}{z}=A_{\lambda_{n}}(R\ell-Qz)$, $z(0)=z_{0}$, is
denoted by

$\mathcal{S}_{\lambda_{n}}\colon L^{2}(0,T;\mathcal{X})\to
H^{1}(0,T;\mathcal{H}).$

######  Proposition 4.5 (Existence of Optimal Solutions of the Regularized
Problems).

Let $n\in\mathbb{N}$, $z_{0}\in\mathcal{H}$, and $M$ a closed subset of
$D(A)$. Then, under 4.4, there exists a global solution of Eq. Pn.

######  Proof 4.6.

Let $\ell_{1},\ell_{2}\in L^{2}(0,T;\mathcal{X})$ be arbitrary and define
$z_{i}$, $i=1,2$. Then, due to the Lipschitz continuity of $A_{n}$, we have
for almost all $t\in[0,T]$

(19)
$\displaystyle\|\overset{\boldsymbol{.}}{z}_{1}(t)-\overset{\boldsymbol{.}}{z}_{2}(t)\|_{\mathcal{H}}$
$\displaystyle=\|A_{n}(R\ell_{1}(t)-Qz_{1}(t))-A_{n}(R\ell_{2}(t)-Qz_{2}(t))\|_{\mathcal{H}}$
$\displaystyle\leq
c\big{(}\|l_{1}(t)-l_{2}(t)\|_{\mathcal{X}}+\|z_{1}(t)-z_{2}(t)\|_{\mathcal{H}}\big{)},$

which yields, thanks to Gronwall’s inequality , the Lipschitz continuity of
$\mathcal{S}_{n}$. Using this together with the fact that $\mathcal{X}_{c}$ is
compactly embedded into $\mathcal{X}$, one can argue as in the proof of
Theorem 4.2 to obtain the existence of a global solution of Eq. Pn for all
$n\in\mathbb{N}$.

######  Theorem 4.7 (Weak Approximation of Global Minimizers).

Let $z_{0}\in\mathcal{H}$ and $M$ be a closed subset of $D(A)$. Suppose
moreover that 4.4 holds and let $\\{\overline{\ell}_{n}\\}_{n\in\mathbb{N}}$
be a sequence of globally optimal controls of Eq. Pn, $n\in\mathbb{N}$. Then
there exists a weak accumulation point and every weak accumulation point is a
global solution of Eq. P.

######  Proof 4.8.

Due to $M\subset D(A)$, Proposition 3.6 gives
$\mathcal{S}_{\lambda_{n}}(\overline{\ell}_{1})\rightarrow\mathcal{S}(\overline{\ell}_{1})$
in $H^{1}(0,T;\mathcal{H})$ so that Lemma 3.21 yields
$\mathcal{S}_{n}(\overline{\ell}_{1})\rightarrow\mathcal{S}(\overline{\ell}_{1})$
in $H^{1}(0,T;\mathcal{H})$ and thus

$\displaystyle\limsup_{n\rightarrow\infty}\Psi(\mathcal{S}_{n}(\overline{\ell}_{n}),\overline{\ell}_{n})+\Phi(\overline{\ell}_{n})=\limsup_{n\rightarrow\infty}J(\mathcal{S}_{n}(\overline{\ell}_{n}),\overline{\ell}_{n})\leq\limsup_{n\rightarrow\infty}J(\mathcal{S}_{n}(\overline{\ell}_{1}),\overline{\ell}_{1})=J(\mathcal{S}(\overline{\ell}_{1}),\overline{\ell}_{1}).$

Hence, by virtue of the boundedness of $\Psi$ from below and the radial
unboundedness of $\Phi$, $\\{\overline{\ell}_{n}\\}$ is bounded and therefore
admits a weak accumulation point in $H^{1}(0,T;\mathcal{X}_{c})$.

Let us now assume that a given subsequence of
$\\{\overline{\ell}_{n}\\}_{n\in\mathbb{N}}$, denoted by the same symbol for
simplicity, converges weakly to $\tilde{\ell}$ in
$H^{1}(0,T;\mathcal{X}_{c})$. Since $\mathcal{X}_{c}$ is compactly embedded in
$\mathcal{X}$, we obtain $\overline{\ell}_{n}\rightarrow\tilde{\ell}$ in
$C([0,T];\mathcal{X})$ and consequently,
$\tilde{\ell}\in\mathcal{U}(z_{0};M)$. In addition, the strong convergence in
$C([0,T];\mathcal{X})$ in combination with Theorem 3.16 and Lemma 3.21 yields
weak convergence of the states, i.e.,
$\mathcal{S}_{n}(\overline{\ell}_{n})\rightharpoonup\mathcal{S}(\tilde{\ell})$
in $H^{1}(0,T;\mathcal{H})$ and thus also in $H^{1}(0,T;\mathcal{W})$. Now,
let $\overline{\ell}$ be a global solution of Eq. P. We can again use
Proposition 3.6 and Lemma 3.21 to obtain
$\mathcal{S}_{n}(\overline{\ell})\rightarrow\mathcal{S}(\overline{\ell})$ in
$H^{1}(0,T;\mathcal{H})$. This, together with the weak lower semicontinuity of
$\Psi$ and $\Phi$, implies

(20) $\displaystyle J(\mathcal{S}(\tilde{\ell}),\tilde{\ell})$
$\displaystyle=\Psi(\mathcal{S}(\tilde{\ell}),\tilde{\ell})+\Phi(\tilde{\ell})\leq\liminf_{n\rightarrow\infty}\Psi(\mathcal{S}_{n}(\overline{\ell}_{n}),\overline{\ell}_{n})+\Phi(\overline{\ell}_{n})$
$\displaystyle\leq\limsup_{n\rightarrow\infty}J(\mathcal{S}_{n}(\overline{\ell}_{n}),\overline{\ell}_{n})\leq\limsup_{n\rightarrow\infty}J(\mathcal{S}_{n}(\overline{\ell}),\overline{\ell})=J(\mathcal{S}(\overline{\ell}),\overline{\ell}),$

giving in turn the optimality of the weak limit.

######  Corollary 4.9 (Strong Approximation of Global Minimizers).

In addition to 4.4, assume that $\Phi\colon
H^{1}(0,T;\mathcal{X}_{c})\to\mathbb{R}$ is such that, if a sequence
$\\{\ell_{n}\\}_{n\in\mathbb{N}}$ satisfies $\ell_{n}\rightharpoonup\ell$ in
$H^{1}(0,T;\mathcal{X}_{c})$ and $\Phi(\ell_{n})\to\Phi(\ell)$, then
$\ell_{n}\to\ell$ in $H^{1}(0,T;\mathcal{X}_{c})$. Then every weak
accumulation point of a sequence of globally optimal controls of (Pn) is also
a strong one.

Moreover, if in addition, at least one of the following holds

  * •

3.13 is satisfied, that is $A=\partial\phi$, and $\phi$ is continuous on $M$
or

  * •

$\Psi\colon H^{1}(0,T;\mathcal{W})\times
H^{1}(0,T;\mathcal{X}_{c})\to\mathbb{R}$ is such that, if sequences
$\\{z_{n}\\}_{n\in\mathbb{N}}$ and $\\{\ell_{n}\\}_{n\in\mathbb{N}}$ satisfy
$z_{n}\rightharpoonup z$ in $H^{1}(0,T;\mathcal{H})$ and
$\ell_{n}\rightarrow\ell$ in $H^{1}(0,T;\mathcal{X}_{c})$ and
$\Psi(z_{n},\ell_{n})\to\Psi(z,\ell)$, then $z_{n}\to z$ in
$H^{1}(0,T;\mathcal{H})$,

then the associated sequence of state also converges strongly in
$H^{1}(0,T;\mathcal{H})$.

######  Proof 4.10.

Consider an arbitrary accumulation point $\tilde{\ell}$ of a sequence of
global minimizers of (Pn), i.e.,
$\overline{\ell}_{n}\rightharpoonup\tilde{\ell}$ in
$H^{1}(0,T;\mathcal{X}_{c})$. From the previous proof, we know that then (20)
holds, giving in turn

$\Psi(\mathcal{S}_{n}(\overline{\ell}_{n}),\overline{\ell}_{n})+\Phi(\overline{\ell}_{n})\to\Psi(\mathcal{S}(\tilde{\ell}),\tilde{\ell})+\Phi(\tilde{\ell}).$

Since
$\mathcal{S}_{n}(\overline{\ell}_{n})\rightharpoonup\mathcal{S}(\tilde{\ell})$,
as seen in the previous proof, and both, $\Psi$ and $\Phi$, are weakly lower
semicontinuous by assumption, this implies
$\Phi(\overline{\ell}_{n})\to\Phi(\tilde{\ell})$ and
$\Psi(\mathcal{S}_{n}(\overline{\ell}_{n}),\overline{\ell}_{n})\to\Psi(\mathcal{S}(\tilde{\ell}),\tilde{\ell})$.
The hypothesis on $\Phi$ thus yields $\overline{\ell}_{n}\to\tilde{\ell}$ in
$H^{1}(0,T;\mathcal{X}_{c})$ so that $\tilde{\ell}$ is indeed a strong
accumulation point as claimed.

Due to $\mathcal{X}_{c}\hookrightarrow\mathcal{X}$, the strong convergence
carries over to $H^{1}(0,T;\mathcal{X})$ and therefore, we deduce from Theorem
3.16 that
$\mathcal{S}_{\lambda_{n}}(\overline{\ell}_{n})\to\mathcal{S}(\tilde{\ell})$
in $H^{1}(0,T;\mathcal{H})$, provided that 3.13 is fulfilled and
$\phi(R\overline{\ell}_{n}(0)-Qz_{0})\rightarrow\phi(R\tilde{\ell}(0)-Qz_{0})$
holds. If the additional requirements on $\Psi$ are fulfilled, we also obtain
the strong convergence
$\mathcal{S}_{\lambda_{n}}(\overline{\ell}_{n})\to\mathcal{S}(\tilde{\ell})$
in $H^{1}(0,T;\mathcal{H})$, since we already showed
$\overline{\ell}_{n}\to\tilde{\ell}$ in $H^{1}(0,T;\mathcal{X}_{c})$. Thus, in
both cases, Lemma 3.21 gives
$\mathcal{S}_{n}(\overline{\ell}_{n})\to\mathcal{S}(\tilde{\ell})$ in
$H^{1}(0,T;\mathcal{H})$, which is the second assertion.

######  Example 4.11.

Let us assume that $\mathcal{X}_{c}$ is a Hilbert space. Then a possible
objective functional fulfilling the requirements on $\Phi$ in Corollary 4.9
reads as follows:

$J(z,\ell)=\Psi(z,\ell)+\frac{\alpha}{2}\,\|\ell\|_{H^{1}(0,T;\mathcal{X}_{c})}^{2},$

i.e., $\Phi(\ell)$. Herein, $\Psi\colon H^{1}(0,T;\mathcal{H})\times
H^{1}(0,T;\mathcal{X}_{c})\to\mathbb{R}$ is again lower semicontinuous and
bounded from below and $\alpha>0$ is a given constant. Since
$H^{1}(0,T;\mathcal{X}_{c})$ is a Hilbert space, too, weak convergence and
norm convergence give strong convergence and consequently, this specific
choice of $\Phi$ fulfills the condition in Corollary 4.9.

######  Remark 4.12 (Approximation of Local Minimizers).

By standard localization arguments, the above convergence analysis can be
adapted to approximate local minimizers. Following the lines of, for instance,
[12], one can show that, under the assumptions of Corollary 4.9, every strict
local minimum of Eq. P can be approximated by a sequence of local minima of
Eq. Pn. A local minimizer $\overline{\ell}$ of Eq. P, which is not necessarily
strict, can be approximated by replacing the objective in (Pn) by
$\overline{J}(z,l)$, which is of course only of theoretical interest, cf. e.g.
[4]. Since these results and their proofs are standard, we omitted them.

Now that we answered the question of approximation of optimal controls via
regularization, we turn to the regularized problems and derive optimality
conditions for these in the next two sections.

##  5 First-Order Optimality Conditions

In the following, we consider a single element of the sequence of regularized
problems. The associated regularized operator is denoted by $A_{s}$ so that
the regularized optimal control problems reads as follows:

(Ps) $\left\\{\quad\begin{aligned} \min\text{ \ \ }&J(z,\ell),\\\ \text{ s.t.
}\text{ \ \ }&\overset{\boldsymbol{.}}{z}=A_{s}(R\ell-Qz),\text{ \ \ \ \
}z(0)=z_{0},\\\ &(z,\ell)\in
H^{1}(0,T;\mathcal{H})\times\big{(}H^{1}(0,T;\mathcal{X}_{c})\cap\mathcal{U}(z_{0};M)\big{)}.\end{aligned}\right.$

Beside our standing assumption and the Lipschitz continuity required in 4.4,
we need the following additional assumptions for the derivation of first-order
necessary optimality conditions for (Ps). Recall the continuous embeddings
$\mathcal{Y}\hookrightarrow\mathcal{Z}\hookrightarrow\mathcal{H}$ from Section
2.

######  Assumption 5.1.

  * (i)

$J\colon H^{1}(0,T;\mathcal{W})\times
H^{1}(0,T;\mathcal{X}_{c})\rightarrow\mathbb{R}$ is Fréchet differentiable.

  * (ii)

$A_{s}\colon\mathcal{Y}\rightarrow\mathcal{Y}$ is Lipschitz continuous and
Fréchet differentiable from $\mathcal{Y}$ to $\mathcal{Z}$. Moreover,
$A_{s}^{\prime}(y)$ can be extended to elements of
$L(\mathcal{Z};\mathcal{Z})$ and $L(\mathcal{H};\mathcal{H})$, respectively,
denoted by the same symbol. There exists a constant $C$ such that these
extensions satisfy $\|A_{s}^{\prime}(y)z\|_{\mathcal{Z}}\leq
C\,\|z\|_{\mathcal{Z}}$ and $\|A_{s}^{\prime}(y)h\|_{\mathcal{H}}\leq
C\|h\|_{\mathcal{H}}$ for all $y\in\mathcal{Y}$, $z\in\mathcal{Z}$, and
$h\in\mathcal{H}$.

######  Remark 5.2.

It is well known that a norm gap is often indispensable to ensure Fréchet
differentiability. This is also the case in our application example in Section
7.4. This is the reason for considering two different spaces $\mathcal{Y}$ and
$\mathcal{Z}$ in context of the Fréchet differentiability of $A_{s}$ in 5.1.

We start the derivation of optimality conditions for (Ps) with the Fréchet-
derivative of the associated control-to-state mapping.

###  5.1 Differentiability of the Regularized Control-to-State Mapping

As $A_{s}\colon\mathcal{Y}\to\mathcal{Y}$ is supposed to be Lipschitz
continuous and $R$ and $Q$ are not only linear and continuous as operators
with values in $\mathcal{H}$, but also in $\mathcal{Y}$ according to our
standing assumptions, Banach’s fixed point theorem immediately implies that
the state equation in (Ps), i.e.,

(21) $\overset{\boldsymbol{.}}{z}=A_{s}(R\ell-Qz),\quad z(0)=z_{0},$

admits a unique solution $z\in H^{1}(0,T;\mathcal{Y})$ for every right hand
side $\ell\in L^{2}(0,T;\mathcal{X})$, provided that $z_{0}\in\mathcal{Y}$.
Therefore, similar to above, we can define the associated solution operator
$\mathcal{S}_{s}\colon L^{2}(0,T;\mathcal{X})\rightarrow
H^{1}(0,T;\mathcal{Y})$ (for fixed $z_{0}\in\mathcal{Y}$). We will frequently
consider this operator with different domains, e.g. $H^{1}(0,T;\mathcal{X})$,
and ranges, in particular $H^{1}(0,T;\mathcal{Z})$. With a little abuse of
notation, these operators are denoted by the same symbol.

######  Lemma 5.3 (Lipschitz Continuity of $\mathcal{S}_{s}$).

The solution operator $\mathcal{S}_{s}$ is globally Lipschitz continuous from
$L^{2}(0,T;\mathcal{X})$ to $H^{1}(0,T;\mathcal{Y})$.

######  Proof 5.4.

This can be proven completely analogously to the Lipschitz continuity of
$\mathcal{S}_{n}$ from $L^{2}(0,T;\mathcal{X})$ to $H^{1}(0,T;\mathcal{Y})$ in
Proposition 4.5.

######  Lemma 5.5.

Assume that 5.1(ii) is fulfilled and let $y\in L^{2}(0,T;\mathcal{Y})$ and
$w\in L^{2}(0,T;\mathcal{Z})$ be given. Then there exists a unique solution
$\eta\in H^{1}(0,T;\mathcal{Z})$ of

(22)
$\overset{\boldsymbol{.}}{\eta}=A_{s}^{\prime}(y)(w-Q\eta),\quad\eta(0)=0.$

######  Proof 5.6.

Let us define

$\displaystyle
B\colon[0,T]\times\mathcal{Z}\rightarrow\mathcal{Z},\quad(t,\eta)\mapsto
A_{s}^{\prime}(y(t))(w(t)-Q\eta)$

so that (22) becomes $\overset{\boldsymbol{.}}{\eta}(t)=B(t,\eta(t))$ a.e. in
$(0,T)$, $\eta(0)=0$. Now, given $\eta\in L^{2}(0,T;\mathcal{Z})$, $[0,T]\ni
t\mapsto B(t,\eta(t))\in\mathcal{Z}$ is Bochner measurable as a pointwise
limit of Bochner measurable functions. Furthermore, 5.1(ii) implies for almost
all $t\in[0,T]$ and all $\eta_{1},\eta_{2}\in\mathcal{Z}$ that
$\|B(t,\eta_{1})-B(t,\eta_{2})\|_{\mathcal{Z}}\leq
C\,\|\eta_{1}-\eta_{2}\|_{\mathcal{Z}}$. Therefore, we can apply Banach’s
fixed point argument to the integral equation associated with (22), which
gives the assertion.

######  Theorem 5.7 (Fréchet differentiability of the regularized solution
operator).

Under 5.1(ii), the solution operator $\mathcal{S}_{s}$ is Fréchet
differentiable from $H^{1}(0,T;\mathcal{X})$ to $H^{1}(0,T;\mathcal{Z})$. Its
directional derivative at $\ell\in H^{1}(0,T;\mathcal{X})$ in direction $h\in
H^{1}(0,T;\mathcal{X})$ is given by the unique solution of

(23) $\displaystyle\overset{\boldsymbol{.}}{\eta}=A_{s}^{\prime}(R\ell-
Qz)(Rh-Q\eta),\quad\eta(0)=0,$

where $z$. Moreover, there exists a constant $C$ such that
$\|\mathcal{S}_{s}^{\prime}(\ell)h\|_{H^{1}(0,T;\mathcal{Z})}\leq
C\|h\|_{L^{2}(0,T;\mathcal{X})}$ holds for all $\ell,h\in
H^{1}(0,T;\mathcal{X})$.

######  Proof 5.8.

Let $\ell,h\in H^{1}(0,T;\mathcal{X})$ be arbitrary and abbreviate $z_{h}$.
Thanks to Lemma 5.5, there exists a unique solution $\eta\in
H^{1}(0,T;\mathcal{Z})$ of Eq. 23. Clearly, the solution operator of Eq. 23 is
linear with respect to $h$. Moreover, 5.1(ii) implies for almost all
$t\in[0,T]$ that

$\|\overset{\boldsymbol{.}}{\eta}(t)\|_{\mathcal{Z}}\leq
C\big{(}\|h(t)\|_{\mathcal{X}}+\|\eta(t)\|_{\mathcal{Z}}\big{)},$

so that Gronwall’s inequality gives $\|\eta\|_{H^{1}(0,T;\mathcal{Z})}\leq
C\|h\|_{L^{2}(0,T;\mathcal{X})}$, i.e., the continuity of the solution
operator of Eq. 23. This also proves the asserted inequality (after having
proved that $\eta=\mathcal{S}_{s}^{\prime}(\ell)h$, which we do next).

It remains to verify the remainder term property. For this purpose, let us
denote the remainder term of $A_{s}$ by $r_{1}$, i.e.,

$A_{s}(y+\zeta)=A_{s}(y)+A_{s}^{\prime}(y)\zeta+r_{1}(y;\zeta)\quad\text{with}\quad\frac{\|r_{1}(y;\zeta)\|_{\mathcal{Z}}}{\|\zeta\|_{\mathcal{Y}}}\to
0\text{ as }\zeta\to 0\text{ in }\mathcal{Y}.$

Moreover, we abbreviate

Definition 20 $y$

Then, in view of the definition of $z$, $z_{h}$, and $\eta$ (as solution of
(23)), we find for almost all $t\in[0,T]$

$\displaystyle\|\overset{\boldsymbol{.}}{z}_{h}(t)-\overset{\boldsymbol{.}}{z}(t)-\overset{\boldsymbol{.}}{\eta}(t)\|_{\mathcal{Z}}$
$\displaystyle=\|A_{s}(y(t)+\zeta(t))-A_{s}(y(t))-A_{s}^{\prime}(y(t))(\zeta(t)+Q(z_{h}(t)-z(t)-\eta(t)))\|_{\mathcal{Z}}$
$\displaystyle\leq\|A_{s}^{\prime}(y(t))Q(z_{h}(t)-z(t)-\eta(t))\|_{\mathcal{Z}}+\|r_{1}(y(t);\zeta(t))\|_{\mathcal{Z}}.$

Hence, 5.1(ii) and Gronwall’s inequality yield

(24) $\|z_{h}-z-\eta\|_{H^{1}(0,T;\mathcal{Z})}\leq
C\,\|r_{1}(y;\zeta)\|_{L^{2}(0,T;\mathcal{Z})}.$

(note that $r_{1}(y;\zeta)\in L^{2}(0,T;\mathcal{Z})$ by its definition as
remainder term). Furthermore, thanks to Lemma 5.3 and the definition of
$\zeta$, we obtain

(25) $\|\zeta\|_{H^{1}(0,T;\mathcal{Y})}\leq
C\,\|h\|_{H^{1}(0,T;\mathcal{X})}$

such that $h\to 0$ in $H^{1}(0,T;\mathcal{X})$ implies $\zeta\to 0$ in
$H^{1}(0,T;\mathcal{Y})$. The continuous embedding
$H^{1}(0,T;\mathcal{Y})\hookrightarrow C([0,T];\mathcal{Y})$ and the remainder
term property of $r_{1}$ thus give for almost all $t\in(0,T)$ that

(26)
$\frac{\|r_{1}(y(t);\zeta(t))\|_{\mathcal{Z}}}{\|h\|_{H^{1}(0,T;\mathcal{X})}}\leq
C\,\frac{\|r_{1}(y(t);\zeta(t))\|_{\mathcal{Z}}}{\|\zeta(t)\|_{\mathcal{Y}}}\,\frac{\|\zeta\|_{H^{1}(0,T;\mathcal{Y})}}{\|h\|_{H^{1}(0,T;\mathcal{X})}}\to
0$

as $h\to 0$ in $H^{1}(0,T;\mathcal{X})$. Moreover, the Lipschitz continuity of
$A_{s}\colon\mathcal{Y}\to\mathcal{Y}$ together with 5.1(ii),
$\mathcal{Y}\hookrightarrow\mathcal{Z}$, and (25) yield for almost all
$t\in(0,T)$ that

$\frac{\|r_{1}(y(t);\zeta(t))\|_{\mathcal{Z}}}{\|h\|_{H^{1}(0,T;\mathcal{X})}}=\frac{\|(A_{s}(y+\zeta)-A_{s}(y)-A_{s}^{\prime}(y)\zeta)(t)\|_{\mathcal{Z}}}{\|h\|_{H^{1}(0,T;\mathcal{X})}}\leq
C\,\frac{\|\zeta(t)\|_{\mathcal{Y}}}{\|h\|_{H^{1}(0,T;\mathcal{X})}}\leq C.$

In combination with (26) and Lebesgue’s dominated convergence theorem, this
yields

$\frac{\|r_{1}(y;\zeta)\|_{L^{2}(0,T;\mathcal{Z})}}{\|h\|_{H^{1}(0,T;\mathcal{X})}}\to
0$

as $h\rightarrow 0$ in $H^{1}(0,T;\mathcal{X})$, which, in view of (24)
finishes the proof.

######  Remark 5.9.

It is to be noted that we did not employ the implicit function theorem to show
the differentiability of $\mathcal{S}_{s}$. The reason is that $H\colon
z\mapsto\overset{\boldsymbol{.}}{z}-A_{s}(R\ell-Qz)$ is Fréchet differentiable
from $H^{1}(0,T;\mathcal{Y})$ to $L^{2}(0,T;\mathcal{Z})$, but the derivative
$H^{\prime}(z)$ is not continuously invertible in these spaces, cf. Lemma 5.5.
On the other hand, $H$ is not differentiable from $H^{1}(0,T;\mathcal{Y})$ to
$L^{2}(0,T;\mathcal{Y})$ (due to the differentiability properties of $A_{s}$,
see Remark 5.2), which would be the right spaces for the existence result from
Lemma 5.5. The same observation for a more abstract setting was already made
in [41].

###  5.2 Adjoint Equation

Now that we know that the (regularized) control-to-state map is Gâteaux
differentiable, we can apply the standard adjoint approach to derive first-
order necessary optimality conditions in form of a Karush-Kuhn-Tucker (KKT)
system. To keep the discussion concise, we restrict our analysis to the case
without further control constraints. To be more precise, we require the
following:

######  Assumption 5.10.

Let $z_{0}\in\mathcal{Y}$ such that $-Qz_{0}\in D(A)$. The set $M$ in the
definition of the set of admissible controls is given by the singleton
$M=\\{-Qz_{0}\\}$ such that

Definition 22 $\mathcal{U}$

Note that $\mathcal{U}$ is a linear subspace of $H^{1}(0,T;\mathcal{X})$.

######  Remark 5.11 (Additional Control Constraints).

One could allow for additional control constraints in our analysis, even more
complex ones than the ones covered by $\mathcal{U}(z_{0};M)$ such as for
instance box constraints over the whole time interval or vanishing initial and
final loading, i.e., $\ell(0)=\ell(T)=0$, which is certainly meaningful for
many practically relevant problems. However, since the differentiability of
the control-to-state map is the essential issue in the derivation of
optimality conditions and additional (convex and closed) control constraints
can be incorporated by standard argument, we decided to leave them out in
order to keep the discussion as concise as possible.

However, without any further assumptions, the existence of solutions to the
unregularized state equation (3) cannot be guaranteed. To be more precise, one
needs that $R\ell(0)-Qz_{0}\in D(A)$, see Theorem 3.3, which holds in case of
$\mathcal{U}$, provided that $-Qz_{0}\in D(A)$. This is the reason for
considering the set $H^{1}(0,T;\mathcal{X}_{C})\cap\mathcal{U}$ as set of
admissible controls in the rest of the paper.

Note moreover that, if the operator $R$ is injective (which is the case in
Section 7), then $\mathcal{U}=\\{\ell\in
H^{1}(0,T;\mathcal{X})\colon\ell(0)=0\\}$.

The chain rule immediately gives that the reduced objective defined by

(27) $F\colon H^{1}(0,T;\mathcal{X}_{c})\to\mathbb{R},\quad\ell\mapsto
J(\mathcal{S}_{s}(\ell),\ell)$

is Fréchet differentiable, too. Thus, by standard arguments, one derives the
following

######  Lemma 5.12 (Purely Primal Necessary Optimality Conditions).

Let 5.1 and 5.10 hold. Then, if a control $\overline{\ell}\in
H^{1}(0,T;\mathcal{X}_{c})\cap\mathcal{U}$ with associated state
$\overline{z}=\mathcal{S}_{s}(\overline{\ell})$ is locally optimal for Eq. Ps,
then

(28)
$F^{\prime}(\overline{\ell})h=J^{\prime}_{z}(\overline{z},\overline{\ell})\mathcal{S}^{\prime}_{s}(\overline{\ell})h+J^{\prime}_{\ell}(\overline{z},\overline{\ell})h=0,$

for all $h\in H^{1}(0,T;\mathcal{X}_{c})\cap\mathcal{U}$.

Next, we reformulate (28) in terms of a KKT-system by introducing an adjoint
equation, which formally reads

(29)
$\overset{\boldsymbol{.}}{\varphi}=QA^{\prime}_{s}(y)^{*}\varphi+v,\quad\varphi(T)=0.$

Depending on the regularity of the right hand side $v$, we define different
notions of solutions:

######  Definition 5.13.

Let $y\in L^{2}(0,T;\mathcal{Y})$ and $v\in H^{1}(0,T;\mathcal{H})^{*}$ be
given. A function $\varphi\in L^{2}(0,T;\mathcal{H})$ is called _weak
solution_ of (29), if

(30)
$-\left(\varphi,\overset{\boldsymbol{.}}{\eta}\right)_{L^{2}(0,T;\mathcal{H})}=\left(\varphi,A^{\prime}_{s}(y)Q\eta\right)_{L^{2}(0,T;\mathcal{H})}+v(\eta)$

holds for all $\eta\in H^{1}(0,T;\mathcal{H})$ with $\eta(0)=0$.

If $v$ takes the form

(31)
$v(\eta)=(v_{1},\eta)_{L^{2}(0,T;\mathcal{H})}+(v_{2},\eta(T))_{\mathcal{H}}$

with some $v_{1}\in L^{2}(0,T;\mathcal{H})$ and $v_{2}\in\mathcal{H}$, then we
call $\varphi\in H^{1}(0,T;\mathcal{H})$ _strong solution_ of (29), if, for
almost all $t\in(0,T)$,

(32)
$\overset{\boldsymbol{.}}{\varphi}(t)=\big{(}QA^{\prime}_{s}(y)^{*}\varphi\big{)}(t)+v_{1}(t)\quad\text{in
}\mathcal{H},\quad\varphi(T)=-v_{2}\quad\text{in }\mathcal{H}.$

In the following, we will—as usual—identify $v\in L^{2}(0,T;\mathcal{H})$ with
an element of $H^{1}(0,T;\mathcal{H})^{*}$ via
$(v,\cdot\,)_{L^{2}(0,T;\mathcal{H})}$ and denote this element with a slight
abuse of notation by the same symbol.

######  Lemma 5.14.

Let $y\in L^{2}(0,T;\mathcal{Y})$ and $v\in H^{1}(0,T;\mathcal{H})^{*}$. Then
there is a unique weak solution of Eq. 29, which is given by $\varphi$, where
$\mathcal{S}_{y}\colon L^{2}(0,T;\mathcal{H})\rightarrow
H^{1}(0,T;\mathcal{H})$ is the solution operator of

(33)
$\displaystyle\overset{\boldsymbol{.}}{\eta}=-A^{\prime}_{s}(y)Q\eta+w,\quad\eta(0)=0,$

that is, $\mathcal{S}_{y}(w)=\eta$.

Moreover, if $v$ is of the form (31), then there exists a unique strong
solution of (29), and the weak and the strong solution coincide.

######  Proof 5.15.

At first note that the existence of a solution of Eq. 33 can be proven exactly
as in Lemma 5.5. Let $\eta\in H^{1}(0,T;\mathcal{H})$ with $\eta(0)=0$ be
arbitrary and define $w$, hence, $\eta=\mathcal{S}_{y}(w)$. By the definition
of $w$ and $\varphi$, it follows that

$(\overset{\boldsymbol{.}}{\eta}+A^{\prime}_{s}(y)Q\eta,\varphi)_{L^{2}(0,T;\mathcal{H})}=(\varphi,w)_{L^{2}(0,T;\mathcal{H})}=-v(\mathcal{S}_{y}(w))=-v(\eta),$

i.e., Eq. 30 holds. Since $\eta$ was arbitrary, we see that $\varphi$ is a
weak solution of Eq. 29.

To prove uniqueness, let $\tilde{\varphi}\in L^{2}(0,T;\mathcal{H})$ be
another weak solution. Then, we choose an arbitrary $w\in
L^{2}(0,T;\mathcal{H})$ and set $\eta$ to see that

$\left(\varphi,w\right)_{L^{2}(0,T;\mathcal{H})}=-v(\eta)=\left(\tilde{\varphi},\overset{\boldsymbol{.}}{\eta}+A^{\prime}_{s}(y)Q\eta\right)_{L^{2}(0,T;\mathcal{H})}=\left(\tilde{\varphi},w\right)_{L^{2}(0,T;\mathcal{H})},$

and therefore $\varphi=\tilde{\varphi}$.

Now we turn to the strong solution and suppose that $v$ is as given in (31).
Existence and uniqueness of a strong solution can again be shown by means of
Banach’s fixed point theorem. To this end, let us consider the affine-linear
operator

$B\colon[0,T]\times\mathcal{Z}\to\mathcal{Z},\quad
B(t,\varphi)=QA^{\prime}_{s}(y(t))^{*}\varphi+v_{1}(t).$

Since $\mathcal{H}$ is separable by our standing assumptions, we can apply
[20, Chap. IV, Thm. 1.4] to obtain that, for every $\varphi\in
L^{2}(0,T;\mathcal{H})$, the mapping $(0,T)\mapsto B(t,\varphi(t))$ is Bochner
measureable. Moreover, since
$\|A_{s}^{\prime}(y)^{*}\|_{L(\mathcal{H};\mathcal{H})}=\|A_{s}^{\prime}(y)\|_{L(\mathcal{H};\mathcal{H})}$,
5.1(ii) yields that $B$ is also Lipschitz continuous w.r.t. the second
variable for almost all $t\in(0,T)$. Therefore, similarly to the proof of
Lemma 5.5, one can apply Banach’s fixed point theorem to the integral equation
associated with (32) to establish the existence of a unique strong solution.

Finally, if we test (32) with an arbitrary $\eta\in H^{1}(0,T;\mathcal{H})$
with $\eta(0)=0$ and integrate by parts, then we see that every strong
solution is also a weak solution. Since the latter one is unique, as seen
above, we deduce that weak and strong solution coincide.

######  Theorem 5.16 (KKT-Conditions for (Ps)).

Assume that 5.1 and 5.10 hold and let $\overline{\ell}\in
H^{1}(0,T;\mathcal{X}_{c})\cap\mathcal{U}$ be a locally optimal control for
Eq. Ps with associated state $\overline{z}=\mathcal{S}_{s}(\overline{\ell})$.
Then there exists a unique adjoint state $\varphi\in L^{2}(0,T;\mathcal{H})$
such that the following optimality system is fulfilled

(34a)
$\displaystyle\overset{\boldsymbol{.}}{\overline{z}}=A_{s}(R\overline{\ell}-Q\overline{z}),$
$\displaystyle\overline{z}(0)=z_{0},$ (34b)
$\displaystyle\left\\{\begin{aligned}
&-\left(\varphi,\overset{\boldsymbol{.}}{\eta}\right)_{L^{2}(0,T;\mathcal{H})}\\\
&\quad=\left(\varphi,A^{\prime}_{s}(R\overline{\ell}-Q\overline{z})Q\eta\right)_{L^{2}(0,T;\mathcal{H})}+J^{\prime}_{z}(\overline{z},\overline{\ell})\eta\end{aligned}\right.$
$\displaystyle\begin{aligned} \forall\,\eta\in
H^{1}(0,T;\mathcal{H})\colon\quad&\\\ \eta(0)=0&\end{aligned}$ (34c)
$\displaystyle\left(\varphi,A_{s}^{\prime}(R\overline{\ell}-Q\overline{z})Rh\right)_{L^{2}(0,T;\mathcal{H})}=J^{\prime}_{\ell}(\overline{z},\overline{\ell})h$
$\displaystyle\forall\,h\in H^{1}(0,T;\mathcal{X}_{c})\cap\mathcal{U}.$

If $J$ enjoys extra regularity, namely

(35) $J(z,\ell)=\Psi_{1}(z,\ell)+\Psi_{2}(z(T),\ell(T))+\Phi(\ell)$

with two Fréchet differentiable functionals $\Psi_{1}\colon
L^{2}(0,T;\mathcal{H})\times H^{1}(0,T;\mathcal{X}_{c})\to\mathbb{R}$ and
$\Psi_{2}\colon\mathcal{H}\times\mathcal{X}_{c}\to\mathbb{R}$, then
$\varphi\in H^{1}(0,T;\mathcal{H})$ is a strong solution of

(36)
$\overset{\boldsymbol{.}}{\varphi}(t)=\big{(}QA^{\prime}_{s}(R\overline{\ell}-Q\overline{z})^{*}\varphi\big{)}(t)+\frac{\partial\Psi_{1}}{\partial
z}(\overline{z},\overline{\ell}),\qquad\varphi(T)=-\frac{\partial\Psi_{2}}{\partial
z}(\overline{z}(T),\overline{\ell}(T)).$

######  Remark 5.17.

The exemplary objective functionals in Section 7 are precisely of the form in
(35).

######  Proof 5.18 (Proof of Theorem 5.16).

Since $J^{\prime}_{z}(\overline{z},\overline{\ell})\in
H^{1}(0,T;\mathcal{W})^{*}\hookrightarrow H^{1}(0,T;\mathcal{H})^{*}$, Lemma
5.14 gives the existence of a unique solution of (34b). Now, let $h\in
H^{1}(0,T;\mathcal{X}_{c})\cap\mathcal{U}$ be arbitrary and define $\eta$. The
weak form of the adjoint equation then implies

(37)
$\displaystyle\left(\varphi,A_{s}^{\prime}(R\overline{\ell}-Q\overline{z})Rh\right)_{L^{2}(0,T;\mathcal{H})}=\left(\varphi,\overset{\boldsymbol{.}}{\eta}+A_{s}^{\prime}(R\overline{\ell}-Q\overline{z})Q\eta\right)_{L^{2}(0,T;\mathcal{H})}=-J^{\prime}_{z}(\mathcal{S}_{s}(\overline{\ell}),\overline{\ell})\eta.$

This together with Lemma 5.12 shows that
$(\overline{z},\overline{\ell},\varphi)$ fulfills the optimality system (34).
If $J$ is of the form in (35), then Lemma 5.14 implies that the weak solution
of the adjoint equation is in fact a strong solution and solves (36).

######  Corollary 5.19.

Let 5.1 and 5.10 hold. Then $\bar{\ell}\in
H^{1}(0,T;\mathcal{X}_{c})\cap\mathcal{U}$ with associated state
$\overline{z}=\mathcal{S}_{s}(\overline{\ell})$ fulfills (28) if and only if
there exists an adjoint state $\varphi\in L^{2}(0,T;\mathcal{H})$ such that
$(\overline{z},\overline{\ell},\varphi)$ satisfies the optimality system (34).

######  Proof 5.20.

The proof of Theorem 5.16 already shows that (28) implies the optimality
system in (34).

To prove the reverse implication, assume that
$(\overline{z},\overline{\ell},\varphi)$ fulfills the optimality system Eq.
34. Then choose an arbitrary $h\in H^{1}(0,T;\mathcal{H})$, define $\eta$, and
use the fact that $\varphi$ is the weak solution of (34b) to obtain Eq. 37.
This together with (34c) finally give Eq. 28.

######  Example 5.21.

Under suitable additional assumptions, it is possible to further simplify the
gradient equation (34c). For this purpose assume that $R$ is injective (so
that $\mathcal{U}=\\{\ell\in H^{1}(0,T,\mathcal{X})\colon\ell(0)=0\\}$),
$\mathcal{X}_{c}$ is a Hilbert space, and

(38)
$J(z,\ell)=\Psi_{1}(z,\ell)+\Psi_{2}(z(T),\ell(T))+\frac{\gamma}{2}\,\|\overset{\boldsymbol{.}}{\ell}\|_{L^{2}(0,T;\mathcal{X}_{c})}^{2},$

where $\Psi_{1}\colon H^{1}(0,T;\mathcal{W})\times
L^{2}(0,T;\mathcal{X}_{c})\to\mathbb{R}$ and
$\Psi_{2}\colon\mathcal{W}\times\mathcal{X}_{c}\to\mathbb{R}$ are Fréchet
differentiable and $\gamma>0$. This type of objective will also appear in the
application problem in Section 7. Then (34c) becomes

(39)
$\displaystyle\gamma(\partial_{t}\overline{\ell},\partial_{t}\overline{h})_{L^{2}(0,T;\mathcal{X}_{c})}$
$\displaystyle-\int_{0}^{T}\langle
R^{*}A_{s}^{\prime}(R\overline{\ell}-Q\overline{z})^{*}\varphi,h\rangle_{\mathcal{X}^{*},\mathcal{X}}\,dt$
$\displaystyle+\int_{0}^{T}\frac{\partial\Psi_{1}}{\partial\ell}(\overline{z},\overline{\ell})h\,dt+\frac{\partial\Psi_{2}}{\partial\ell}(\overline{z}(T),\overline{\ell}(T))h(T)=0$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\forall\,h\in
H^{1}(0,T;\mathcal{X}_{c})\text{ with }h(0)=0,$

where we identified
$\partial_{\ell}\Psi_{1}(\overline{z},\overline{\ell})\in(L^{2}(0,T;\mathcal{X}_{c}))^{*}=L^{2}(0,T;\mathcal{X}_{c})$.
Note that $\mathcal{X}_{c}$ as a Hilbert space satisfies the Radon-Nikodým-
property. Since $\mathcal{X}_{c}\hookrightarrow\mathcal{X}$, we may identify
$R^{*}A_{s}^{\prime}(R\overline{\ell}-Q\overline{z})^{*}\varphi$ with an
element of $L^{2}(0,T;\mathcal{X}_{c})$, too, which we denote by the same
symbol. Then, if we choose $h(t)=\psi(t)\,\xi$ with $\psi\in
C^{\infty}_{c}(0,T)$ and $\xi\in\mathcal{X}_{c}$ arbitrary, we obtain

$\Big{(}-\int_{0}^{T}\Big{[}\gamma\,\partial_{t}\psi\,\partial_{t}\overline{\ell}+R^{*}A_{s}^{\prime}(R\overline{\ell}-Q\overline{z})^{*}\varphi\,\psi-\frac{\partial\Psi_{1}}{\partial\ell}(\overline{z},\overline{\ell})\psi\Big{]}dt,\xi\Big{)}_{\mathcal{X}_{c}}=0.$

Now, since $\xi\in\mathcal{X}_{c}$ was arbitrary, we find that the second
distributional time derivative of $\overline{\ell}$ is a regular distribution
in $L^{2}(0,T;\mathcal{X}_{c})$, i.e., $\overline{\ell}\in
H^{2}(0,T;\mathcal{X}_{c})$, satisfying for almost all $t\in(0,T)$

(40)
$\gamma\,\partial_{t}^{2}\overline{\ell}(t)+R^{*}A_{s}^{\prime}\big{(}R\overline{\ell}(t)-Q\overline{z}(t)\big{)}^{*}\varphi(t)=\frac{\partial\Psi_{1}}{\partial\ell}(\overline{z},\overline{\ell})(t)\quad\text{in
}\mathcal{X}_{c}.$

Since $\mathcal{X}_{c}$ is supposed to be a Hilbert space, we can apply
integration by parts to (39). Together with
$\overline{\ell}\in\mathcal{U}=\\{\ell\in H^{1}(0,T;\mathcal{X}):\ell(0)=0\\}$
and (40), this implies the following boundary conditions:

(41)
$\overline{\ell}(0)=0,\quad\gamma\,\partial_{t}\overline{\ell}(T)=-\frac{\partial\Psi_{2}}{\partial\ell}(\overline{z}(T),\overline{\ell}(T)),$

where we again identified
$\partial_{\ell}\Psi_{2}(\overline{z}(T),\overline{\ell}(T))\in\mathcal{X}_{c}^{*}$
with its Riesz representative. In summary, we have thus seen that the gradient
equation in (34c) becomes an operator boundary value problem in
$\mathcal{X}_{c}$, namely (40)–(41).

##  6 Second-Order Sufficient Conditions

The next section is devoted to the derivation of second-order sufficient
optimality conditions for the regularized problem (Ps). As it was the case for
the first-order conditions, the main part concerns the differentiability
properties of the control-to-state map $\mathcal{S}_{s}$ and the reduced
objective, to be more precise to show that these are twice continuously
Fréchet differentiable. For this purpose, we need the following sharpened
assumptions on the objective and the regularized operator $A_{s}$:

######  Assumption 6.1.

  * (i)

$J\colon H^{1}(0,T;\mathcal{W})\times
H^{1}(0,T;\mathcal{X}_{c})\rightarrow\mathbb{R}$ is twice continuously Fréchet
differentiable.

  * (ii)

The Fréchet-derivative $A_{s}^{\prime}$ is Lipschitz continuous from
$\mathcal{Y}$ to $L(\mathcal{Z};\mathcal{Z})$. Moreover, for every
$y\in\mathcal{Y}$, $A_{s}^{\prime}(y)$ can be extended to an element of
$L(\mathcal{W};\mathcal{W})$. The mapping arising in this way is Lipschitz
continuous from $\mathcal{Y}$ to $L(\mathcal{W};\mathcal{W})$. Furthermore,
there is a constant $C>0$ such that $\|A_{s}^{\prime}(y)w\|_{\mathcal{W}}\leq
C\|w\|_{\mathcal{W}}$ hold for all $y\in\mathcal{Y}$ and all
$w\in\mathcal{W}$.

  * (iii)

$A_{s}^{\prime}$ is Fréchet differentiable from $\mathcal{Y}$ to
$L(\mathcal{Z};\mathcal{W})$. For all $y\in\mathcal{Y}$, its derivative
$A_{s}^{\prime\prime}(y)$ can be extended to an element of
$L(\mathcal{Z};L(\mathcal{Z};\mathcal{W}))$ and the mapping $y\mapsto
A^{\prime\prime}_{s}(y)$ is continuous in these spaces. Moreover, there exists
a constant $C$ such that
$\|A_{s}^{\prime\prime}(y)[z_{1},z_{2}]\|_{\mathcal{W}}\leq
C\|z_{1}\|_{\mathcal{Z}}\|z_{2}\|_{\mathcal{Z}}$ for all $y\in\mathcal{Y}$ and
all $z_{1},z_{2}\in\mathcal{Z}$.

######  Remark 6.2.

We point out that a second norm gap arises in 6.1, since $A_{s}^{\prime}$ is
only Fréchet differentiable as an operator with values in $\mathcal{W}$ and
not in $\mathcal{Z}\hookrightarrow\mathcal{W}$. This assumption is again
motivated by the application problem in Section 7. The example given there
demonstrates that such as second norm gap is indeed necessary in general,
since, given a concrete application, one cannot expect $A_{s}$ to be twice
Fréchet differentiable in $\mathcal{Y}$, and even not as an operator from
$\mathcal{Y}$ to $\mathcal{Z}$.

The following proposition addresses the second derivative of the solution
operator under the above assumptions. Its proof is in principle completely
along the lines of the proof of Theorem 5.7 on the first derivative of
$\mathcal{S}$. We therefore postpone it to Appendix A.

######  Proposition 6.3 (Second Derivative of the Solution Operator).

Under 5.1(ii) and 6.1(ii) & (iii), the solution operator
$\mathcal{S}_{s}\colon H^{1}(0,T;\mathcal{X})\rightarrow
H^{1}(0,T;\mathcal{W})$ is twice Fréchet differentiable. Given
$\ell,h_{1},h_{2}\in H^{1}(0,T;\mathcal{X})$, its second derivative
$\mathcal{S}^{\prime\prime}_{s}(\ell)[h_{1},h_{2}]\in H^{1}(0,T;\mathcal{W})$
is given by the unique solution of

(42) $\displaystyle\overset{\boldsymbol{.}}{\xi}$
$\displaystyle=A_{s}^{\prime\prime}(R\ell-
Qz)[Rh_{1}-Q\eta_{1},Rh_{2}-Q\eta_{2}]-A_{s}^{\prime}(R\ell-
Qz)Q\xi,\quad\xi(0)=0,$

where $z$ and $\eta_{i}$, $i=1,2$.

Moreover, there exists a constant $C$ such that

(43)
$\|\mathcal{S}^{\prime\prime}_{s}(\ell)[h_{1},h_{2}]\|_{H^{1}(0,T;\mathcal{W})}\leq
C\|h_{1}\|_{H^{1}(0,T;\mathcal{X})}\|h_{2}\|_{H^{1}(0,T;\mathcal{X})}$

for all $\ell,h_{1},h_{2}\in H^{1}(0,T;\mathcal{X})$.

######  Lemma 6.4.

Assume that 5.1 (ii) and 6.1 (ii) and (iii) are fulfilled. Then there exists a
constant $C$ such that

$\displaystyle\|\mathcal{S}_{s}^{\prime\prime}(\ell_{1})$
$\displaystyle-\mathcal{S}_{s}^{\prime\prime}(\ell_{2})\|_{L(H^{1}(0,T;\mathcal{X});L(H^{1}(0,T;\mathcal{X});H^{1}(0,T;\mathcal{W})))}$
$\displaystyle\leq
C\big{(}\|A_{s}^{\prime\prime}(R\ell_{1}-Qz_{1})-A_{s}^{\prime\prime}(R\ell_{2}-Qz_{2})\|_{L^{2}(0,T;L(\mathcal{Z};L(\mathcal{Z};\mathcal{W})))}+\|\ell_{1}-\ell_{2}\|_{H^{1}(0,T;\mathcal{X})}\big{)}$

holds for all $\ell_{1},\ell_{2}\in H^{1}(0,T;\mathcal{X})$, where $z_{i}$,
$i=1,2$.

######  Proof 6.5.

Let $\ell_{1},\ell_{2},h_{1},h_{2}\in H^{1}(0,T;\mathcal{X})$ be arbitrary. We
again abbreviate $z_{i}$, $\eta_{i,j}$, $\xi_{i}$, and $y_{i}$ for
$i,j\in\\{1,2\\}$. By the equation for $\mathcal{S}_{s}^{\prime\prime}$, we
obtain for almost all $t\in[0,T]$

$\displaystyle\overset{\boldsymbol{.}}{\xi}_{1}-\overset{\boldsymbol{.}}{\xi}_{2}$
$\displaystyle=A_{s}^{\prime\prime}(y_{1})[Rh_{1}-Q\eta_{1,1},Rh_{2}-Q\eta_{1,2}]-A_{s}^{\prime}(y_{1})Q\xi_{1}$
$\displaystyle\quad-
A_{s}^{\prime\prime}(y_{2})[Rh_{1}-Q\eta_{2,1},Rh_{2}-Q\eta_{2,2}]-A_{s}^{\prime}(y_{2})Q\xi_{2}$
$\displaystyle=\big{(}A_{s}^{\prime\prime}(y_{1})(Rh_{1}-Q\eta_{1,1})-A_{s}^{\prime\prime}(y_{2})(Rh_{1}-Q\eta_{2,1})\big{)}(Rh_{2}-Q\eta_{1,2})$
$\displaystyle\quad+A_{s}^{\prime\prime}(y_{2})[Rh_{1}-Q\eta_{2,1},Q(\eta_{2,2}-\eta_{1,2})]+\big{(}A_{s}^{\prime}(y_{2})-A_{s}^{\prime}(y_{1})\big{)}Q\xi_{1}+A_{s}^{\prime}(y_{2})Q(\xi_{2}-\xi_{1}).$

With the help of

$A_{s}^{\prime\prime}(y_{1})(Rh_{1}-Q\eta_{1,1})-A_{s}^{\prime\prime}(y_{2})(Rh_{1}-Q\eta_{2,1})\\\
=\big{(}A_{s}^{\prime\prime}(y_{1})-A_{s}^{\prime\prime}(y_{2})\big{)}(Rh_{1}-Q\eta_{1,1})+A_{s}^{\prime\prime}(y_{2})Q(\eta_{2,1}-\eta_{1,1}),$

and Gronwall’s inequality, we thus arrive at

$\displaystyle\|\xi_{1}-\xi_{2}\|_{H^{1}(0,T;\mathcal{W})}$ $\displaystyle\leq
C\Big{[}\|Rh_{1}-Q\eta_{2,1}\|_{H^{1}(0,T;\mathcal{Z})}\|\eta_{1,2}-\eta_{2,2}\|_{H^{1}(0,T;\mathcal{Z})}+\|y_{1}-y_{2}\|_{H^{1}(0,T;\mathcal{Y})}\|\xi_{1}\|_{H^{1}(0,T;\mathcal{W})}$
$\displaystyle\qquad\;+\Big{(}\|A_{s}^{\prime\prime}(y_{1})-A_{s}^{\prime\prime}(y_{2})\|_{L^{2}(0,T;L(\mathcal{Z};L(\mathcal{Z};\mathcal{W})))}\|Rh_{1}-Q\eta_{1,1}\|_{H^{1}(0,T;\mathcal{Z})}$
$\displaystyle\hskip
129.16626pt+\|\eta_{1,1}-\eta_{2,1}\|_{H^{1}(0,T;\mathcal{Z})}\Big{)}\|Rh_{2}-Q\eta_{1,2}\|_{H^{1}(0,T;\mathcal{Z})}\Big{]}$
$\displaystyle\leq
C\Big{(}\|A_{s}^{\prime\prime}(y_{1})-A_{s}^{\prime\prime}(y_{2})\|_{L^{2}(0,T;L(\mathcal{Z};L(\mathcal{Z};\mathcal{W})))}+\|\ell_{1}-\ell_{2}\|_{H^{1}(0,T;\mathcal{X})}\Big{)}\|h_{1}\|_{H^{1}(0,T;\mathcal{X})}\|h_{2}\|_{H^{1}(0,T;\mathcal{X})},$

where we used the estimate in Theorem 5.7, (43), and the Lipschitz continuity
of $\mathcal{S}_{s}^{\prime}$ by Lemma A.1 in the Appendix.

If $A_{s}^{\prime\prime}$ were Lipschitz continuous from $\mathcal{Y}$ to
$L(\mathcal{Z};L(\mathcal{Z};\mathcal{W}))$, then Lemma 6.4 would immediately
imply the Lipschitz continuity of $\mathcal{S}_{s}^{\prime\prime}$. However,
to obtain the continuity of the second derivative, this additional assumption
is not necessary as the following theorem shows:

######  Theorem 6.6 (Second-Order Continuous Fréchet Differentiability of the
Solution Operator).

Suppose that 5.1(ii) and 6.1(ii) & (iii) are fulfilled. Then
$\mathcal{S}_{s}\colon H^{1}(0,T;\mathcal{X})\rightarrow
H^{1}(0,T;\mathcal{W})$ is twice continuously Fréchet differentiable. Its
second derivative at $\ell\in H^{1}(0,T;\mathcal{X})$ in directions
$h_{1},h_{2}\in H^{1}(0,T;\mathcal{X})$ is given by the unique solution of Eq.
42.

######  Proof 6.7.

Thanks to Proposition 6.3, we only have to show that
$\mathcal{S}_{s}^{\prime\prime}$ is continuous from $H^{1}(0,T;\mathcal{X})$
to
$L(H^{1}(0,T;\mathcal{X});L(H^{1}(0,T;\mathcal{X});H^{1}(0,T;\mathcal{W})))$.
For this let $\\{\ell_{n}\\}_{n\in\mathbb{N}}\subset H^{1}(0,T;\mathcal{X})$
and $\ell\in H^{1}(0,T;\mathcal{X})$ be given such that
$\ell_{n}\rightarrow\ell$ in $H^{1}(0,T;\mathcal{X})$ so that in particular
$\ell_{n}\rightarrow\ell$ in $C([0,T];\mathcal{X})$. Then, Lemma 5.3 implies
$z_{n}$ in $C([0,T];\mathcal{Y})$. With this convergence results at hand, we
can apply Lemma C.3 with $M=[0,T]$, $N=\mathcal{Y}$, $G_{n}=R\ell_{n}-Qz_{n}$
and $G=R\ell-Qz$ to see that

Definition 36 $U$

is compact. Therefore, thanks to the continuity assumption in 6.1(iii),
$A_{s}^{\prime\prime}\colon\mathcal{Y}\rightarrow
L(\mathcal{Z};L(\mathcal{Z};\mathcal{W}))$ is uniformly continuous on $U$.
Consequently, $A_{s}^{\prime\prime}(R\ell_{n}-Qz_{n})$ converges to
$A_{s}^{\prime\prime}(R\ell-Qz)$ in
$C([0,T];L(\mathcal{Z};L(\mathcal{Z};\mathcal{W})))$, which, together with
Lemma 6.4, yields the assertion.

######  Remark 6.8.

It is to be noted that the regularized state equation (21) and the equations
corresponding to the derivatives of $\mathcal{S}_{s}$, i.e., (23) and (42),
provide more regular solutions under the hypotheses of 5.1(ii) and 6.1(ii) &
(iii). Indeed, if $\ell,h_{1},h_{2}\in H^{1}(0,T;\mathcal{X})$, then the
solutions of all three equations can be shown to be continuously
differentiable in time with values in the respective spaces ($\mathcal{Y}$,
$\mathcal{Z}$, and $\mathcal{W}$, respectively). Moreover, the time
derivatives of $z$ and $\eta$ are absolutely continuous and the same would
hold for $\xi$, if $A_{s}^{\prime\prime}$ were Lipschitz continuous. However,
we did not exploit this additional regularity, since the original
unregularized problem (P) does not provide this property in general.

With the above differentiability result at hand, it is now standard to derive
the following:

######  Theorem 6.9 (Second-Order Sufficient Optimality Conditions for (Ps)).

Assume that 5.1, 5.10, and 6.1 hold. Let
$(\overline{z},\overline{\ell},\varphi)\in
H^{1}(0,T;\mathcal{Y})\times(H^{1}(0,T;\mathcal{X}_{c})\cap\mathcal{U})\times
L^{2}(0,T;\mathcal{H})$ be a solution of the optimality system (34). Moreover,
suppose that there is a $\delta>0$ such that

(SSC)
$F^{\prime\prime}(\overline{\ell})h^{2}\geq\delta\|h\|_{H^{1}(0,T;\mathcal{X}_{c})}^{2}$

for all $h\in H^{1}(0,T;\mathcal{X}_{c})\cap\mathcal{U}$, where $F$ is the
reduced objective from (27). Then $(\overline{z},\overline{\ell})$ is locally
optimal for Eq. Ps and there exist $\varepsilon>0$ and $\tau>0$ such that the
following _quadratic growth condition_

(44) $F(\ell)\geq
F(\overline{\ell})+\tau\|\ell-\overline{\ell}\|_{H^{1}(0,T;\mathcal{X}_{c})}^{2}$

holds for all $\ell\in H^{1}(0,T;\mathcal{X}_{c})\cap\mathcal{U}$ with
$\|\ell-\overline{\ell}\|_{H^{1}(0,T;\mathcal{X}_{c})}\leq\varepsilon$.

######  Proof 6.10.

Thanks to the assumptions on $J$ and Theorem 6.6, the chain rule implies that
the reduced objective function $F(\cdot)=J(\mathcal{S}_{s}(\cdot),\cdot)\colon
H^{1}(0,T;\mathcal{X}_{c})\rightarrow\mathbb{R}$ is twice continuously Fréchet
differentiable and, according to Corollary 5.19, the equation in (28) holds
for all $h\in H^{1}(0,T;\mathcal{X}_{c})\cap\mathcal{U}$. Since $\mathcal{U}$
is a linear subspace, the claim then follows from standard arguments, see e.g.
[39, Satz 4.23].

######  Remark 6.11.

As already mentioned in Remark 5.11, one could also account for additional
control constraints. In this case, a critical cone would arise in the second-
order conditions, cf. e.g. the survey article [13].

Using the adjoint equation, the second derivative of the reduced objective in
(SSC) can be reformulated as follows:

######  Corollary 6.12.

Assume in addition to the hypotheses of 6.1(iii) that
$\|A^{\prime\prime}_{s}(y)[z_{1},z_{2}]\|_{\mathcal{H}}\leq
C\,\|z_{1}\|_{\mathcal{Z}}\|z_{2}\|_{\mathcal{Z}}$ for all $y\in\mathcal{Y}$
and $z_{1},z_{2}\in\mathcal{Z}$, i.e., the last inequality in 6.1 holds in
$\mathcal{H}$ instead of the weaker space $\mathcal{W}$. Then it holds for all
$\ell,h\in H^{1}(0,T;\mathcal{H})$ that

$F^{\prime\prime}(\ell)h^{2}=\Psi^{\prime\prime}(z,\ell)(\eta,h)^{2}+\Phi^{\prime\prime}(\ell)h^{2}-\big{(}\varphi,A_{s}^{\prime\prime}(R\ell-
Qz)(Rh-Q\eta)^{2}\big{)}_{L^{2}(0,T;\mathcal{H})},$

where $z=\mathcal{S}_{s}(\ell)$, $\eta=\mathcal{S}_{s}^{\prime}(\ell)h$, and
$\varphi$ solves the adjoint equation in (34b).

######  Proof 6.13.

Let us again abbreviate $y=R\ell-Qz$. According to the chain rule, the second
derivative of the reduced objective is given by

$\displaystyle F^{\prime\prime}(\ell)h^{2}$
$\displaystyle=\tfrac{\partial^{2}}{\partial\ell^{2}}J(z,\ell)h^{2}+\tfrac{\partial^{2}}{\partial
z^{2}}J(z,\ell)\eta^{2}+2\,\tfrac{\partial^{2}}{\partial\ell\partial
z}J(z,\ell)[h,\eta]+\tfrac{\partial}{\partial z}J(z,\ell)\xi$
$\displaystyle=\Psi^{\prime\prime}(z,\ell)(\eta,h)^{2}+\Phi^{\prime\prime}(\ell)h^{2}+\tfrac{\partial}{\partial
z}J(z,\ell)\xi$

with $z=\mathcal{S}_{s}(\ell)$, $\eta=\mathcal{S}_{s}^{\prime}(\ell)h$, and
$\xi=\mathcal{S}_{s}^{\prime\prime}(\ell)h^{2}$. Now, since
$A^{\prime\prime}_{s}(y)$ is a bilinear form on $\mathcal{H}$ by assumption,
we obtain that $\xi\in H^{1}(0,T;\mathcal{H})$. Therefore, we are allowed to
test the adjoint equation in (34b) (in its weak form) with $\xi$, which
results in

$\displaystyle\tfrac{\partial}{\partial
z}J(z,\ell)\xi=-(\varphi,\overset{\boldsymbol{.}}{\xi}+A^{\prime}_{s}(y)Q\xi)_{L^{2}(0,T;\mathcal{H})}=-(\varphi,A_{s}^{\prime\prime}(y)(Rh-Q\eta)^{2})_{L^{2}(0,T;\mathcal{H})},$

where we used the precise form of $\mathcal{S}_{s}^{\prime\prime}(\ell)$ in
(42) for the last identity.

##  7 Application to Optimal Control of Homogenized Elastoplasticity

In the upcoming sections, we apply the analysis from the previous sections to
an optimal control problem governed by a system of equations that arise as
homogenization limit in elastoplasticity and was derived in [33, Theorem.
2.2]. It describes the evolution of plastic deformation in a material with
periodic microstructure and formally (i.e., in its strong form) reads as
follows:

(45a) $\displaystyle-\nabla_{x}\cdot\pi\Sigma$ $\displaystyle=f$
$\displaystyle\text{ in }\Omega,$ (45b) $\displaystyle\Sigma$
$\displaystyle=\mathbb{C}(\nabla_{x}^{s}u+\nabla_{y}^{s}v-Bz)$
$\displaystyle\text{ in }\Omega\times Y,$ (45c)
$\displaystyle-\nabla_{y}\cdot\Sigma$ $\displaystyle=0$ $\displaystyle\text{
in }\Omega\times Y,$ (45d) $\displaystyle\overset{\boldsymbol{.}}{z}$
$\displaystyle\in A(B^{\top}\Sigma-\mathbb{B}z)$ $\displaystyle\text{ in
}\Omega\times Y,$ (45e) $\displaystyle u$ $\displaystyle=0$
$\displaystyle\text{ on }\Gamma_{D},$ (45f) $\displaystyle\nu\cdot\pi\Sigma$
$\displaystyle=g$ $\displaystyle\text{ on }\Gamma_{N},$ (45g) $\displaystyle
z(0)$ $\displaystyle=z_{0}$ $\displaystyle\text{ in }\Omega\times Y.$

Herein, $\Omega\subset\mathbb{R}^{d}$, $d=2,3$, is a given domain occupied by
the body under consideration, while $Y=[0,1]^{d}$ is the unit cell. The
boundary of $\Omega$ consists of two disjoint parts, the Dirichlet boundary
$\Gamma_{D}$ and the Neumann boundary $\Gamma_{N}$. Furthermore,
$u\colon(0,T)\times\Omega\to\mathbb{R}^{d}$ is the displacement on the macro
level, while $v\colon(0,T)\times\Omega\times Y\to\mathbb{R}^{d}$ is the
displacement reflecting the micro structure. The stress tensor is denoted by
$\Sigma\colon(0,T)\times\Omega\times Y\to\mathbb{R}^{d\times
d}_{\textup{sym}}$ and $z\colon(0,T)\times\Omega\times Y\to\mathbb{V}$ is the
internal variable describing changes in the material behavior under plastic
deformation (such as hardening), where $\mathbb{V}$ is a finite dimensional
Banach space. Moreover, $\nabla_{x}^{s}$ is the linearized strain in $\Omega$
and $\nabla^{s}_{y}$ is defined analogously. The elasticity tensor
$\mathbb{C}\colon\Omega\times Y\to\mathcal{L}(\mathbb{R}^{d\times
d}_{\textup{sym}})$ and the hardening parameter $\mathbb{B}\colon\Omega\times
Y\to\mathcal{L}(\mathbb{V})$ are given linear and coercive mappings and, by
$B\colon\Omega\times Y\rightarrow\mathcal{L}(\mathbb{V};\mathbb{R}^{d\times
d}_{\textup{sym}})$, one recovers the plastic strain from the internal
variables $z$. The evolution of the internal variables is determined by a
maximal monotone operator $A\colon\mathbb{V}\to 2^{\mathbb{V}}$. In Section
7.4 below, we present a concrete example for such an operator, namely the case
of linear kinematic hardening with von Mises yield condition. Finally, $z_{0}$
is a given initial state and $\pi$ is the averaging over the unit cell, i.e.,

(46) $\pi\colon\Sigma\mapsto\fint_{Y}\Sigma(\cdot,y)\,dy$

The precise assumptions on these data as well as the precise notion of
solutions to (45) are given below.

The volume force $f\colon(0,T)\times\Omega\to\mathbb{R}^{d}$ and the boundary
loads $g\colon(0,T)\times\Gamma_{N}\to\mathbb{R}^{d}$, serve as control
variables. In the following, we will frequently write $\ell$ for the tuple
$(f,g)$. Possible objectives could include a desired displacement or stress
distribution at end time, i.e.,

$\displaystyle
J(u,\Sigma,\ell)\defn\frac{\alpha}{2}\int_{\Omega}|u(T)-u_{d}|^{2}\,dx+\frac{\beta}{2}\int_{\Omega}|(\pi\Sigma)(T)-\sigma_{d}|^{2}\,dx+\Phi(\ell),$

where $u_{d}\colon\Omega\to\mathbb{R}^{d}$ and
$\sigma_{d}\colon\Omega\to\mathbb{R}^{d\times d}_{\textup{sym}}$ are given
desired displacement and stress field, respectively, $\alpha,\beta\geq 0$, and
$\Phi$ is a regularization term depending on the choice of the control space
that will be specified below, see Remark 7.16.

###  7.1 Homogenized Plasticity – Notation and Standing Assumptions

Before discussing the optimal control problem, we first have to introduce the
precise notion of solution for homogenized elastoplasticity system in (45).
For this purpose, we need several assumptions and definitions. We start with
the following

######  Assumption 7.1 (Hypotheses on the data in (45)).

  * •

_Regularity of the domain:_ The domain $\Omega\subset\mathbb{R}^{d}$,
$d\in\\{2,3\\}$, is bounded with Lipschitz boundary $\Gamma$. The boundary
consists of two disjoint measurable parts $\Gamma_{N}$ and $\Gamma_{D}$ such
that $\Gamma=\Gamma_{N}\cup\Gamma_{D}$. While $\Gamma_{N}$ is a relatively
open subset, $\Gamma_{D}$ is a relatively closed subset of $\Gamma$ with
positive measure. In addition, the set $\Omega\cup\Gamma_{N}$ is regular in
the sense of Gröger, cf. [23].

  * •

_Assumptions on the coefficients:_ The elasticity tensor and the hardening
parameter satisfy $\mathbb{C}\in L^{\infty}(\Omega\times
Y;\mathcal{L}(\mathbb{R}^{d\times d}_{\textup{sym}}))$ and $\mathbb{B}\in
L^{\infty}(\Omega\times Y;\mathcal{L}(\mathbb{V}))$ and are symmetric and
uniformly coercive, i.e., there exist constants $\underline{c}>0$ and
$\underline{b}>0$ such that

$\displaystyle\mathbb{C}(x,y)\sigma\colon\sigma$
$\displaystyle\geq\underline{c}\,\|\sigma\|_{\mathbb{R}^{d\times d}}^{2}$
$\displaystyle\forall\,\sigma\in\mathbb{R}^{d\times d}_{\textup{sym}},$
$\displaystyle\text{f.a.a.~{}}(x,y)\in\Omega\times Y,$
$\displaystyle\mathbb{B}(x,y)\zeta\colon\zeta$
$\displaystyle\geq\underline{b}\,\|\zeta\|_{\mathbb{V}}^{2}$
$\displaystyle\forall\,\zeta\in\mathbb{V},$
$\displaystyle\text{f.a.a.~{}}(x,y)\in\Omega\times Y.$

In addition, $B\in L^{\infty}(\Omega\times
Y;\mathcal{L}(\mathbb{V};\mathbb{R}^{d\times d}_{\textup{sym}}))$ is a given
linear mapping.

Next, we define the function spaces for the various variables in (45):

######  Definition 7.2 (Function spaces).

Let $s\in[1,\infty)$. For the quantities in (45), we define the following
spaces:

  * •

space for the macro displacement $u$:

Definition 40 $U^{s}$

  * •

space for the internal variable $z$:

Definition 42 $Z^{s}$

  * •

stress space for $\Sigma$:

Definition 43 $S^{s}$

  * •

space for the micro displacement $v$:

Definition 44 $V^{s}$

For the latter, we denote by $C^{\infty}_{\textup{per}}(Y;\mathbb{R}^{d})$ the
space of $C^{\infty}(\mathbb{R}^{d};\mathbb{R}^{d})$ functions which are
$Y$-periodic, identified with their restriction on $Y$, and define
$W^{1,s}_{\textup{per}}(Y;\mathbb{R}^{d})$ to be the closure of
$C^{\infty}_{\textup{per}}(Y;\mathbb{R}^{d})$ with respect to the
$W^{1,s}(Y;\mathbb{R}^{d})$ norm. Further, $W^{1,s}_{\perp}(Y;\mathbb{R}^{d})$
is the closed subspace of $W^{1,s}(Y;\mathbb{R}^{d})$ consisting of functions
of mean $0$, and

$W^{1,s}_{\textup{per},\perp}(Y;\mathbb{R}^{d})=W^{1,s}_{\textup{per}}(Y;\mathbb{R}^{d})\cap
W^{1,s}_{\perp}(Y;\mathbb{R}^{d}).$

We set the norm on $V^{s}$ to be

Definition 45 $\|v\|_{V^{s}}$

with which $V^{s}$ becomes a Banach space and for the case $s=2$ a Hilbert
space with the obvious scalar product.

######  Assumption 7.3 (Maximal monotone operator).

The maximal monotone operator $A$ from the evolution law in (45d) is a set-
valued map in the Hilbert space $\mathcal{H}=Z^{2}$, i.e., $A\colon Z^{2}\to
2^{Z^{2}}$. It is assumed to satisfy our standing assumptions from Section 2.
Moreover, we assume that there is a sequence of operators $\\{A_{n}\\}$ from
$Z^{2}$ to $Z^{2}$ satisfying 4.4.

In Section 7.4 below, we will investigate the maximal monotone operator
arising in the case of linear kinematic hardening with von Mises yield
condition and show how to construct the approximating sequence of smooth
operators for this particular case. With the above definitions at hand, we are
now in the position to define our precise notion of solutions to (45):

######  Definition 7.4 (Weak solutions).

Let $\ell\in H^{1}(0,T;(U^{2})^{*})$ and $z_{0}\in Z^{2}$. Then we say that a
tuple

$(u,v,z,\Sigma)\in H^{1}(0,T;U^{s})\times H^{1}(0,T;V^{2})\times
H^{1}(0,T;Z^{2})\times H^{1}(0,T;S^{2})$

is a solution of Eq. 45, if, for almost all $t\in(0,T)$, there holds

(47a)
$\displaystyle\int_{\Omega}(\pi\Sigma(t))(x)\cdot\nabla_{x}^{s}\varphi(x)\,dx$
$\displaystyle={\langle\ell(t),\varphi\rangle}$
$\displaystyle\forall\,\varphi\in U^{2},$ (47b) $\displaystyle\Sigma(t)$
$\displaystyle=\mathbb{C}(\pi_{r}^{-1}\nabla_{x}^{s}u(t)+\nabla_{y}^{s}v(t)-Bz(t))$
$\displaystyle\text{in }S^{2},$ (47c) $\displaystyle\int_{\Omega\times
Y}\Sigma(t,x,y)\cdot\nabla_{y}^{s}\psi(x,y)\,dy$ $\displaystyle=0$
$\displaystyle\forall\,\psi\in V^{2},$ (47d)
$\displaystyle\overset{\boldsymbol{.}}{z}(t)$ $\displaystyle\in
A(B^{\top}\Sigma(t)-\mathbb{B}z(t))\quad$ $\displaystyle\text{in }Z^{2},$
(47e) $\displaystyle z(0)$ $\displaystyle=z_{0}$ $\displaystyle\text{in
}Z^{2},$

where $\pi\colon S^{2}\to L^{2}(\Omega;\mathbb{R}^{d\times d}_{\textup{sym}})$
is the average mapping from (46) and

$\pi^{-1}_{r}\colon L^{2}(\Omega;\mathbb{R}^{d\times
d}_{\textup{sym}})\ni\varepsilon\mapsto\Big{(}\Omega\times
Y\ni(x,y)\mapsto\varepsilon(x)\in\mathbb{R}^{d\times
d}_{\textup{sym}}\Big{)}\in S^{2}.$

In the following, we will frequently consider $\pi_{r}^{-1}$ in different
domains and ranges, for simplicity denoted by the same symbol.

###  7.2 Reduction of the System

In the following, we reduce the system (47) to an equation in the internal
variable $z$ only and it will turn out that this equation has exactly the form
of our general equation (3). To this end we proceed analog to [24, Chapter 4].
For this purpose, let us define the following operators:

######  Definition 7.5.

Let $s\in[1,\infty)$. Then we define

Definition 46 $\nabla^{s}_{(x,y)}\colon U^{s}\times V^{s}\to
S^{s},\quad\nabla^{s}_{(x,y)}(u,v)$

For its adjoint, we write

$\displaystyle\operatorname{Div}_{(x,y)}\colon
S^{s^{\prime}}\to(U^{s})^{*}\times(V^{s})^{*},$
$\displaystyle\langle\operatorname{Div}_{(x,y)}\sigma,(\varphi,\psi)\rangle\begin{aligned}
&\defn-\langle{\nabla^{s}_{(x,y)}}^{*}\sigma,(\varphi,\psi)\rangle=-\int_{\Omega\times
Y}\sigma(x,y)\colon(\nabla^{s}_{x}\varphi(x)+\nabla^{s}_{y}\psi(x,y))\,d(x,y).\end{aligned}$

With a slight abuse of notation, we denote these operators for different
values of $s$ always by the same symbol.

######  Lemma 7.6.

Let 7.1 be fulfilled. Then there is an index $\bar{s}>2$ such that, for every
$s\in[\bar{s}^{\prime},\bar{s}]$ and every
$(\mathfrak{f},\mathfrak{g})\in(U^{s^{\prime}})^{*}\times(V^{s^{\prime}})^{*}$,
there exists a unique solution $(u,v)\in U^{s}\times V^{s}$ of

(48)
$-\operatorname{Div}_{(x,y)}\big{(}\mathbb{C}\nabla^{s}_{(x,y)}(u,v)\big{)}=(\mathfrak{f},\mathfrak{g})\quad\text{in
}(U^{s^{\prime}})^{*}\times(V^{s^{\prime}})^{*}$

and there is a constant $C_{s}>0$, independent of $\mathfrak{f}$ and
$\mathfrak{g}$, such that

$\|(u,v)\|_{U^{s}\times V^{s}}\leq
C_{s}\big{(}\|\mathfrak{f}\|_{(U^{s^{\prime}})^{*}}+\|\mathfrak{g}\|_{(V^{s^{\prime}})^{*}}\big{)}.$

######  Proof 7.7.

The claim is equivalent to
$-\operatorname{Div}_{(x,y)}\mathbb{C}\nabla^{s}_{(x,y)}$ being a topological
isomorphism between $U^{s}\times V^{s}$ and its dual space
$(U^{s})^{*}\times(V^{s})^{*}$ for $s\in[\bar{s}^{\prime},\bar{s}]$. We start
with the case $s=2$. For this, the left hand side of (48) gives rise to a
bilinear form $\mathfrak{b}$ on the Hilbert space $U^{2}\times V^{2}$:

Definition 48 $\mathfrak{b}\bigl{(}(u,v),(\varphi,\psi)\bigr{)}$

Clearly, $\mathfrak{b}$ is bounded. Due to Poincaré’s inequality for functions
with zero mean value, which implies

$\displaystyle\mathfrak{b}\bigl{(}(u,v),(u,v)\bigr{)}$
$\displaystyle\geq\underline{c}\,|Y|\int_{\Omega}|\nabla^{s}_{x}u(x)|^{2}\,dx+\underline{c}\int_{\Omega\times
Y}|\nabla^{s}_{y}v(x,y)|^{2}\,d(x,y)$ $\displaystyle\geq
C\big{(}\|u\|_{H^{1}_{D}(\Omega;\mathbb{R}^{d})}+\|v\|_{V^{2}(\Omega\times
Y;\mathbb{R}^{d})}\big{)}^{2}\quad\forall\,(u,v)\in U^{2}\times V^{2},$

it is also coercive so that the claim and isomorphism property for
$-\operatorname{Div}_{(x,y)}\mathbb{C}\nabla^{s}_{(x,y)}$ for $s=2$ follows
from the Lax-Milgram lemma.

We next extrapolate this isomorphism property to $U^{s}\times V^{s}$ for $s$
around $2$ using the fundamental stability theorem by Šneĭberg [36]. (See also
the more accessible and extensive [3, Appendix A].) More precisely, we show
that the spaces $U^{s}\times V^{s}$ and their duals form complex interpolation
scales in $s$. Then the stability theorem shows that the set of scale
parameters $s$ such that
$-\operatorname{Div}_{(x,y)}\mathbb{C}\nabla^{s}_{(x,y)}$ is a topological
isomorphism between $U^{s}\times V^{s}$ and its dual space is open. Since the
set includes $2$, as seen above, this then implies the claim.

To establish the interpolation scale, it is enough to consider the primal
case, since the dual interpolation scale is inherited from the primal one by
duality properties of the complex interpolation functor [38, Theorem 1.11.3].
So, we show that

$U^{s_{\theta}}\times V^{s_{\theta}}=\bigl{[}U^{s_{0}}\times
V^{s_{0}},U^{s_{1}}\times
V^{s_{1}}\bigr{]}_{\theta}\quad\text{for}\quad\frac{1}{s}=\frac{1-\theta}{s_{0}}+\frac{\theta}{s_{1}}$

for all $s_{0},s_{1}\in(1,\infty)$ and $\theta\in(0,1)$. It is moreover
sufficient to consider each component in the interpolation separately.

For the $U^{s}=W^{1,s}_{D}(\Omega;\mathbb{R}^{d})$ spaces, the interpolation
scale property is well known by now in the setting of 7.1 and even much more
general ones; we refer to [6]. The result for $V^{s}$ is proven by reducing
the problem to the $W^{1,s}_{\textup{per},\perp}(Y;\mathbb{R}^{d})$ spaces and
showing that these are complemented subspaces of $W^{1,s}(Y;\mathbb{R}^{d})$
and thus inherit the latter’s interpolation properties. This is done in the
appendix, Theorem B.5, and finishes the proof.

######  Remark 7.8.

In general, one cannot expect $\bar{s}$ to be significantly larger than 2, due
to both the irregular coefficient tensors and the mixed boundary conditions,
see e.g. [18, 34, 28]. This issue will become crucial in the discussion of
second-order necessary optimality conditions in 7.3 below.

Now we are in the position to reduce (47) to an equation in the variable $z$
only. For this purpose, we need the following

######  Definition 7.9 ($Q$ and $R$ for the case of homogenized plasticity).

Let $s\in[\bar{s}^{\prime},\bar{s}]$ be given. By Lemma 7.6, the solution
operator associated with (48), denoted by

Definition 49 $\mathcal{G}$

is well defined, linear and bounded. The components of $\mathcal{G}$ are
abbreviated by

Definition 50 $\mathcal{G}_{u}$

Based on this solution operator, we moreover define

$T\colon Z^{s}\ni z\mapsto
B^{\top}\mathbb{C}\nabla^{s}_{(x,y)}\mathcal{G}(-\operatorname{Div}_{(x,y)}(\mathbb{C}Bz))\in
Z^{s}.$

Now, we have everything at hand to define the mappings $R$ and $Q$ from our
general equation (3) for the special case of homogenized plasticity:

(49) $\displaystyle R\colon(U^{s^{\prime}})^{*}\ni\ell\mapsto
B^{\top}\mathbb{C}\nabla^{s}_{(x,y)}\mathcal{G}(\ell,0)\in Z^{s},$ (50)
$\displaystyle Q\colon Z^{s}\ni z\mapsto(B^{\top}\mathbb{C}B+\mathbb{B}-T)z\in
Z^{s}.$

Again, with a slight abuse of notation, we denote all of the above operators
for different values of $s\in[\bar{s}^{\prime},\bar{s}]$ always by the same
symbol.

The reason for defining the operators $Q$ and $R$ in the way we did in
Definition 7.9 is the following: Owing to Lemma 7.6, given $z\in Z^{2}$, one
can solve (47a)–(47c) for $u$, $v$, and $\Sigma$ so that the tuple
$(u,v,\Sigma)\in U^{2}\times V^{2}\times S^{2}$ is uniquely determined by $z$.
Even more, using the operators from Definition 7.9, we see that the solution
of (47a)–(47c) for given $z$ is

(51) $\displaystyle(u,v)$
$\displaystyle=\mathcal{G}(-\operatorname{Div}_{(x,y)}(\mathbb{C}Bz))+(\ell,0)),$
(52) $\displaystyle\Sigma$
$\displaystyle=\mathbb{C}\big{[}\nabla^{s}_{(x,y)}\mathcal{G}\big{(}-\operatorname{Div}_{(x,y)}(\mathbb{C}Bz))+(\ell,0)\big{)}-Bz\big{]}.$

Inserting the last equation in (47d) and employing the definition of $Q$ and
$R$ in (50) and (49) then yields

$\overset{\boldsymbol{.}}{z}\in A(B^{\top}\Sigma-\mathbb{B}z)=A(R\ell-Qz),$

i.e., exactly an evolution equation of the general form in (3). This shows
that the system (47) of homogenized elastoplasticity can equivalently be
rewritten as an abstract operator evolution equation of the form (3).

For the differentiability properties needed in sections 5 and 6, a norm gap is
required such that it is no longer sufficient to consider just the Hilbert
space $\mathcal{H}=Z^{2}$. In accordance with the definitions of $R$ and $Q$,
we therefore define the spaces in the abstract setting in our concrete
application problem as follows:

######  Definition 7.10 (Spaces in case of homogenized plasticity).

The spaces $\mathcal{Y}$, $\mathcal{Z}$, $\mathcal{H}$, and $\mathcal{W}$ from
Section 2 are set to

(53) $\mathcal{Y}$

with $s_{1}\geq s_{2}\geq 2\geq s_{3}$. The integrability indices $s_{1}$,
$s_{2}$, and $s_{3}$ depend crucially on the differentiability properties of
the regularized version of $A$ and will be specified for a concrete
realization of $A$ in Section 7.4 below. Moreover, we choose

Definition 55 $\mathcal{X}$

Furthermore, the control space is given by

(54) $\mathcal{X}_{c}$

Due to $s_{1}\geq 2$, $\mathcal{X}_{c}$ is reflexive and embeds compactly in
$\mathcal{X}$ by Sobolev embedding and trace theorems. Therefore, all our
standing assumptions on the spaces in Section 2 are fulfilled.

Of course, elements in $\mathcal{X}_{c}$ are identified with those in
$\mathcal{X}$ by

Definition 57 $\langle(f,g),u\rangle_{(U^{s_{1}})^{*},U^{s_{1}}}$

In order to apply our general theory to the present setting, we need the
following assumption on the regularity of the linear equation (48). As we will
see in subsections 7.3 and 7.4 below, this assumption may become fairly
restrictive, if one aims to establish second-order sufficient optimality
conditions, since, in this case, $s_{1}$ and the conjugate index
$s_{3}^{\prime}$ may be rather large.

######  Assumption 7.11 (Critical regularity condition).

The index $\bar{s}$ from Lemma 7.6 satisfies
$\bar{s}\geq\max\\{s_{1},s_{3}^{\prime}\\}$, where $s_{1}$ and $s_{3}$ are the
integrability indices from (53).

######  Proposition 7.12.

Under 7.11 and with the spaces defined in Definition 7.10, the operators $R$
and $Q$ from (49) and (50), respectively, satisfy the standing assumptions
from Section 2, that is, $R$ is linear and bounded from
$(U^{s_{1}^{\prime}})^{*}$ to $Z^{s_{1}}$ and $Q$ is a linear and bounded
operator from $Z^{s}$ to $Z^{s}$ for all $s\in[s_{3},s_{1}]$ and, considered
as an operator in $Z^{2}$, coercive and self-adjoint.

######  Proof 7.13.

The required mapping properties of $Q$ and $R$ directly follow from their
construction in Definition 7.9 in combination with Lemma 7.6 and 7.11,
respectively. It remains to show that $Q$ is coercive and self-adjoint. Since
$\mathbb{B}$ is symmetric and coercive according to 7.1, it is sufficient to
prove that the operator $B^{\top}\mathbb{C}B-T\colon Z^{2}\rightarrow Z^{2}$
is symmetric and positive. To prove the symmetry, first observe that
$B^{\top}\mathbb{C}B$ is symmetric by the symmetry of $\mathbb{C}$. The
symmetry of $\mathbb{C}$ moreover implies that
$\mathcal{G}\colon(U^{2})^{*}\times(V^{2})^{*}\to U^{2}\times V^{2}$, i.e.,
the solution operator of (48), is self-adjoint. Therefore, the construction of
$T$ in Definition 7.9 implies for all $z_{1},z_{2}\in Z^{2}$ that

$\displaystyle(Tz_{1},z_{2})_{Z^{2}}$
$\displaystyle=\langle-\operatorname{Div}_{(x,y)}(\mathbb{C}Bz_{2}),\mathcal{G}(-\operatorname{Div}_{(x,y)}(\mathbb{C}Bz_{1}))\rangle$
$\displaystyle=\langle\mathcal{G}(-\operatorname{Div}_{(x,y)}(\mathbb{C}Bz_{2})),-\operatorname{Div}_{(x,y)}(\mathbb{C}Bz_{1})\rangle=(z_{1},Tz_{2})_{Z^{2}}$

so that $T$ is also symmetric. To show the positivity of
$B^{\top}\mathbb{C}B-T$, let $z\in Z^{2}$ be arbitrary. To shorten the
notation, we abbreviate $(u_{z},v_{z})$. Then, by testing the equation for
$(u_{z},v_{z})$, i.e., (48) with
$(\mathfrak{f},\mathfrak{g})=-\operatorname{Div}_{(x,y)}(\mathbb{C}Bz)$, with
$(-u_{z},-v_{z})$, we arrive at

$\big{(}\mathbb{C}(Bz-\nabla^{s}_{(x,y)}(u_{z},v_{z}),-\nabla^{s}_{(x,y)}(u_{z},v_{z})\big{)}_{S^{2}}=0.$

Since, by construction,
$Tz=B^{\top}\mathbb{C}\nabla^{s}_{(x,y)}(u_{z},v_{z})$, the coercivity of
$\mathbb{C}$ therefore implies

$\displaystyle\left((B^{\top}\mathbb{C}B-T)z,z\right)_{Z^{2}}$
$\displaystyle=\big{(}\mathbb{C}(Bz-\nabla^{s}_{(x,y)}(u_{z},v_{z}),Bz\big{)}_{S^{2}}$
$\displaystyle=\big{(}\mathbb{C}(Bz-\nabla^{s}_{(x,y)}(u_{z},v_{z}),Bz-\nabla^{s}_{(x,y)}(u_{z},v_{z})\big{)}_{S^{2}}\geq
0.$

As $z$ was arbitrary, this proves the positivity.

We point out that the whole analysis in sections 3 and 4 is carried out in the
Hilbert space $\mathcal{H}=Z^{2}$. Therefore, for the mere existence and
approximation results from these two sections, the critical regularity
condition in 7.11 is _automatically fulfilled_ by setting
$s_{1}=s_{2}=s_{3}=2$ (so that
$\mathcal{Y}=\mathcal{Z}=\mathcal{W}=\mathcal{H}=Z^{2}$). Note that, in this
case, the Lax-Milgram lemma guarantees the assertion of 7.11 without any
further regularity assumptions, see the proof of Lemma 7.6. The additional
crucial regularity assumption only comes into play, when first- and second-
order optimality conditions are investigated, see Remark 2.1. In Section 7.4
below, we will elaborate in detail, where the critical 7.11 is needed to
ensure the required differentiability properties of the regularized control-
to-state map for the example of a specific yield condition.

We collect our findings so far in the following

######  Theorem 7.14 (Homogenized plasticity as abstract evolution VI).

Under the Assumptions 7.1 and 7.3, the system of homogenized elastoplasticty
in its weak form in (47) is equivalent to an abstract operator differential
equation of the form

(55) $\overset{\boldsymbol{.}}{z}\in A(R\ell-Qz),\text{ \ \ \ \ }z(0)=z_{0},$

with $Q$ and $R$ as defined in (49) and (50) in the following sense: If
$(u,v,z,\Sigma)$ solves (47), then $z$ is a solution (55), and vice versa, if
$z$ solves (55), then $z$ together with $(u,v)$ and $\Sigma$ as defined in
(51) and (52), respectively, form a solution of (47).

In addition, $Q$ and $R$ satisfy the standing assumptions from Section 2
(provided that the function spaces are chosen according to Definition 7.10).
Therefore, the existence and approximation results of Sections 3 and 4 hold
for (55), in particular:

  * •

For every $\ell\in\mathcal{U}(z_{0},D(A))$, there is a unique solution
$(u,v,z,\Sigma)$ of the weak system of homogenized plasticity in (47), cf.
Theorem 3.3.

  * •

Optimal control problems governed by the weak system of homogenized
elastoplasticity admit globally optimal solutions, provided that the standing
assumptions on the objective are fulfilled, cf. Theorem 4.2.

  * •

The approximation results of Theorem 4.7 and Corollary 4.9 apply in case of
homogenized elastoplasticity.

######  Remark 7.15.

Existence and uniqueness of solutions to (47) was already established in [33].

######  Remark 7.16.

The example for objective functionals mentioned above, i.e.,

(56) $\displaystyle
J(u,\Sigma,f,g)\defn\tfrac{\alpha}{2}\int_{\Omega}|u(T)-u_{d}|^{2}\,dx+\tfrac{\beta}{2}\int_{\Omega}|(\pi\Sigma)(T)-\sigma_{d}|^{2}\,dx+\tfrac{\gamma}{2}\|(f,g)\|_{H^{1}(0,T;\mathcal{X}_{c})}^{2}$

with $u_{D}\in L^{2}(\Omega;\mathbb{R}^{d})$ and $\sigma_{D}\in
L^{2}(\Omega;\mathbb{R}^{d\times d}_{\textup{sym}})$ and $\alpha,\beta\geq 0$,
$\gamma>0$, satisfies the standing assumptions on the objective functional, as
we will see in the following. In this case, the functional
$\Psi\colon(z,\ell)\to\mathbb{R}$ in the general setting consists of the two
integrals at end point $T$. Let us consider the first one containing the
displacement $u$. Since the latter is given by the first component of the
solution operator of (51), which maps $H^{1}(0,T;Z^{2})\times
H^{1}(0,T;\mathcal{X}_{c})$ to $H^{1}(0,T;U^{2})\hookrightarrow
C([0,T];L^{2}(\Omega;\mathbb{R}^{d}))$, this integral is well defined. (Note
that the operators in (51) just act pointwise in time and the time regularity
of $z$ and $\ell$ carries over to $u$ and $v$.) Clearly, this solution
operator is linear and bounded. In case of the second integral involving
$\Sigma$, one can argue completely analogously based on the solution operator
of (52). Thus, $\Psi$ is convex and continuous, hence weakly lower
semicontinuous, and in addition, bounded from below by zero. Moreover, the
purely control part of the objective is given by
$\Phi(f,g)=\frac{\gamma}{2}\|(f,g)\|_{H^{1}(0,T;\mathcal{X}_{c})}^{2}$ and
therefore clearly weakly lower semicontinuous and coercive as required. Thus
all standing assumptions are fulfilled as claimed. Of course, various other
objective functionals are possible as well, such as tracking type objectives
over the whole space-time-cylinder, but to keep the discussion concise, we
just mention the example above.

###  7.3 Optimality System

In the following section, we establish necessary and sufficient optimality
conditions for the optimal control of regularized homogenized
elastoplasticity. To be more precise, we consider a single element of the
sequence of regularizations of the maximal monotone operator $A$ from 7.3,
which we again denote by $A_{s}$, and apply the general theory from Section 5
and Section 6. In view of the norm gap needed for the differentiability of
$A_{s}$, we will consider $A_{s}$ in different domains and ranges (denoted by
the same symbol) and assume that $A_{s}$ maps $\mathcal{Y}=Z^{s_{1}}$ to
itself. Accordingly, we treat the regularized version of the state equation
(with $A_{s}$ instead of $A$) in the same manner, i.e., with integrability
index $s_{1}$ instead of $2$, see (57) below. To keep the discussion concise,
we moreover assume in all what follows that $s_{1}\geq 2$ is such that $p=r=2$
satisfy the conditions in (54). For $d=\dim(\Omega)=3$, this implies $s_{1}<3$
and, in case without boundary control, i.e., $g\equiv 0$, $s_{1}<6$ is
sufficient. This will become important in the discussion of second-order
sufficient conditions, as we will see below. Motivated by (56), we consider an
optimal control problem of the form

(57) $\left\\{\quad\begin{aligned} \min\quad&\begin{aligned}
J(u,\Sigma,f,g)&\defn F_{1}(u,\Sigma)+F_{2}(u(T),\Sigma(T))\\\
&\qquad+\frac{\gamma}{2}\Big{(}\|\overset{\boldsymbol{.}}{f}\|_{L^{2}(0,T;L^{2}(\Omega;\mathbb{R}^{d}))}^{2}+\|\overset{\boldsymbol{.}}{g}\|_{L^{2}(0,T;L^{2}(\Gamma_{N};\mathbb{R}^{d}))}^{2}\Big{)}\end{aligned}\\\
\text{s.t.}\quad&(f,g)\in H^{1}(0,T;L^{2}(\Omega;\mathbb{R}^{d})\times
L^{2}(\Gamma_{N};\mathbb{R}^{d})),\\\ &(u,v,z,\Sigma)\in
H^{1}(0,T;U^{s_{1}}\times V^{s_{1}}\times Z^{s_{1}}\times S^{s_{1}}),\\\
\text{and}\quad&\begin{aligned}
-\operatorname{Div}_{(x,y)}\Sigma&=((f,g),0),\\\
\Sigma&=\mathbb{C}\big{(}\nabla^{s}_{(x,y)}(u,v)-Bz\big{)},\\\
\overset{\boldsymbol{.}}{z}&=A_{s}(B^{\top}\Sigma-\mathbb{B}z),\quad
z(0)=z_{0},\end{aligned}\\\ &\ell(0)=0.\end{aligned}\right.$

Since, in many applications, displacement and stress on the macro level are of
special interest, especially at end time, we focus on objectives with this
particular structure with continuously Fréchet differentiable mappings

(58) $F_{1}\colon L^{2}(0,T;U^{2})\times L^{2}(0,T;S^{2})\to\mathbb{R},\quad
F_{2}\colon U^{2}\times S^{2}\to\mathbb{R}.$

In order to apply our general theory, we not only have to reduce the state
system to an equation of the form (55), but also have to reduce the objective.
For this purpose, let us denote the solution operators of (51) and (52) by
$\mathfrak{u}\colon(\ell,z)\mapsto u$ and
$\mathfrak{S}\colon(\ell,z)\mapsto\Sigma$. To shorten the notation, we will
consider $\mathfrak{u}$ and $\mathfrak{S}$ with different domains and ranges,
e.g. $\mathfrak{u}\colon(U^{s^{\prime}})^{*}\times Z^{s}\to U^{s}$ and
$\mathfrak{u}\colon L^{2}(0,T;(U^{s^{\prime}})^{*})\times L^{2}(0,T;Z^{s})\to
L^{2}(0,T;U^{s})$ with $s\in[\bar{s}^{\prime},\bar{s}]$ and analogously for
$\mathfrak{S}$. Note again that the time regularity of $z$ and $\ell$ directly
carries over to the time regularity of $u$ and $\Sigma$. Given these
operators, we define

$\displaystyle\Psi_{1}\colon L^{2}(0,T;Z^{2})\times
L^{2}(0,T;(U^{2})^{*})\to\mathbb{R},$ $\displaystyle\Psi_{1}(z,\ell)$
$\displaystyle\Psi_{2}\colon Z^{2}\times(U^{2})^{*}\to\mathbb{R},$
$\displaystyle\Psi_{2}(z,\ell)$

so that the objective in (57) becomes

(59)
$J(z,\ell)=\Psi_{1}(z,\ell)+\Psi_{2}(z(T),\ell(T))+\tfrac{\gamma}{2}\|(\overset{\boldsymbol{.}}{f},\overset{\boldsymbol{.}}{g})\|_{L^{2}(0,T;\mathcal{X}_{c})}^{2},$

i.e., exactly an objective of the form in (35) and (38), respectively. Since
$\mathfrak{u}$ and $\mathfrak{S}$ are linear and bounded and $F_{1}$ and
$F_{2}$ are assumed to be continuously Fréchet differentiable, the chain rule
implies the differentiability of $\Psi_{1}$ and $\Psi_{2}$ so that 5.1(i) is
met, if we set $s_{3}=2$ so that $\mathcal{W}=Z^{2}$. To apply the results of
Section 5 in order to establish an optimality system for (57), we additionally
need that $A_{s}$ satisfies 5.1(ii), which is ensured by the following

######  Assumption 7.17.

We set $s_{2}=s_{3}=2$, i.e., $\mathcal{W}=\mathcal{Z}=Z^{2}$, and assume that
$A_{s}$ fulfills 5.1(ii) with $\mathcal{Y}=Z^{s_{1}}$, $s_{1}\geq 2$, i.e., in
particular that $A_{s}$ is Fréchet differentiable from $\mathcal{Z}^{s_{1}}$
to $\mathcal{Z}^{2}$.

In light of Lemma 7.6, 7.17 does not impose any restriction for practical
realizations of $A_{s}$, as we will see in Section 7.4 below. Given this
assumption, Theorem 5.16 and Example 5.21 imply for a locally optimal solution
$(\overline{\ell})=(\overline{f},\overline{g})$ with associated optimal
internal variable $\overline{z}$:

(60a)
$\displaystyle\overset{\boldsymbol{.}}{\overline{z}}(t)=A_{s}(R\overline{\ell}-Q\overline{z})(t),\quad\overline{z}(0)=z_{0},$
(60b)
$\displaystyle\overset{\boldsymbol{.}}{\varphi}(t)=\big{(}QA^{\prime}_{s}(R\overline{\ell}-Q\overline{z})^{*}\varphi\big{)}(t)+\frac{\partial\Psi_{1}}{\partial
z}(\overline{z},\overline{\ell})(t),\quad\varphi(T)=-\frac{\partial\Psi_{2}}{\partial
z}(\overline{z}(T),\overline{\ell}(T)),$ (60c)
$\displaystyle\left\\{\quad\begin{aligned}
\gamma\,\partial_{t}^{2}\overline{\ell}(t)+R^{*}A_{s}^{\prime}\big{(}R\overline{\ell}(t)-Q\overline{z}(t)\big{)}^{*}\varphi(t)&=\frac{\partial\Psi_{1}}{\partial\ell}(\overline{z},\overline{\ell})(t),\\\
\overline{\ell}(0)=0,\quad\partial_{t}\overline{\ell}(T)&=-\frac{\partial\Psi_{2}}{\partial\ell}(\overline{z}(T),\overline{\ell}(T)).\end{aligned}\right.$

Then, owing to the precise structure of $R$, $Q$, $\Psi_{1}$, and $\Psi_{2}$,
this leads us to the following

######  Theorem 7.18 (KKT-system for optimal control of homogenized
plasticity).

Let 7.1 be satisfied and assume that $A_{s}$ fulfills 7.17. Suppose moreover
that the regularity condition in 7.11 is satisfied, i.e., $\bar{s}\geq s_{1}$.
Then, if $(\overline{f},\overline{g})\in H^{1}(0,T;L^{2}(\Omega))\times
H^{1}(0,T;L^{2}(\Gamma_{N}))$ is locally optimal for (57) with associated
state $(\overline{u},\overline{v},\overline{z},\overline{\Sigma})\in
H^{1}(0,T;U^{s_{1}})\times H^{1}(0,T;V^{s_{1}})\times
H^{1}(0,T;Z^{s_{1}})\times H^{1}(0,T;S^{s_{1}})$, then there exists an adjoint
state

$\begin{gathered}(w,q,\varphi,\Upsilon)\in H^{1}(0,T;U^{2})\times
H^{1}(0,T;V^{2})\times H^{1}(0,T;Z^{2})\times H^{1}(0,T;S^{2}),\\\
(w_{T},q_{T},\Upsilon_{T})\in U^{2}\times V^{2}\times S^{2}\end{gathered}$

such that the following _optimality system_ is satisfied:

State equation: (61b)
$\displaystyle-\operatorname{Div}_{(x,y)}\overline{\Sigma}$
$\displaystyle=((\overline{f},\overline{g}),0),$ (61c)
$\displaystyle\overline{\Sigma}$
$\displaystyle=\mathbb{C}\big{(}\nabla^{s}_{(x,y)}(\overline{u},\overline{v})-B\overline{z}\big{)},$
(61d) $\displaystyle\overset{\boldsymbol{.}}{\overline{z}}$
$\displaystyle=A_{s}(B^{\top}\overline{\Sigma}-\mathbb{B}\overline{z}),\quad\overline{z}(0)=z_{0},$
Adjoint equation: (61e) $\displaystyle-\operatorname{Div}_{(x,y)}\Upsilon$
$\displaystyle=\big{(}\tfrac{\partial}{\partial
u}F_{1}(\overline{u},\overline{\Sigma}),0\big{)}-\operatorname{Div}_{(x,y)}\big{(}\mathbb{C}BA_{s}^{\prime}(B^{\top}\overline{\Sigma}-\mathbb{B}\overline{z})^{*}\varphi\big{)},$
(61f) $\displaystyle\Upsilon$
$\displaystyle=\mathbb{C}\big{(}\nabla^{s}_{(x,y)}(w,q)-\tfrac{\partial}{\partial\Sigma}F_{1}(\overline{u},\overline{\Sigma})\big{)},$
(61g) $\displaystyle\overset{\boldsymbol{.}}{\varphi}$
$\displaystyle=(B^{\top}\mathbb{C}B+\mathbb{B})A_{s}^{\prime}(B^{\top}\overline{\Sigma}-\mathbb{B}\overline{z})^{*}\varphi+B^{\top}\Upsilon,\;\;\varphi(T)=-B^{\top}\Upsilon_{T},$
(61h) $\displaystyle-\operatorname{Div}_{(x,y)}\Upsilon_{T}$
$\displaystyle=\big{(}\tfrac{\partial}{\partial
u}F_{2}(\overline{u}(T),\overline{\Sigma}(T)),0\big{)},$ (61i)
$\displaystyle\Upsilon_{T}$
$\displaystyle=\mathbb{C}\big{(}\nabla^{s}_{(x,y)}(w_{T},q_{T})-\tfrac{\partial}{\partial\Sigma}F_{2}(\overline{u}(T),\overline{\Sigma}(T))\big{)},$
Gradient equation: (61j) $\displaystyle\gamma\,\partial_{t}^{2}\overline{f}+w$
$\displaystyle=0,\qquad\overline{f}(0)=0,\quad\gamma\,\partial_{t}\overline{f}(T)+w_{T}=0,$
(61k) $\displaystyle\gamma\,\partial_{t}^{2}\overline{g}+w$
$\displaystyle=0,\qquad\overline{g}(0)=0,\quad\gamma\,\partial_{t}\overline{g}(T)+w_{T}=0.$

######  Remark 7.19.

A passage to the limit w.r.t. the regularization in order to obtain an
optimality system for the original optimal control problem involving the
maximal monotone operator $A$ would of course be of particular interest. The
results of [43] however indicate that the optimality conditions obtained in
this way are in general rather weak. In [43], an optimal control problem
governed by quasi-static elastoplasticity (without homogenization) is
considered, which provides substantial similarities to (57). This system could
also be treated by means of a reduction to the internal variable similar to
our procedure for (45). In [43] however, a time discretization followed by a
regularization was employed for the derivation of first-order optimality
conditions. The reason for the comparatively weak optimality conditions
obtained for the original (non-smooth) problem is the poor regularity of the
dual variables in the limit, in particular the adjoint state. We expect a
similar behavior in case of (61), when the regularization is driven to zero.
This however is subject to future research.

Next, we turn to second-order sufficient optimality conditions. Now,
$\Psi_{1}$, $\Psi_{2}$, and $A_{s}$ have to fulfill 6.1. For this purpose, we
require the following

######  Assumption 7.20.

The mappings $F_{1}$ and $F_{2}$ from (58) are twice Fréchet differentiable.
Moreover, $A_{s}$ satisfies 5.1 (ii) and 6.1(ii) and (iii) with
$\mathcal{W}=\mathcal{H}=Z^{2}$ (i.e., $s_{3}=2$), $\mathcal{Z}=Z^{s_{2}}$,
$s_{2}\geq 2$, and $\mathcal{Y}=Z^{s_{1}}$, $s_{1}\geq s_{2}$.

In contrast to 7.17, this assumption is very restrictive. If we assume that
$A_{s}$ arises as a Nemyzki operator from a nonlinear function
$A_{s}\colon\mathbb{V}\to\mathbb{V}$ (for simplicity denoted by the same
symbol), then the last condition in 6.1, i.e.,

(62) $\|A_{s}^{\prime\prime}(z)[z_{1},z_{2}]\|_{\mathcal{W}}\leq
C\,\|z_{1}\|_{Z^{s_{2}}}\|z_{2}\|_{Z^{s_{2}}}\quad\forall\,z\in
Z^{s_{1}},\;z_{1},z_{2}\in Z^{s_{2}}$

with $\mathcal{W}=Z^{2}$, may only hold—even in case
$A_{s}^{\prime\prime}(z)\in L^{\infty}(\Omega\times
Y;\mathcal{L}(\mathbb{V},\mathbb{V}))$—provided that $s_{2}\geq 2\,s_{3}=4$.
In order to have that $A_{s}$ is Fréchet differentiable from $Z^{s_{1}}$ to
$Z^{s_{2}}$, we therefore need $s_{1}>4$ such that 7.11 indeed becomes very
restrictive, see Remark 7.8 and Remark 7.33 below. Moreover, as described at
the beginning of this subsection, if one sets $r=2$, i.e., considers to
boundary loads in $H^{1}(0,T;L^{2}(\Gamma_{N};\mathbb{R}^{d}))$, then, in view
of (54), $s_{1}<3$ has to hold (at least in three spatial dimensions) so that
our second-order analysis cannot be applied in case of boundary controls (at
least if $r=2$ and $d=3$). Therefore, we restrict to volume forces, i.e.,
controls in $H^{1}(0,T;L^{2}(\Omega;\mathbb{R}^{d}))$ in what follows.

Then, given all these assumptions, we can apply Theorem 6.9 and Corollary 6.12
to (57) to obtain the following

######  Theorem 7.21 (Second-order sufficient conditions for optimal control
of homogenized plasticity).

Let Assumptions 7.1 and 7.20 hold and suppose that $\bar{s}\geq s_{1}$ so that
7.11 is fulfilled. Assume moreover that $\overline{f}\in
H^{1}(0,T;L^{2}(\Omega;\mathbb{R}^{d}))$ together with its associated state
$(\overline{u},\overline{v},\overline{z},\overline{\Sigma})$ and an adjoint
state $(w,q,\varphi,\Upsilon,w_{T},q_{T},\Upsilon_{T})$ satisfies the
optimality system (61b)–(61j) (without (61k) because boundary controls are
omitted) and, in addition, that there exists $\delta>0$ such that

$\displaystyle\tfrac{\partial^{2}}{\partial
u^{2}}F_{1}(\overline{u},\overline{\Sigma})\eta_{u}^{2}+\tfrac{\partial^{2}}{\partial\Sigma^{2}}F_{1}(\overline{u},\overline{\Sigma})\eta_{\Sigma}^{2}$
$\displaystyle\;+\tfrac{\partial^{2}}{\partial
u^{2}}F_{2}(\overline{u}(T),\overline{\Sigma}(T))\eta_{u}(T)^{2}+\tfrac{\partial^{2}}{\partial\Sigma^{2}}F_{2}(\overline{u}(T),\overline{\Sigma}(T))\eta_{\Sigma}(T)^{2}+\gamma\,\|\overset{\boldsymbol{.}}{h}\|_{L^{2}(0,T;L^{2}(\Omega;\mathbb{R}^{d}))}^{2}$
$\displaystyle\qquad\qquad+\int_{0}^{T}\int_{\Omega}\big{(}\varphi,A_{s}^{\prime\prime}(B^{\top}\overline{\Sigma}-\mathbb{B}\overline{z})(B^{\top}\eta_{\Sigma}-\mathbb{B}\eta_{z})^{2}\big{)}_{\mathbb{V}}\,dx\,dt\geq\delta\,\|h\|_{H^{1}(0,T;L^{2}(\Omega;\mathbb{R}^{d}))}^{2}$

holds for all $h\in H^{1}(0,T;L^{2}(\Omega;\mathbb{R}^{d}))$ with $h(0)=0$,
where $(\eta_{u},\eta_{v},\eta_{z},\eta_{\Sigma})\in H^{1}(0,T;U^{2}\times
V^{2}\times Z^{2}\times S^{2})$ is the solution of the linearized state system
associated with $h$, i.e.,

$\displaystyle-\operatorname{Div}_{(x,y)}\eta_{\Sigma}$ $\displaystyle=(h,0),$
$\displaystyle\eta_{\Sigma}$
$\displaystyle=\mathbb{C}\big{(}\nabla^{s}_{(x,y)}(\eta_{u},\eta_{v})-B\eta_{z}\big{)},$
$\displaystyle\overset{\boldsymbol{.}}{\eta_{z}}$
$\displaystyle=A_{s}^{\prime}(B^{\top}\overline{\Sigma}-\mathbb{B}\overline{z})(B^{\top}\eta_{\Sigma}-\mathbb{B}\eta_{z}),\quad\eta_{z}(0)=0.$

Then $\overline{f}$ is a strict local minimizer of (57) and satisfies the
quadratic growth condition (44).

######  Remark 7.22.

As indicated above, 7.20 in combination with 7.11 is very restrictive. One can
however weaken these assumptions, if the objective provides certain
properties. Let us for instance consider an objective of the form

(63) $J(u,f)$

with a twice Fréchet differentiable functional

$F_{3}\colon H^{1}(0,T;L^{2}(\Omega;\mathbb{R}^{d}))\to\mathbb{R}.$

In this case, it is sufficient to choose $s_{3}$ such that
$\mathfrak{u}\colon(z,\ell)\mapsto u$ maps $\mathcal{W}\times\mathcal{X}_{c}$
with $\mathcal{W}=Z^{s_{3}}$ to $W^{1,p}(\Omega;\mathbb{R}^{d})\hookrightarrow
L^{2}(\Omega;\mathbb{R}^{d})$, i.e., $p\geq 6/5$ for $d=3$. Moreover, as seen
above, in order to have that the Nemyzki operator $A_{s}$ fulfills (62), we
need $s_{1}>2s_{3}$. Thus, we are tempted to choose $s_{3}$ as small as
possible. However, the crucial, limiting condition is the regularity
assumption in 7.11 involving the conjugate exponent, i.e.,
$\bar{s}\geq\max\\{s_{1},s_{3}^{\prime}\\}$, and this leads to the following
equilibration of $s_{1}$ and $s_{3}$ in the case $d=3$: $s_{1}>3$, $s_{3}=3/2$
(such that $s_{3}^{\prime}=3$). Then, in view of (51), $\mathfrak{u}$ maps
$Z^{s_{3}}$ to $W^{1,s_{3}}(\Omega;\mathbb{R}^{d})$, which is continuously
embedded in $L^{2}(\Omega;\mathbb{R}^{d})$ for $d\leq 3$ as desired. In the
next subsection, we will present an example for a Nemyzki operator $A_{s}$
fulfilling all assumptions for $s_{1}$ arbitrarily close to $3$. But
nonetheless, $\bar{s}>3$ in 7.11 is still a rather restrictive assumption and
will not be satisfied in general (depending on the regularity of $\mathbb{C}$
and the boundary of $\Omega$). This shows that the second-order analysis for
problems of type (P) (resp. its regularized counterparts, to be precise) is in
general a delicate issue, mainly due to the quasi-linear structure of the
state equation.

###  7.4 A Concrete Flow Rule