remarkRemark hypothesisHypothesis claimClaim Optimal Control of Perfect
PlasticityC. Meyer and S. Walther

# Optimal Control of Perfect Plasticity
Part I: Stress Tracking††thanks: Submitted to the editors DATE. 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).

Christian Meyer TU Dortmund, Faculty of Mathematics, Vogelpothsweg 87, 44227
Dortmund, Germany (, , http://www.mathematik.tu-dortmund.de/lsx).
<EMAIL_ADDRESS><EMAIL_ADDRESS>Stephan
Walther22footnotemark: 2

###### Abstract

The paper is concerned with an optimal control problem governed by the rate-
independent system of quasi-static perfect elasto-plasticity. The objective is
to optimize the stress field by controlling the displacement at prescribed
parts of the boundary. The control thus enters the system in the Dirichlet
boundary conditions. Therefore, the safe load condition is automatically
fulfilled so that the system admits a solution, whose stress field is unique.
This gives rise to a well defined control-to-state operator, which is
continuous but not Gâteaux-differentiable. The control-to-state map is
therefore regularized, first by means of the Yosida regularization and then by
a second smoothing in order to obtain a smooth problem. The approximation of
global minimizers of the original non-smooth optimal control problem is shown
and optimality conditions for the regularized problem are established. A
numerical example illustrates the feasibility of the smoothing approach.

###### keywords:

Optimal control of variational inequalities, perfect plasticity, rate-
independent systems, Yosida regularization, first-order necessary optimality
conditions, Dirichlet control problems

49J20, 49K20, 74C05

##  1 Introduction

We consider the following optimal control problem governed by the equations of
_quasi-static perfect plasticity_ at small strain:

(P) $\left\\{\quad\begin{aligned} \min\text{ \ \
}&J(\sigma,\ell):=\Psi(\sigma,\ell)+\frac{\alpha}{2}\|\overset{\boldsymbol{.}}{\ell}\|_{L^{2}(\mathcal{X}_{c})}^{2},\\\
\text{s.t.}\text{ \ \ }&\begin{aligned} -\operatorname{div}\sigma&=0&&\text{
in }\Omega,\\\ \sigma&=\mathbb{C}(\nabla^{s}u-z)&&\text{ in }\Omega,\\\
\overset{\boldsymbol{.}}{z}&\in\partial
I_{\mathcal{K}(\Omega)}(\sigma)&&\text{ in }\Omega,\\\ u&=u_{D}&&\text{ on
}\Gamma_{D},\\\ \sigma\nu&=0&&\text{ on }\Gamma_{N},\\\
u(0)&=u_{0},\quad\sigma(0)=\sigma_{0}&&\text{ in }\Omega.\end{aligned}\\\
\text{and}\text{ \ \
}&u_{D}=\mathcal{G}\ell+\mathfrak{a},\quad\ell(0)=\ell(T)=0.\end{aligned}\right.$

Herein, $u:(0,T)\times\Omega\to\mathbb{R}^{n}$, $n=2,3$, is the displacement
field, while $\sigma,z:(0,T)\times\Omega\to\mathbb{R}^{n\times n}$ are stress
tensor and plastic strain. The boundary of $\Omega$ is split in two disjoint
parts $\Gamma_{D}$ and $\Gamma_{N}$ with outward unit normal $\nu$. Moreover,
$\mathbb{C}$ is the elasticity tensor and $\mathcal{K}(\Omega)$ denotes the
set of feasible stresses. The initial data $u_{0}$ and $\sigma_{0}$ are given
and fixed. The Dirichlet data $u_{D}$ arises from an artificial control
variable $\ell$ through a linear operator $\mathcal{G}$ in combination with a
given offset $\mathfrak{a}$. In principle, $\mathcal{G}$ could be an arbitrary
linear operator (fulfilling certain assumptions, see below), but in Section 6
it is chosen to be the solution operator of linear elasticity which is the
reason for calling $\ell$ _pseudo forces_. Finally, $\mathcal{X}_{c}$ is a
suitably chosen control space and $\alpha>0$ a fixed Tikhonov regularization
parameter. The objective $\Psi$ only contains the stress field and neither the
displacement nor the plastic strain. This is why the optimal control problem
(P) is termed _stress tracking problem_. A mathematically rigorous version of
(P) involving the functions space and a rigorous notion of solutions for the
state equation will be formulated in Section 4 below. The precise assumptions
on the data are given in Section 2. Regarding to a detailed description and
derivation of the plasticity model, we refer to [19].

Let us shortly comment on our choice of the control variable $\ell$. It is
well known that the system of perfect plasticity only admits a solution under
a certain additional assumption, also known as _safe load condition_ , see
e.g. [21, 5]. This condition roughly says that the applied loads must allow
for the existence of a stress field that fulfills the balance of momentum and
at the same time stays in the interior of the feasible set
$\mathcal{K}(\Omega)$. Thus, if one uses exterior loads as control variables,
the safe load condition arises as additional constraint in the optimal control
problem, but, at least up to our knowledge, it is an open question how to deal
with this additional constraint. We therefore choose the Dirichlet
displacement as control variables and set the exterior loads in the balance of
momentum to zero. Then the safe load condition is automatically fulfilled, but
we are faced with a Dirichlet boundary control problem. Problems of this kind
provide a particular challenge, since “standard” $L^{2}$-type spaces lead to
regularity issues, see e.g. [3, 15]. To overcome this challenge, we introduce
the Dirichlet data as the trace of an $H^{1}$-function in the domain $\Omega$,
as also proposed e.g. in [4, 7]. In our approach, the $H^{1}$-function arises
as a solution of another linear elliptic equation hidden behind the operator
$\mathcal{G}$. The inhomogeneity in this equation, i.e., the pseudo force
$\ell$, then serves as control variable. By the last constraints in (P), it is
forced to vanish at the beginning and in the end time. These additional
constraints are motivated by the application we have in mind: in practice, one
is often interested in reaching a desired shape and, at the same time,
optimizing the stress distribution at end time (e.g., keeping it as small as
possible). The desired shape is given in form of the offset $\mathfrak{a}$ and
the condition $\ell(T)=0$ ensures that it is indeed reached at end time. At
the beginning of the process, control variable is also assumed to vanish
($\ell(0)=0$), but in between it is allowed to alter the process in order to
optimize the stress distribution. More general control constraints are
possible as well and can easily be incorporated into our analysis, but, to
keep the discussion concise, we restrict ourselves to this particular setting.

The present paper is the first of two papers. In a companion paper [17], we
draw our attention to the displacement tracking problem. While the stress
tracking may be seen more important from an application point of view and
allows a comparatively comprehensive analysis, the displacement tracking is
mathematically more interesting and by far more challenging. This is due to
the lack of uniqueness and regularity of the displacement field in case of
perfect plasticity, see e.g. [21, 22].

Let us put our work into perspective. Optimal control of elasto-plastic
deformation has been considered from a mathematical perspective in various
articles, in particular concerning the static case, see e.g. [12, 14] and the
references therein. When it comes to the (physically much more reasonable)
quasi-static case however, the literature becomes rather scarce. The only
contributions in this field we are aware of are [23, 24, 25, 26, 16]. However,
all of these works deal with problems involving hardening, which essentially
simplifies the analysis. Quasi-static elasto-plasticity falls into the class
of rate-independent systems. The mathematical properties of such a system
strongly depend on the underlying energy functional. If the latter is
uniformly convex, then the system admits a unique and time-continuous
(differential) solution in the energy space. This however changes, if the
energy lacks convexity, and it is even not clear how to define a solution in
this case. For an overview over rate-independent processes and the various
notions of solutions, we refer to [18]. Hardening leads to a uniform convex
energy functional. In contrast to this, perfect plasticity may be seen as
limit case in this respect, since the energy is convex, but not uniformly
convex. Therefore, as already mentioned above, parts of the solution, namely
displacement and plastic strain, lack uniqueness and regularity, whereas the
stress is unique and provides the regularity expected for the uniformly convex
case. This behavior carries over to the optimal control problem. It turns out
that, as long as the stress tracking is considered, the optimal control
problem can be treated by similar techniques as in case with hardening and one
obtains comparable results concerning existence of optimal solution and their
approximation via regularization. For the case with hardening, this has been
elaborated in [24, 25, 26]. This however changes, if the displacement tracking
is considered, as we will see in the companion paper. To the best of our
knowledge, our two papers are the first contributions dealing with optimal
control of perfect plasticity, and it is remarkable that the stress tracking
allows for similar results as in the case with hardening, whereas the non-
uniform convexity of the energy takes its full effect when it comes to the
displacement tracking.

The paper is organized as follows: After introducing our notation and standing
assumptions in Section 2, we turn to the analysis of the state system in
Section 3. We establish the existence of a solution by means of the Yosida
regularization of the convex subdifferential $\partial
I_{\mathcal{K}(\Omega)}$, which is afterwards also used for the regularization
of the optimal control problem. The underlying analysis follows the lines of
[21], but we slightly extend the known results and therefore present the
arguments in detail. Section 4 is then devoted to the proof of existence of an
optimal solution and its approximation via Yosida regularization. The
regularized optimal control problems are still not smooth, since the control-
to-state map is not Gâteaux-differentiable in general. Therefore, we show for
the special case of the von Mises yield condition how to obtain a
differentiable problem by means of a second smoothing. This allows us to
derive optimality conditions involving an adjoint equation in Section 5. In
Section 6, we first specify the operator $\mathcal{G}$ and deduce the
particular form of the gradient of the objective functional reduced to the
control variable only. Based on that, we have implemented a gradient descent
method. The paper ends with an illustrative numerical example.

##  2 Notation and Standing Assumptions

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

#### Notation

Given two vector spaces $X$ and $Y$, we denote the space of linear and
continuous functions from $X$ into $Y$ by $\mathcal{L}(X,Y)$. If $X=Y$, we
simply write $\mathcal{L}(X)$. The dual space of $X$ is denoted by
$X^{*}=\mathcal{L}(X,\mathbb{R})$. If $H$ is a Hilbert space, we denote its
scalarproduct by $\left(\cdot,\cdot\right)_{H}$. For the whole paper, we fix
the final time $T>0$. For $t>0$ we denote the Bochner space of square-
integrable functions on the time interval $[0,t]$ by $L^{2}(0,t;X)$, the
Bochner-Sobolev space by $H^{1}(0,t;X)$ and the space of continuous functions
by $C([0,t];X)$ and abbreviate $L^{2}(X):=L^{2}(0,T;X)$,
$H^{1}(X):=H^{1}(0,T;X)$ and $C(X):=C([0,T];X)$. When $G\in\mathcal{L}(X;Y)$
is a linear and continuous operator, we can define an operator in
$\mathcal{L}(L^{2}(X);L^{2}(Y))$ by $G(u)(t):=G(u(t))$ for all $u\in L^{2}(X)$
and for almost all $t\in[0,T]$, we denote this operator also by $G$, that is,
$G\in\mathcal{L}(L^{2}(X);L^{2}(Y))$, and analog for Bochner-Sobolev spaces,
i.e., $G\in\mathcal{L}(H^{1}(X);H^{1}(Y))$. Given a coercive operator
$G\in\mathcal{L}(H)$ in a Hilbert space $H$, we denote its coercivity constant
by $\gamma_{G}$, i.e., $\left(Gh,h\right)_{H}\geq\gamma_{G}\|h\|_{H}^{2}$ for
all $h\in H$. With this operator we can define a new scalar product, which
induces an equivalent norm, by $H\times
H\ni(h_{1},h_{2})\mapsto\left(Gh_{1},h_{2}\right)_{H}\in\mathbb{R}$. We denote
the Hilbert space equipped with this scalar product by $H_{G}$, that is
$\left(h_{1},h_{2}\right)_{H_{G}}=\left(Gh_{1},h_{2}\right)_{H}$ for all
$h_{1},h_{2}\in H$. If $p\in[1,\infty]$, then we denote its conjugate exponent
by $p^{\prime}$, that is $\frac{1}{p}+\frac{1}{p^{\prime}}=1$. Finally, by
$\mathbb{R}^{n\times n}_{s}$, we denote the space of symmetric matrices and
$c,C>0$ are generic constants.

### Standing Assumptions

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

#### Domain

The domain $\Omega\subset\mathbb{R}^{n}$, $n\in\mathbb{N}$, $n\geq 2$, 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. [6].

#### Spaces

Throughout the paper, by $L^{p}(\Omega;M)$ we denote Lebesgue spaces with
values in $M$, where $p\in[1,\infty]$ and $M$ is a finite dimensional space.
To shorten notation, we abbreviate

$\mathbf{L}^{p}(\Omega):=L^{p}(\Omega;\mathbb{R}^{n})\quad\text{and}\quad\mathbb{L}^{p}(\Omega):=L^{p}(\Omega;\mathbb{R}^{n\times
n}_{s})$

and define $\mathbf{L}^{p}(\Lambda)$ and $\mathbb{L}^{p}(\Lambda)$ analogously
for a measurable subset $\Lambda$ of the boundary $\Gamma$. Given
$s\in\mathbb{N}$ and $p\in[1,\infty]$, the Sobolev spaces of vector- resp.
tensor-valued functions are denoted by

$\displaystyle\mathbf{W}^{s,p}(\Omega)$
$\displaystyle:=W^{s,p}(\Omega;\mathbb{R}^{n}),\quad$
$\displaystyle\mathbf{H}^{s}(\Omega)$
$\displaystyle:=\mathbf{W}^{s,2}(\Omega),$
$\displaystyle\mathbb{W}^{s,p}(\Omega)$
$\displaystyle:=W^{s,p}(\Omega;\mathbb{R}^{n\times n}_{s}),\quad$
$\displaystyle\mathbb{H}^{s}(\Omega)$
$\displaystyle:=\mathbb{W}^{s,2}(\Omega).$

Furthermore, set

(1) $\mathbf{W}_{D}^{1,p}(\Omega):=\overline{\\{\psi|_{\Omega}:\psi\in
C^{\infty}_{c}(\mathbb{R}^{n};\mathbb{R}^{n}),\;\operatorname{supp}(\psi)\cap\Gamma_{D}=\emptyset\\}}^{\mathbf{W}^{1,p}(\Omega)}$

and define $\mathbf{H}^{1}_{D}(\Omega)$ analogously. The dual of
$\mathbf{W}^{1,p^{\prime}}_{D}(\Omega)$ and $\mathbf{H}^{1}_{D}(\Omega)$ are
denoted by $\mathbf{W}^{-1,p}_{D}(\Omega)$ and $\mathbf{H}^{-1}_{D}(\Omega)$,
respectively.

Moreover, we assume that $\mathcal{X}$ is a real Banach space,
$\mathcal{X}_{c}$ is a Hilbert space and that $\mathcal{X}_{c}$ is compactly
embedded into $\mathcal{X}$. The elements in $\mathcal{X}$ and
$\mathcal{X}_{c}$ are called _pseudo forces_. Based on these spaces, the
control space is defined by

$\displaystyle H^{1}_{0}(\mathcal{X}_{c}):=\\{\ell\in
H^{1}(\mathcal{X}_{c}):\ell(0)=\ell(T)=0\\}.$

#### Coefficients

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

$\displaystyle\left(\mathbb{C}\sigma,\sigma\right)_{\mathbb{R}^{n\times
n}_{s}}\geq\underline{c}\ \|\sigma\|_{\mathbb{R}^{n\times n}_{s}}^{2}\text{ \
\ \ \ }\text{and}\text{ \ \ \ \
}\left(\mathbb{B}\sigma,\sigma\right)_{\mathbb{R}^{n\times
n}_{s}}\geq\underline{b}\ \|\sigma\|_{\mathbb{R}^{n\times n}_{s}}^{2}$

for all $\sigma\in\mathbb{R}^{n\times n}_{s}$. In addition we set
$\mathbb{A}:=\mathbb{C}^{-1}$ and note that
$\left(\mathbb{A}\sigma,\sigma\right)_{\mathbb{R}^{n\times
n}_{s}}\geq\frac{\underline{c}}{\|\mathbb{C}\|^{2}}\|\sigma\|_{\mathbb{R}^{n\times
n}_{s}}^{2}$ for all $\sigma\in\mathbb{R}^{n\times n}_{s}$ holds. Let us note
that $\mathbb{C}$ and $\mathbb{B}$ could also depend on the space, however, to
keep the discussion concise, we restrict ourselves to this setting.

#### Initial data

For the initial stress field $\sigma_{0}$, we assume that
$\sigma_{0}\in\mathbb{L}^{\overline{p}}(\Omega)$, where $\overline{p}>2$ is
specified in Lemma 3.17 below. The initial displacement will be given by the
initial Dirichlet data (at least in the regularized case), see Section 3.2
below.

#### Operators

Throughout the paper,
$\nabla^{s}:=\frac{1}{2}(\nabla+\nabla^{\top}):\mathbf{W}^{1,p}(\Omega)\to\mathbb{L}^{p}(\Omega)$
denotes the linearized strain. Its restriction to
$\mathbf{W}^{1,p}_{D}(\Omega)$ is denoted by the same symbol and, for the
adjoint of this restriction, we write
$-\operatorname{div}:=(\nabla^{s})^{*}:\mathbb{L}^{p^{\prime}}(\Omega)\to\mathbf{W}^{-1,p^{\prime}}_{D}(\Omega)$.

Let $\mathcal{K}\subset\mathbb{L}^{2}(\Omega)$ be a closed and convex set. We
denote the indicator function by

$I_{\mathcal{K}}:\mathbb{L}^{2}(\Omega)\rightarrow\\{0,\infty\\},\text{ \ \ \
\ \ \ }\tau\mapsto\begin{cases}0,&\tau\in\mathcal{K},\\\
\infty,&\tau\notin\mathcal{K}.\end{cases}$

By $\partial I_{\mathcal{K}}:\mathbb{L}^{2}(\Omega)\rightarrow
2^{\mathbb{L}^{2}(\Omega)}$ we denote the subdifferential of the indicator
function. For $\lambda>0$, the Yosida regularization is given by

$\displaystyle I_{\lambda}:\mathbb{L}^{2}(\Omega)\rightarrow\mathbb{R},\text{
\ \ \ \ \ \
}\tau\mapsto\frac{1}{2\lambda}\|\tau-\pi_{\mathcal{K}}(\tau)\|_{\mathbb{L}^{2}(\Omega)}^{2},$

where $\pi_{\mathcal{K}}$ is the projection onto $\mathcal{K}$ in
$\mathbb{L}^{2}(\Omega)$, and its Fréchet derivative is

$\displaystyle\partial
I_{\lambda}(\tau)=\frac{1}{\lambda}(\tau-\pi_{\mathcal{K}}(\tau)).$

When $\lambda=0$ we define $I_{\lambda}=I_{0}:=I_{\mathcal{K}}$. For a
sequence $\\{\lambda_{n}\\}_{n\in\mathbb{N}}\subset(0,\infty)$ we abbreviate
$I_{n}:=I_{\lambda_{n}}$.

#### Optimization Problem

By

$J:H^{1}(\mathbb{L}^{2}(\Omega))\times
H^{1}(\mathcal{X}_{c})\rightarrow\mathbb{R},\quad
J(\sigma,\ell):=\Psi(\sigma,\ell)+\frac{\alpha}{2}\|\overset{\boldsymbol{.}}{\ell}\|_{L^{2}(\mathcal{X}_{c})}$

we denote the objective function. We assume that
$\Psi:H^{1}(\mathbb{L}^{2}(\Omega))\times H^{1}(\mathcal{X}_{c})\to\mathbb{R}$
is weakly lower semicontinuous, continuous and bounded from below and that the
Tikhonov paramenter $\alpha$ is a positive constant. Finally, $\mathcal{G}$ is
a linear and continuous operator from $\mathcal{X}$ to
$\mathbf{H}^{1}(\Omega)$ and $\mathfrak{a}\in H^{1}(\mathbf{H}^{1}(\Omega))$
is given.

##  3 State Equation

We begin our investigation with the state equation. At first we give the
definition of a _reduced solution_ , that is, a notion of solutions involving
only the stress. Then we provide some results concerning this definition. In
Section 3.2 we prove the existence of such a solution by regularization.

The formal strong formulation of the state equation reads

(2a) $\displaystyle-\operatorname{div}\sigma$ $\displaystyle=0\text{ \ \ \ \ \
\ }\text{ \ \ \ \ \ \ }$ $\displaystyle\text{ in }\Omega,$ (2b)
$\displaystyle\sigma$ $\displaystyle=\mathbb{C}(\nabla^{s}u-z)\text{ \ \ \ \ \
\ }\text{ \ \ \ \ \ \ }$ $\displaystyle\text{ in }\Omega,$ (2c)
$\displaystyle\overset{\boldsymbol{.}}{z}$ $\displaystyle\in\partial
I_{\mathcal{K}(\Omega)}(\sigma)\text{ \ \ \ \ \ \ }\text{ \ \ \ \ \ \ }$
$\displaystyle\text{ in }\Omega,$ (2d) $\displaystyle u$
$\displaystyle=u_{D}\text{ \ \ \ \ \ \ }\text{ \ \ \ \ \ \ }$
$\displaystyle\text{ on }\Gamma_{D},$ (2e) $\displaystyle\sigma\nu$
$\displaystyle=0\text{ \ \ \ \ \ \ }\text{ \ \ \ \ \ \ }$ $\displaystyle\text{
on }\Gamma_{N},$ (2f) $\displaystyle u(0)$ $\displaystyle=u_{0},\text{ \ \ \ \
}\sigma(0)=\sigma_{0}\text{ \ \ \ \ \ \ }$ $\displaystyle\text{ in }\Omega.$

Herein, equation Eq. 2a is the _balance of momentum_ , Eq. 2b is the additive
split of the symmetric gradient of the displacement (the strain) into an
elastic part $e=\mathbb{A}\sigma$ and a plastic part $z$. The inclusion Eq. 2c
is the _flow rule_ , saying that the plastic part of the strain only changes
when the stress $\sigma$ has reached the _yield boundary_ , that is, the
boundary of $\mathcal{K}(\Omega)$.

###  3.1 Definitions and Auxiliary Results

The definition of a _reduced solution_ of Eq. 2 consists of two parts, the
_equilibrium condition_ and the _flow rule_ (resp. flow rule inequality). The
equilibrium condition is the weak formulation of Eq. 2a and Eq. 2e, while the
flow rule can be seen as a weak formulation of Eq. 2c.

######  Definition 3.1 (Equilibrium condition).

We define the set of stresses which fulfill the _equilibrium condition_ as

$\displaystyle\mathcal{E}(\Omega):=\operatorname{ker}(\operatorname{div})=\\{\tau\in\mathbb{L}^{2}(\Omega):\left(\tau,\nabla^{s}\varphi\right)_{\mathbb{L}^{2}(\Omega)}=0\
\forall\varphi\in\mathbf{H}^{1}_{D}(\Omega)\\}.$

######  Definition 3.2 (Admissible stresses).

Let $K\subset\mathbb{R}^{n\times n}_{s}$ be a closed and convex set. We define
the set of _admissible stresses_ as

$\displaystyle\mathcal{K}(\Omega):=\\{\tau\in\mathbb{L}^{2}(\Omega):\tau(x)\in
K\text{ f.a.a. }x\in\Omega\\}.$

For the rest of this section, we impose the following

######  Assumption 3.3 (Dirichlet data and initial condition).

  * (i)

We fix the Dirichlet displacement $u_{D}\in H^{1}(\mathbf{H}^{1}(\Omega))$ and
assume that the initial condition fulfills
$\sigma_{0}\in\mathcal{E}(\Omega)\cap\mathcal{K}(\Omega)$.

  * (ii)

The sequence $\\{u_{D,n}\\}_{n\in\mathbb{N}}\subset
H^{1}(\mathbf{H}^{1}(\Omega))$ fulfills $u_{D,n}\rightharpoonup u_{D}$ in
$H^{1}(\mathbf{H}^{1}(\Omega))$, $u_{D,n}\rightarrow u_{D}$ in
$L^{2}(\mathbf{H}^{1}(\Omega))$ and $u_{D,n}(T)\rightarrow u_{D}(T)$ in
$\mathbf{H}^{1}(\Omega)$.

We are now in a position to give the definition of a _reduced solution_ to Eq.
2.

######  Definition 3.4 (Reduced solution of the state equation).

A function $\sigma\in H^{1}(\mathbb{L}^{2}(\Omega))$ is called _reduced
solution_ of Eq. 2 (with respect to $u_{D}$), if, for almost all $t\in(0,T)$,
it holds

(3a) $\displaystyle\sigma(t)$
$\displaystyle\in\mathcal{E}(\Omega)\cap\mathcal{K}(\Omega),$ (3b)
$\displaystyle\left(\mathbb{A}\overset{\boldsymbol{.}}{\sigma}(t)-\nabla^{s}\overset{\boldsymbol{.}}{u_{D}}(t),\tau-\sigma(t)\right)_{\mathbb{L}^{2}(\Omega)}$
$\displaystyle\geq
0\quad\forall\tau\in\mathcal{E}(\Omega)\cap\mathcal{K}(\Omega),$ (3c)
$\displaystyle\sigma(0)$ $\displaystyle=\sigma_{0}.$

The inequality in Eq. 3b will be frequently termed as _flow rule inequality_.

Note that the definitions above correspond to [13, Plasticity Problem II] and
the definition given in [21, 1.4 Formulations. Résultats]. In order to
_formally_ derive the flow rule from Eq. 2c, one replaces $z$ by
$\nabla^{s}u-\mathbb{A}\sigma$ and use the definition of the subdifferential
to obtain the variational inequality

$\displaystyle\left(\mathbb{A}\overset{\boldsymbol{.}}{\sigma}(t)-\nabla^{s}\overset{\boldsymbol{.}}{u}(t),\tau-\sigma(t)\right)_{\mathbb{L}^{2}(\Omega)}\geq
0\text{ \ \ \ \ }\forall\tau\in\mathcal{K}(\Omega)\text{ and f.a.a.
}t\in[0,T].$

Restricting now the test functions to
$\mathcal{E}(\Omega)\cap\mathcal{K}(\Omega)$, one can exchange
$\nabla^{s}\overset{\boldsymbol{.}}{u}$ with
$\nabla^{s}\overset{\boldsymbol{.}}{u}_{D}$, which eliminates the unknown
displacement.

We also mention that in [5] the problem of perfect plasticity was analyzed in
the context of _quasistatic evolutions_ , also called _energetic solutions_ of
_rate-independent systems_. The definition given therein is equivalent to the
one in [21, 1.4 Formulations. Résultats] (cf. also [5, Theorem 6.1 and Remark
6.3]) and thus equivalent to ours. This definition was also used in [1].

Let us proceed with some results concerning the definition above. We start
with the uniqueness of the stress.

######  Lemma 3.5 (Uniqueness of the stress).

Assume that $\sigma_{1},\sigma_{2}\in H^{1}(\mathbb{L}^{2}(\Omega))$ are two
reduced solutions of Eq. 2. Then $\sigma_{1}=\sigma_{2}$.

######  Proof 3.6.

This can be easily seen as in [13, Theorem 1] by testing Eq. 3b with
$\sigma_{1}$ respectively $\sigma_{2}$, adding both equations and integrating
over time.

######  Lemma 3.7.

Let $\sigma\in H^{1}(\mathbb{L}^{2}(\Omega))$ be a reduced solution of Eq. 2.
Then

$\displaystyle\|\overset{\boldsymbol{.}}{\sigma}(t)\|_{\mathbb{L}^{2}(\Omega)_{\mathbb{A}}}^{2}=\left(\nabla^{s}\overset{\boldsymbol{.}}{u}_{D}(t),\overset{\boldsymbol{.}}{\sigma}(t)\right)_{\mathbb{L}^{2}(\Omega)}$

holds for almost all $t\in[0,T]$.

######  Proof 3.8.

There exists a set $N\subset[0,T]$ with measure zero, such that

$\displaystyle\lim_{h\rightarrow
0}\frac{\sigma(t+h)-\sigma(h)}{h}=\overset{\boldsymbol{.}}{\sigma}(t)\text{ \
\ \ \ }\text{and}\text{ \ \ \ \
}\left(\mathbb{A}\overset{\boldsymbol{.}}{\sigma}(t)-\nabla^{s}\overset{\boldsymbol{.}}{u}_{D}(t),\tau-\sigma(t))\right)_{\mathbb{L}^{2}(\Omega)}\geq
0$

for all $t\in[0,T]\setminus N$ and all
$\tau\in\mathcal{K}(\Omega)\cap\mathcal{E}(\Omega)$ (for the first property we
refer to [23, Theorem 3.1.40]). Testing this inequality with $\sigma(t\pm h)$
for a fixed $t\in(0,T)\setminus N$ and a sufficient small $h$, dividing by $h$
and letting $h\rightarrow 0$, we obtain the desired equation.

Since the conditions in $\mathcal{K}(\Omega)$ and $\mathcal{E}(\Omega)$ are
pointwise in time and independent of the time, one immediately deduces the
following

######  Lemma 3.9 (Time dependent flow rule inequality).

Let $\sigma\in H^{1}(\mathbb{L}^{2}(\Omega))$. Then

(4)
$\displaystyle\begin{split}&\left(\mathbb{A}\overset{\boldsymbol{.}}{\sigma}-\nabla^{s}\overset{\boldsymbol{.}}{u}_{D},\tau-\sigma\right)_{L^{2}(\mathbb{L}^{2}(\Omega))}\geq
0\\\ &\text{ \ \ \ \ \ \ }\forall\tau\in L^{2}(\mathbb{L}^{2}(\Omega))\text{
with }\tau(t)\in\mathcal{E}(\Omega)\cap\mathcal{K}(\Omega)\text{ f.a.a.
}t\in[0,T]\end{split}$

holds if and only if Eq. 3b holds.

We end this section with a continuity result for reduced solutions (supposed
they exists, which will be shown in the next section by means of
regularization). For this purpose, we need two auxiliary results.

######  Lemma 3.10.

Let $\\{a_{n}\\}_{n\in\mathbb{N}}\subset\mathbb{R}$ and
$\\{\tau_{n}\\}_{n\in\mathbb{N}}\subset H^{1}(\mathbb{L}^{2}(\Omega))$ such
that $\tau_{n}(0)=\sigma_{0}$ for all $n\in\mathbb{N}$ and $a_{n}\rightarrow
a$ in $\mathbb{R}$ and $\tau_{n}\rightharpoonup\tau$ in
$H^{1}(\mathbb{L}^{2}(\Omega))$. Moreover, assume that
$a_{n}\leq-\left(\mathbb{A}\overset{\boldsymbol{.}}{\tau}_{n},\tau_{n}\right)_{L^{2}(\mathbb{L}^{2}(\Omega))}$
for all $n\in\mathbb{N}$. Then
$a\leq-\left(\mathbb{A}\overset{\boldsymbol{.}}{\tau},\tau\right)_{L^{2}(\mathbb{L}^{2}(\Omega))}$
holds.

######  Proof 3.11.

Using the lower weakly semicontinuity of
$\|\cdot\|_{\mathbb{L}^{2}(\Omega)_{\mathbb{A}}}$ and the linear and
continuous embedding $H^{1}(\mathbb{L}^{2}(\Omega))\hookrightarrow
C(\mathbb{L}^{2}(\Omega))$, we deduce

$\displaystyle\liminf_{n\rightarrow\infty}\left(\mathbb{A}\overset{\boldsymbol{.}}{\tau}_{n},\tau_{n}\right)_{L^{2}(\mathbb{L}^{2}(\Omega))}$
$\displaystyle=\frac{1}{2}\liminf_{n\rightarrow\infty}\|\tau_{n}(T)\|_{\mathbb{L}^{2}(\Omega)_{\mathbb{A}}}^{2}-\frac{1}{2}\|\sigma_{0}\|_{\mathbb{L}^{2}(\Omega)_{\mathbb{A}}}^{2}$
$\displaystyle\geq\frac{1}{2}\|\tau(T)\|_{\mathbb{L}^{2}(\Omega)_{\mathbb{A}}}^{2}-\frac{1}{2}\|\sigma_{0}\|_{\mathbb{L}^{2}(\Omega)_{\mathbb{A}}}^{2}=\left(\mathbb{A}\overset{\boldsymbol{.}}{\tau},\tau\right)_{L^{2}(\mathbb{L}^{2}(\Omega))},$

which immediately gives the claim.

######  Lemma 3.12.

Let $H$ be a Hilbert space, $v,\tau\in H^{1}(H)$ and
$\\{v_{n}\\}_{n\in\mathbb{N}},\\{\tau_{n}\\}_{n\in\mathbb{N}}\subset H^{1}(H)$
such that $\tau_{n}\rightharpoonup\tau$ in $H^{1}(H)$,
$\tau_{n}(0)\rightarrow\tau(0)$, $v_{n}\rightarrow v$ in $L^{2}(H)$,
$v_{n}(0)\rightharpoonup v(0)$ and $v_{n}(T)\rightarrow v(T)$ in $H$. Then
$\left(\overset{\boldsymbol{.}}{v}_{n},\tau_{n}\right)_{L^{2}(H)}\rightarrow\left(\overset{\boldsymbol{.}}{v},\tau\right)_{L^{2}(H)}$
holds true.

######  Proof 3.13.

This follows immediately from integration by parts:

$\displaystyle\left(\overset{\boldsymbol{.}}{v}_{n},\tau_{n}\right)_{L^{2}(H)}$
$\displaystyle=-\left(v_{n},\overset{\boldsymbol{.}}{\tau}_{n}\right)_{L^{2}(H)}+\left(v_{n}(T),\tau_{n}(T)\right)_{H}-\left(v_{n}(0),\tau_{n}(0)\right)_{H}$
$\displaystyle\rightarrow-\left(v,\overset{\boldsymbol{.}}{\tau}\right)_{L^{2}(H)}+\left(v(T),\tau(T)\right)_{H}-\left(v(0),\tau(0)\right)_{H}=\left(\overset{\boldsymbol{.}}{v},\tau\right)_{L^{2}(H)},$

where we used the linear and continuous embedding $H^{1}(H)\hookrightarrow
C(H)$ to see that $\tau_{n}(t)\rightharpoonup\tau(t)$ in $H$ for
$t\in\\{0,T\\}$.

######  Proposition 3.14 (Continuity properties of reduced solutions).

Let us assume that $\sigma_{n}\in H^{1}(\mathbb{L}^{2}(\Omega))$ is the
reduced solution of Eq. 2 with respect to $u_{D,n}$ for every
$n\in\mathbb{N}$. Then there exists a reduced solution $\sigma\in
H^{1}(\mathbb{L}^{2}(\Omega))$ of Eq. 2 with respect to $u_{D}$ and
$\sigma_{n}\rightharpoonup\sigma$ in $H^{1}(\mathbb{L}^{2}(\Omega))$.
Moreover, if $u_{D,n}\rightarrow u_{D}$ in $H^{1}(\mathbf{H}^{1}(\Omega))$,
then $\sigma_{n}\rightarrow\sigma$ in $H^{1}(\mathbb{L}^{2}(\Omega))$.

######  Proof 3.15.

According to Lemma 3.7 (and $\sigma_{n}(0)=\sigma_{0}$), $\sigma_{n}$ is
bounded in $H^{1}(\mathbb{L}^{2}(\Omega))$, hence, there exists a subsequence,
again denoted by $\sigma_{n}$, and a weak limit $\sigma$ such that
$\sigma_{n}\rightharpoonup\sigma$ in $H^{1}(\mathbb{L}^{2}(\Omega))$. Thanks
to the linear and continuous embedding
$H^{1}(\mathbb{L}^{2}(\Omega))\hookrightarrow C(\mathbb{L}^{2}(\Omega))$, we
have $\sigma_{n}(t)\rightharpoonup\sigma(t)$ in $\mathbb{L}^{2}(\Omega)$ for
all $t\in[0,T]$, therefore, since $\mathcal{E}(\Omega)$ and
$\mathcal{K}(\Omega)$ are weakly closed,
$\sigma(t)\in\mathcal{E}(\Omega)\cap\mathcal{K}(\Omega)$ for all $t\in[0,T]$
and $\sigma(0)=\sigma_{0}$.

In order to prove that $\sigma$ fulfills the flow rule inequality, we use
Lemma 3.9. To this end we choose an arbitrary $\tau\in
L^{2}(\mathbb{L}^{2}(\Omega))$ with
$\tau(t)\in\mathcal{E}(\Omega)\cap\mathcal{K}(\Omega)$ for almost all
$t\in[0,T]$. Defining

$\displaystyle
a_{n}:=\left(\nabla^{s}\overset{\boldsymbol{.}}{u}_{D,n},\sigma_{n}\right)_{L^{2}(\mathbb{L}^{2}(\Omega))}+\left(\nabla^{s}\overset{\boldsymbol{.}}{u}_{D,n}-\mathbb{A}\overset{\boldsymbol{.}}{\sigma}_{n},\tau\right)_{L^{2}(\mathbb{L}^{2}(\Omega))}$

we see that
$a_{n}\leq-\left(\mathbb{A}\overset{\boldsymbol{.}}{\sigma}_{n},\sigma_{n}\right)_{L^{2}(\mathbb{L}^{2}(\Omega))}$
holds for all $n\in\mathbb{N}$. Thus, using Lemma 3.12 to see that
$\left(\nabla^{s}\overset{\boldsymbol{.}}{u}_{D,n},\sigma_{n}\right)_{L^{2}(\mathbb{L}^{2}(\Omega))}\rightarrow\left(\nabla^{s}\overset{\boldsymbol{.}}{u}_{D},\sigma\right)_{L^{2}(\mathbb{L}^{2}(\Omega))}$
(here we need in particular $u_{D,n}(T)\rightarrow u_{D}(T)$), Lemma 3.10
implies that Eq. 4 holds. Thanks to Lemma 3.5 we obtain the convergence
$\sigma_{n}\rightharpoonup\sigma$ in $H^{1}(\mathbb{L}^{2}(\Omega))$ for the
whole sequence by standard arguments.

If $u_{D,n}\rightarrow u_{D}$ in $H^{1}(\mathbf{H}^{1}(\Omega))$, then we
obtain
$\|\overset{\boldsymbol{.}}{\sigma}_{n}\|_{L^{2}(\mathbb{L}^{2}(\Omega)_{\mathbb{A}})}\rightarrow\|\overset{\boldsymbol{.}}{\sigma}\|_{L^{2}(\mathbb{L}^{2}(\Omega)_{\mathbb{A}})}$
from Lemma 3.7, which gives the strong convergence.

######  Remark 3.16.

It is also possible to consider perturbations in the initial condition, that
is, $\sigma_{n}$ in Proposition 3.14 is a reduced solution of Eq. 2 with
respect to the initial condition $\sigma_{0,n}$ (and the Dirichlet
displacement $u_{D,n}$), where
$\\{\sigma_{0,n}\\}_{n\in\mathbb{N}}\subset\mathcal{E}(\Omega)\cap\mathcal{K}(\Omega)$
is a sequence such that $\sigma_{0,n}\rightarrow\sigma_{0}$ in
$\mathbb{L}^{2}(\Omega)$. In this case Lemma 3.10 can be proven analogously
and the proof of Proposition 3.14 does not change.

###  3.2 Regularization and Existence

In this section, we establish the existence of a reduced solution by means of
regularization. We underline that similar results have already been obtained
in the literature, see e.g. [21, 1.4 Formulations. Résultats, Problème quasi
statique en plasticité parfaite]. However, since we slightly extend these
results (as explained in Remark 3.33 below), we present the full proofs for
the convenience of the reader.

We consider the following regularized version of the state equation Eq. 2:

(5a) $\displaystyle-\operatorname{div}\sigma_{n}$ $\displaystyle=0\text{ \ \ \
\ \ \ }\text{ \ \ \ \ \ \ }$ $\displaystyle\text{ in }\Omega,$ (5b)
$\displaystyle\sigma_{n}$
$\displaystyle=\mathbb{C}(\nabla^{s}u_{n}-z_{n})\text{ \ \ \ \ \ \ }\text{ \ \
\ \ \ \ }$ $\displaystyle\text{ in }\Omega,$ (5c)
$\displaystyle\overset{\boldsymbol{.}}{z}_{n}$ $\displaystyle\in\partial
I_{n}(\sigma_{n}-\varepsilon_{n}\mathbb{B}z_{n})\text{ \ \ \ \ \ \ }\text{ \ \
\ \ \ \ }$ $\displaystyle\text{ in }\Omega,$ (5d) $\displaystyle u_{n}$
$\displaystyle=u_{D,n}\text{ \ \ \ \ \ \ }\text{ \ \ \ \ \ \ }$
$\displaystyle\text{ on }\Gamma_{D},$ (5e) $\displaystyle\sigma_{n}\nu$
$\displaystyle=0\text{ \ \ \ \ \ \ }\text{ \ \ \ \ \ \ }$ $\displaystyle\text{
on }\Gamma_{N},$ (5f) $\displaystyle u_{n}(0)$ $\displaystyle=u_{D,n}(0)\text{
\ \ \ \ }\sigma_{n}(0)=\sigma_{0}\text{ \ \ \ \ \ \ }$ $\displaystyle\text{ in
}\Omega,$

where the sequence
$\\{(\varepsilon_{n},\lambda_{n})\\}_{n\in\mathbb{N}}\subset\mathbb{R}^{2}\setminus\\{0\\}$
fulfills $\varepsilon_{n},\lambda_{n}\geq 0$,
$(\varepsilon_{n},\lambda_{n})\rightarrow 0$ and

(6)
$\displaystyle(\sigma_{0}-\varepsilon_{n}\mathbb{B}(\nabla^{s}u_{D,n}(0)-\mathbb{A}\sigma_{0}))\in\mathcal{K}(\Omega),$

whenever $\lambda_{n}=0$. We emphasize that the following settings are
possible

$\displaystyle\lambda_{n}$ $\displaystyle>0,\text{ \ \ \ \
}\varepsilon_{n}=0\text{ \ \ \ \ \ \ }\text{(vanishing viscosity)},$
$\displaystyle\lambda_{n}$ $\displaystyle=0,\text{ \ \ \ \
}\varepsilon_{n}>0\text{ \ \ \ \ \ \ }\text{(vanishing hardening)},$
$\displaystyle\lambda_{n}$ $\displaystyle>0,\text{ \ \ \ \
}\varepsilon_{n}>0\text{ \ \ \ \ \ \ }\text{(mixed vanishing viscosity and
hardening)}.$

Let us recall that $I_{n}=I_{\lambda_{n}}$ and
$I_{n}=I_{0}=I_{\mathcal{K}(\Omega)}$ when $\lambda_{n}=0$. When
$\lambda_{n}>0$ the inclusion $a\in\partial I_{n}(b)$ is simply an equation,
$a=\partial I_{n}(b)$, for $a,b\in\mathbb{L}^{2}(\Omega)$. In Section 5 below,
we aim to apply the results of [16, section 5] to derive first-order
optimality conditions. For this purpose, because of differentiability reasons,
a norm gap is needed and therefore, we define solutions to Eq. 5 in
$L^{p}$-type spaces (although, in this section, we only need $p=2$). The
following result of [10] serves as a basis therefor:

######  Lemma 3.17.

There exists $\overline{p}>2$, such that for all
$p\in[\overline{p}^{\prime},\overline{p}]$,
$\ell\in\mathbf{W}^{-1,p}_{D}(\Omega)$ and $u_{D}\in\mathbf{W}^{1,p}(\Omega)$,
there exists a unique $u\in\mathbf{W}^{1,p}(\Omega)$ of the following _linear
elasticity equation_ :

$\left(\mathbb{C}\nabla^{s}u,\nabla^{s}\zeta\right)_{\mathbb{L}^{2}(\Omega)}=\langle\ell,\zeta\rangle\quad\forall\zeta\in\mathbf{W}^{1,p^{\prime}}_{D}(\Omega),\qquad
u-u_{D}\in\mathbf{W}^{1,p}_{D}(\Omega).$

We define the associated solution operator

(7)
$\mathcal{T}:\mathbf{W}^{-1,p}_{D}(\Omega)\times\mathbf{W}^{1,p}(\Omega)\rightarrow\mathbf{W}^{1,p}(\Omega),\text{
\ \ \ \ \ \ }(\ell,u_{D})\mapsto u,$

which we denote by the same symbol for different values of $p$. For every
$p\in[\overline{p}^{\prime},\overline{p}]$, it is linear and continuous.

######  Proof 3.18.

For the case $p\geq 2$, the claim is a direct consequence [10, Theorem 1.1 and
Remark 1.3]. The case $p<2$ then follows by duality.

Given the integrability exponent $\overline{p}$, our definition of a solution
to Eq. 9 reads as follows:

######  Definition 3.19.

Let $n\in\mathbb{N}$ and $p\in[2,\overline{p}]$, where $\overline{p}$ is from
Lemma 3.17, when $\lambda_{n}>0$ and $p=2$ when $\lambda_{n}=0$. Moreover,
assume that $u_{D,n}\in H^{1}(\mathbf{W}^{1,p}(\Omega))$. Then a tuple
$(u_{n},\sigma_{n},z_{n})\in
H^{1}(\mathbf{W}^{1,p}_{D}(\Omega)\times\mathbb{L}^{p}(\Omega)\times\mathbb{L}^{p}(\Omega))$
is called solution of Eq. 5, if, for almost all $t\in(0,T)$, it holds

(8a) $\displaystyle-\operatorname{div}\sigma_{n}(t)$ $\displaystyle=0$
$\displaystyle\text{in }\mathbf{W}^{-1,p}_{D}(\Omega),$ (8b)
$\displaystyle\sigma_{n}(t)$
$\displaystyle=\mathbb{C}(\nabla^{s}u_{n}(t)-z_{n}(t))$ $\displaystyle\text{in
}\mathbb{L}^{p}(\Omega),$ (8c)
$\displaystyle\overset{\boldsymbol{.}}{z}_{n}(t)$ $\displaystyle\in\partial
I_{n}(\sigma_{n}(t)-\varepsilon_{n}\mathbb{B}z_{n}(t))$ $\displaystyle\text{in
}\mathbb{L}^{p}(\Omega),$ (8d) $\displaystyle u_{n}(t)-u_{D,n}(t)$
$\displaystyle\in\mathbf{W}^{1,p}_{D}(\Omega),$ (8e)
$\displaystyle(u_{n},\sigma_{n})(0)$ $\displaystyle=(u_{D,n}(0),\sigma_{0})$
$\displaystyle\text{in }\mathbf{W}^{1,p}(\Omega)\times\mathbb{L}^{p}(\Omega).$

In order to analyze Eq. 5 we will apply the results from [16, section 3].

######  Definition 3.20.

Let $p$ be as in Definition 3.19. We define the linear and continuous operator

$Q_{n}:\mathbb{L}^{p}(\Omega)\rightarrow\mathbb{L}^{p}(\Omega),\text{ \ \ \ \
\ \
}z\mapsto(\mathbb{C}+\varepsilon_{n}\mathbb{B})z-\mathbb{C}\nabla^{s}\mathcal{T}(-\operatorname{div}\mathbb{C}z,0),$

where $\mathcal{T}$ is the solution operator from Eq. 7.

Let us note again that for this section only the case $p=2$ is needed.
However, the following holds also when $p\neq 2$, which we will use in Section
5 below.

######  Proposition 3.21 (Transformation into an EVI).

Let $p$ again be as in Definition 3.19 and $\mathcal{T}$ the solution operator
from Eq. 7. Then $(u_{n},\sigma_{n},z_{n})\in
H^{1}(\mathbf{W}^{1,p}(\Omega)\times\mathbb{L}^{p}(\Omega)\times\mathbb{L}^{p}(\Omega))$
is a solution of Eq. 8 if and only if $z_{n}$ is a solution of

(9) $\displaystyle\overset{\boldsymbol{.}}{z}_{n}\in\partial
I_{n}\big{(}\mathbb{C}\nabla^{s}\mathcal{T}(0,u_{D,n})-Q_{n}z_{n}\big{)},\text{
\ \ \ \ }z_{n}(0)=\nabla^{s}u_{D,n}(0)-\mathbb{A}\sigma_{0},$

and $u_{n}$ and $\sigma_{n}$ are defined through
$u_{n}=\mathcal{T}(-\operatorname{div}(\mathbb{C}z_{n}),u_{D,n})$ and
$\sigma_{n}=\mathbb{C}(\nabla^{s}u_{n}-z_{n})$. Moreover, if
$\varepsilon_{n}>0$, then $Q_{n}$ is coercive.

######  Proof 3.22.

In view of the definition of $Q_{n}$ and $\mathcal{T}$, we only have to verify
that the initial conditions are fulfilled. Clearly, if
$(u_{n},\sigma_{n},z_{n})$ is a solution of Eq. 8,
$z_{n}(0)=\nabla^{s}u_{D,n}(0)-\mathbb{A}\sigma_{0}$ follows immediately from
Eq. 8b. On the other hand, if $z_{n}$ is a solution of Eq. 9, then
$\sigma_{0}\in\mathcal{E}(\Omega)$ implies

$\displaystyle
u_{n}(0)=\mathcal{T}(-\operatorname{div}(\mathbb{C}z_{n}(0)),u_{D,n}(0))=\mathcal{T}(-\operatorname{div}(\mathbb{C}\nabla^{s}u_{D,n}(0)),u_{D,n}(0))$

hence, $u_{n}(0)=u_{D,n}(0)$ and
$\sigma_{n}(0)=\mathbb{C}(\nabla^{s}u_{D,n}(0)-z_{n}(0))=\sigma_{0}$.

Let us now investigate the coercivity of $Q_{n}$. Using the definition of
$\mathcal{T}$ one obtains

$\displaystyle\left(\mathbb{C}(z_{n}-\nabla^{s}\mathcal{T}(-\operatorname{div}(\mathbb{C}z_{n}),0)),z_{n}\right)_{\mathbb{L}^{2}(\Omega)}=\|z_{n}-\nabla^{s}\mathcal{T}(-\operatorname{div}(\mathbb{C}z_{n}),0))\|_{\mathbb{L}^{2}(\Omega)_{\mathbb{C}}}^{2},$

which immediately yields the coercivity of $Q_{n}$ when $\varepsilon_{n}>0$.

We are now in the position to deduce existence and uniqueness for Eq. 8. When
$\lambda_{n}=0$, Proposition 3.21 allows us to apply [16, Theorem 3.3] (where
we set $R=\mathcal{T}(0,\cdot)$; note that all requirements for [16, Theorem
3.3] can be easily checked by using Proposition 3.21 and the fact that
$Ru_{D,n}(0)-Q_{n}z_{n}(0)=\sigma_{0}-\varepsilon_{n}\mathbb{B}(\nabla^{s}u_{D,n}(0)-\mathbb{A}\sigma_{0})\in\mathcal{K}(\Omega)$,
see Eq. 6). In case of $\lambda_{n}>0$, existence and uniqueness follows
immediately by Banach’s contraction principle applied to the integral equation
associated with Eq. 9 (so that, in this case, (6) is not needed). Altogether
we obtain

######  Corollary 3.23.

For every $n\in\mathbb{N}$ there exists a unique solution
$(u_{n},\sigma_{n},z_{n})\in
H^{1}(\mathbf{H}^{1}(\Omega)\times\mathbb{L}^{2}(\Omega)\times\mathbb{L}^{2}(\Omega))$,
of Eq. 8. In the rest of this section we tacitly use this notation to denote
the solution of Eq. 8.

######  Remark 3.24.

We note that the existence of a solution for Eq. 8 is a classical result that
can also be found in the literature, see e.g. [8]. However, since we need the
transformation from Proposition 3.21 later anyway in Propositions 4.11 and 5.7
and the existence of a solution is an immediate consequence thereof, we
presented the above corollary for convenience of the reader.

######  Remark 3.25.

We moreover point out that, in case of $\lambda_{n}>0$, the global Lipschitz
continuity of $\partial I_{n}$ allows to establish the existence of a unique
solution to Eq. 8 for less regular data. Since this does however not hold for
the limit problem Eq. 3, we cannot make any use of this in the upcoming
analysis.

Having proved the existence of a solution to Eq. 5 we proceed with the
analysis for the limit case $n\rightarrow\infty$. For this purpose we need the
following result, which is an immediate consequence of [2, Lemme 3.3].

######  Lemma 3.26.

Let $\lambda\geq 0$ and $\tau\in H^{1}(\mathbb{L}^{2}(\Omega))$. Then

$\displaystyle\int_{a}^{b}\left(\xi(t),\overset{\boldsymbol{.}}{\tau}(t)\right)_{\mathbb{L}^{2}(\Omega)}dt=I_{\lambda}(\tau(b))-I_{\lambda}(\tau(a))$

holds for all $\xi:[0,T]\rightarrow\mathbb{L}^{2}(\Omega)$ such that
$\xi(t)\in\partial I_{\lambda}(\tau(t))$ for almost all $t\in[0,T]$ and all
$0\leq a\leq b\leq T$.

Now we will establish a priori estimates and then turn to the existence of a
solution to the state equation Eq. 2.

######  Lemma 3.27 (A priori estimates).

The inequalities

(10)
$\displaystyle\|\overset{\boldsymbol{.}}{\sigma}_{n}\|_{L^{2}(\mathbb{L}^{2}(\Omega)_{\mathbb{A}})}^{2}+\varepsilon_{n}\|\overset{\boldsymbol{.}}{z}_{n}\|_{L^{2}(\mathbb{L}^{2}(\Omega)_{\mathbb{B}})}^{2}\leq\left(\overset{\boldsymbol{.}}{\sigma}_{n},\nabla^{s}\overset{\boldsymbol{.}}{u}_{D,n}\right)_{L^{2}(\mathbb{L}^{2}(\Omega))}$

and

(11) $\displaystyle
I_{n}(\sigma_{n}(t)-\varepsilon_{n}\mathbb{B}z_{n}(t))\leq\|\overset{\boldsymbol{.}}{\sigma}_{n}\|_{L^{2}(\mathbb{L}^{2}(\Omega))}\|\nabla^{s}\overset{\boldsymbol{.}}{u}_{D,n}\|_{L^{2}(\mathbb{L}^{2}(\Omega))}$

hold for all $n\in\mathbb{N}$ and all $t\in[0,T]$.

######  Proof 3.28.

We use the fact that $\sigma_{n}(t)\in\mathcal{E}(\Omega)$ (thus
$\overset{\boldsymbol{.}}{\sigma}_{n}(t)\in\mathcal{E}(\Omega)$) to obtain

$\displaystyle\left(\mathbb{A}\overset{\boldsymbol{.}}{\sigma}_{n}(t),\overset{\boldsymbol{.}}{\sigma}_{n}(t)\right)_{\mathbb{L}^{2}(\Omega)}+\varepsilon_{n}\left(\overset{\boldsymbol{.}}{z}_{n}(t),\mathbb{B}\overset{\boldsymbol{.}}{z}_{n}(t)\right)_{\mathbb{L}^{2}(\Omega)}+\left(\overset{\boldsymbol{.}}{z}_{n}(t),\overset{\boldsymbol{.}}{\sigma}_{n}(t)-\varepsilon_{n}\mathbb{B}\overset{\boldsymbol{.}}{z}_{n}(t)\right)_{\mathbb{L}^{2}(\Omega)}$
$\displaystyle\quad=\left(\mathbb{A}\overset{\boldsymbol{.}}{\sigma}_{n}(t)+\overset{\boldsymbol{.}}{z}_{n}(t),\overset{\boldsymbol{.}}{\sigma}_{n}(t)\right)_{\mathbb{L}^{2}(\Omega)}=\left(\nabla^{s}\overset{\boldsymbol{.}}{u}_{n}(t),\overset{\boldsymbol{.}}{\sigma}_{n}(t)\right)_{\mathbb{L}^{2}(\Omega)}=\left(\nabla^{s}\overset{\boldsymbol{.}}{u}_{D,n}(t),\overset{\boldsymbol{.}}{\sigma}_{n}(t)\right)_{\mathbb{L}^{2}(\Omega)}$

for almost all $t\in[0,T]$. Integrating this equation with respect to time,
applying Lemma 3.26 and using
$(\sigma_{0}-\varepsilon_{n}\mathbb{B}z_{n}(0))\in\mathcal{K}(\Omega)$ yields

(12)
$\displaystyle\|\overset{\boldsymbol{.}}{\sigma}_{n}\|_{L^{2}(0,t;\mathbb{L}^{2}(\Omega)_{\mathbb{A}})}^{2}+\varepsilon_{n}\|\overset{\boldsymbol{.}}{z}_{n}\|_{L^{2}(0,t;\mathbb{L}^{2}(\Omega)_{\mathbb{B}})}^{2}+I_{n}(\sigma_{n}(t)-\varepsilon_{n}\mathbb{B}z_{n}(t))$
$\displaystyle=\left(\overset{\boldsymbol{.}}{\sigma}_{n},\nabla^{s}\overset{\boldsymbol{.}}{u}_{D,n}\right)_{L^{2}(0,t;\mathbb{L}^{2}(\Omega))}$

for all $t\in[0,T]$. The inequalities Eq. 10 and Eq. 11 now follow from this
equation (using $I_{n}\geq 0$ to get Eq. 10).

######  Lemma 3.29.

Let $w\in\mathbb{L}^{2}(\Omega)$ and
$\\{w_{n}\\}_{n\in\mathbb{N}}\subset\mathbb{L}^{2}(\Omega)$ such that
$w_{n}\rightharpoonup w$ in $\mathbb{L}^{2}(\Omega)$ and assume that the
sequence $I_{n}(w_{n})$ is bounded. Then $w\in\mathcal{K}(\Omega)$.

######  Proof 3.30.

Clearly, the mapping
$\mathbb{L}^{2}(\Omega)\ni\tau\mapsto\|\tau-\pi_{\mathcal{K}(\Omega)}(\tau)\|_{\mathbb{L}^{2}(\Omega)}^{2}\in\mathbb{R}$
is convex and continuous and thus weakly lower semicontinuous, hence,

$\displaystyle
0\leq\|w-\pi_{\mathcal{K}(\Omega)}(w)\|_{\mathbb{L}^{2}(\Omega)}^{2}\leq\liminf_{n\rightarrow\infty}\|w_{n}-\pi_{\mathcal{K}(\Omega)}(w_{n})\|_{\mathbb{L}^{2}(\Omega)}^{2}=\liminf_{n\rightarrow\infty}2\lambda_{n}I_{n}(w_{n})=0,$

which implies $w=\pi_{\mathcal{K}(\Omega)}(w)$.

######  Theorem 3.31 (Existence and approximation of a reduced solution).

Under 3.3, there exists a unique reduced solution $\sigma\in
H^{1}(\mathbb{L}^{2}(\Omega))$ of Eq. 2 and it holds
$\sigma_{n}\rightharpoonup\sigma$ in $H^{1}(\mathbb{L}^{2}(\Omega))$.
Furthermore, if $u_{D,n}\rightarrow u_{D}$ in $H^{1}(\mathbf{H}^{1}(\Omega))$,
then $\sigma_{n}\rightarrow\sigma$ in $H^{1}(\mathbb{L}^{2}(\Omega))$.

######  Proof 3.32.

The proof basically follows the lines of the one of Proposition 3.14.
According to Lemma 3.27, the sequences $\\{\sigma_{n}\\}_{n\in\mathbb{N}}$ and
$\\{\sqrt{\varepsilon_{n}}z_{n}\\}_{n\in\mathbb{N}}$ are bounded in
$H^{1}(\mathbb{L}^{2}(\Omega))$ (note that $\sigma_{n}(0)=\sigma_{0}$ and
$\sqrt{\varepsilon_{n}}z_{n}(0)=\sqrt{\varepsilon_{n}}(\nabla^{s}u_{D,n}(0)-\mathbb{A}\sigma_{0})\rightarrow
0$). Therefore there exists a subsequence, again denoted by $\sigma_{n}$, and
a weak limit $\sigma\in H^{1}(\mathbb{L}^{2}(\Omega))$ such that
$\sigma_{n}\rightharpoonup\sigma$ and
$\sigma_{n}+\varepsilon_{n}\mathbb{B}z_{n}\rightharpoonup\sigma$ in
$H^{1}(\mathbb{L}^{2}(\Omega))$. Due to the linear and continuous embedding
$H^{1}(\mathbb{L}^{2}(\Omega))\hookrightarrow C(\mathbb{L}^{2}(\Omega))$ we
arrive at $\sigma_{n}(t)\rightharpoonup\sigma(t)$ and
$\sigma_{n}(t)+\varepsilon_{n}\mathbb{B}z_{n}(t)\rightharpoonup\sigma(t)$ in
$\mathbb{L}^{2}(\Omega)$ for all $t\in[0,T]$. Hence, since
$\mathcal{E}(\Omega)$ is weakly closed and
$\sigma_{n}(t)\in\mathcal{E}(\Omega)$ for all $n\in\mathbb{N}$, we obtain
$\sigma(t)\in\mathcal{E}(\Omega)$ for all $t\in[0,T]$. Moreover, according to
Lemma 3.27, $I_{n}(\sigma_{n}(t)-\varepsilon_{n}\mathbb{B}z_{n}(t))$ is
bounded and thus, Lemma 3.29 gives $\sigma(t)\in\mathcal{K}(\Omega)$ for all
$t\in[0,T]$.

As in the proof of Proposition 3.14, we again employ Lemma 3.12 to verify the
flow rule in the form Eq. 4. To this end we choose an arbitrary $\tau\in
L^{2}(\mathbb{L}^{2}(\Omega))$ with
$\tau(t)\in\mathcal{E}(\Omega)\cap\mathcal{K}(\Omega)$ for almost all
$t\in[0,T]$ and obtain

$\displaystyle 0$
$\displaystyle=\int_{0}^{T}I_{n}(\tau(t))dt\stackrel{{\scriptstyle\lx@cref{creftype~refnum}{eq:regsubdiff}}}{{\geq}}\int_{0}^{T}I_{n}(\sigma_{n}(t)-\varepsilon_{n}\mathbb{B}z_{n}(t))dt+\left(\overset{\boldsymbol{.}}{z}_{n},\tau-\sigma_{n}+\varepsilon_{n}\mathbb{B}z_{n})\right)_{L^{2}(\mathbb{L}^{2}(\Omega))}$
$\displaystyle\stackrel{{\scriptstyle\lx@cref{creftype~refnum}{eq:regstress}}}{{\geq}}\frac{\varepsilon_{n}}{2}\left(z_{n}(T),\mathbb{B}z_{n}(T)\right)_{\mathbb{L}^{2}(\Omega)}-\frac{\varepsilon_{n}}{2}\left(z_{n}(0),\mathbb{B}z_{n}(0)\right)_{\mathbb{L}^{2}(\Omega)}+\left(\nabla^{s}\overset{\boldsymbol{.}}{u}_{n}-\mathbb{A}\overset{\boldsymbol{.}}{\sigma}_{n},\tau-\sigma_{n}\right)_{L^{2}(\mathbb{L}^{2}(\Omega))}$
$\displaystyle\geq-\frac{\varepsilon_{n}}{2}\left(z_{n}(0),\mathbb{B}z_{n}(0)\right)_{\mathbb{L}^{2}(\Omega)}+\left(\nabla^{s}\overset{\boldsymbol{.}}{u}_{D,n}-\mathbb{A}\overset{\boldsymbol{.}}{\sigma}_{n},\tau-\sigma_{n}\right)_{L^{2}(\mathbb{L}^{2}(\Omega))},$

where we have used the monotonicity of the subdifferential, the positivity of
$I_{n}$, the coercivity of $\mathbb{B}$, the fact that
$\tau,\sigma_{n}\in\mathcal{E}(\Omega)$, and
$\overset{\boldsymbol{.}}{u}_{n}-\overset{\boldsymbol{.}}{u}_{D,n}\in
L^{2}(\mathbf{H}^{1}_{D}(\Omega))$. This time we set

$\displaystyle
a_{n}:=-\frac{\varepsilon_{n}}{2}\left(z_{n}(0),\mathbb{B}z_{n}(0)\right)_{\mathbb{L}^{2}(\Omega)}+\left(\nabla^{s}\overset{\boldsymbol{.}}{u}_{D,n},\sigma_{n}\right)_{L^{2}(\mathbb{L}^{2}(\Omega))}+\left(\nabla^{s}\overset{\boldsymbol{.}}{u}_{D,n}-\mathbb{A}\overset{\boldsymbol{.}}{\sigma}_{n},\tau\right)_{L^{2}(\mathbb{L}^{2}(\Omega))}$

and observe that, by means of $\sqrt{\varepsilon}z_{n}(0)\rightarrow 0$ and
Lemma 3.12,

$\displaystyle-\left(\mathbb{A}\overset{\boldsymbol{.}}{\sigma}_{n},\sigma_{n}\right)_{L^{2}(\mathbb{L}^{2}(\Omega))}\geq
a_{n}\rightarrow
a:=\left(\nabla^{s}\overset{\boldsymbol{.}}{u}_{D},\sigma\right)_{L^{2}(\mathbb{L}^{2}(\Omega))}+\left(\nabla^{s}\overset{\boldsymbol{.}}{u}_{D}-\mathbb{A}\overset{\boldsymbol{.}}{\sigma},\tau\right)_{L^{2}(\mathbb{L}^{2}(\Omega))}$

as $n\rightarrow\infty$. Hence, Lemma 3.10 implies that the weak limit
$\sigma$ indeed satisfies Eq. 4. Since the reduced solution is unique by Lemma
3.5, a standard argument gives the weak convergence of the whole sequence.

If $u_{D,n}\rightarrow u_{D}$ in $H^{1}(\mathbf{H}^{1}(\Omega))$, then Lemma
Lemma 3.27 and Lemma 3.7 imply

$\displaystyle\|\overset{\boldsymbol{.}}{\sigma}\|_{L^{2}(\mathbb{L}^{2}(\Omega)_{\mathbb{A}})}^{2}$
$\displaystyle\leq\liminf_{n\rightarrow\infty}\|\overset{\boldsymbol{.}}{\sigma}_{n}\|_{L^{2}(\mathbb{L}^{2}(\Omega)_{\mathbb{A}})}^{2}$
$\displaystyle\leq\limsup_{n\rightarrow\infty}\|\overset{\boldsymbol{.}}{\sigma}_{n}\|_{L^{2}(\mathbb{L}^{2}(\Omega)_{\mathbb{A}})}^{2}$
$\displaystyle\leq\limsup_{n\rightarrow\infty}\left(\overset{\boldsymbol{.}}{\sigma}_{n},\nabla^{s}\overset{\boldsymbol{.}}{u}_{D,n}\right)_{L^{2}(\mathbb{L}^{2}(\Omega))}=\left(\overset{\boldsymbol{.}}{\sigma},\nabla^{s}\overset{\boldsymbol{.}}{u}_{D}\right)_{L^{2}(\mathbb{L}^{2}(\Omega))}=\|\overset{\boldsymbol{.}}{\sigma}\|_{L^{2}(\mathbb{L}^{2}(\Omega)_{\mathbb{A}})}^{2},$

which yields the desired strong convergence.

######  Remark 3.33.

In contrast to Theorem 3.31, the results in [21] only cover the case of
constant Dirichlet data $u_{D}$ and $\lambda_{n}>0$, $\varepsilon_{n}=0$
(i.e., without hardening) and only prove weak convergence of the stresses for
this case.

######  Remark 3.34.

In case of the strong convergence $u_{D,n}\rightarrow u_{D}$ in
$H^{1}(\mathbf{H}^{1}(\Omega))$, one additionally obtains
$\sqrt{\varepsilon_{n}}z_{n}\rightarrow 0$ in $H^{1}(\mathbb{L}^{2}(\Omega))$,
$I_{n}(\sigma_{n}-\varepsilon_{n}\mathbb{B}z_{n})\rightarrow 0$ in
$L^{2}(\Omega)$ and
$I_{n}(\sigma_{n}(t)-\varepsilon_{n}\mathbb{B}z_{n}(t))\rightarrow 0$ for all
$t\in[0,T]$. This follows from Eq. 12 by similar arguments as used at the end
of the proof of Theorem 3.31.

##  4 Existence and Approximation of Optimal Controls

We now turn to the optimization problem Eq. P. Let us first give a rigorous
definition of our optimal control problem based on our previous findings.
Relying on Theorem 3.31, the rigorous counterpart of Eq. P reads as follows:

(P) $\left\\{\quad\begin{aligned} \min\text{ \ \
}&J(\sigma,\ell):=\Psi(\sigma,\ell)+\frac{\alpha}{2}\|\overset{\boldsymbol{.}}{\ell}\|_{L^{2}(\mathcal{X}_{c})},\\\
\text{s.t.}\text{ \ \ }&\ell\in H^{1}_{0}(\mathcal{X}_{c}),\quad\sigma\in
H^{1}(\mathbb{L}^{2}(\Omega))\\\ \text{and}\text{ \ \ }&\sigma\text{ is a
reduced solution of \lx@cref{creftype~refnum}{eq:stateEquation} w.r.t.\
}u_{D}=\mathcal{G}\ell+\mathfrak{a}.\end{aligned}\right.$

For the rest of the paper, we impose the following assumption on the data in
Eq. P:

######  Assumption 4.1 (Initial condition and pseudo force).

We assume that the initial condition fulfills
$\sigma_{0}\in\mathcal{E}(\Omega)\cap\mathcal{K}(\Omega)$ and fix a
“Dirichlet-offset” $\mathfrak{a}\in H^{1}(\mathbf{H}^{1}(\Omega))$.

###  4.1 Existence of Optimal Controls

According to Theorem 3.31 there exists for every $u_{D}\in
H^{1}(\mathbf{H}^{1}(\Omega))$ a unique reduced solution $\sigma\in
H^{1}(\mathbb{L}^{2}(\Omega))$ of Eq. 2 (we can simply choose
$\varepsilon_{n}=0$ and $u_{D,n}=u_{D}$ for every $n\in\mathbb{N}$). This
leads to the following

######  Definition 4.2 (Solution operator for the state equation).

For a given $\ell\in H^{1}_{0}(\mathcal{X}_{c})$ there exists a unique reduced
solution $\sigma$ of Eq. 2 with respect to
$u_{D}=\mathcal{G}\ell+\mathfrak{a}$. We denote the associated solution
operator by

$\displaystyle\mathcal{S}:H^{1}_{0}(\mathcal{X}_{c})\rightarrow
H^{1}(\mathbb{L}^{2}(\Omega)),\text{ \ \ \ \ \ \ }\ell\mapsto\sigma.$

######  Corollary 4.3 (Continuity properties of the solution operator).

The solution operator $\mathcal{S}:H^{1}_{0}(\mathcal{X}_{c})\rightarrow
H^{1}(\mathbb{L}^{2}(\Omega))$ is weakly and strongly continuous, that is,

  * (i)

$\ell_{n}\rightharpoonup\ell$ in $H^{1}_{0}(\mathcal{X}_{c})$
$\Longrightarrow$ $\mathcal{S}(\ell_{n})\rightharpoonup\mathcal{S}(\ell)$ in
$H^{1}(\mathbb{L}^{2}(\Omega))$ and

  * (ii)

$\ell_{n}\rightarrow\ell$ in $H^{1}_{0}(\mathcal{X}_{c})$ $\Longrightarrow$
$\mathcal{S}(\ell_{n})\rightarrow\mathcal{S}(\ell)$ in
$H^{1}(\mathbb{L}^{2}(\Omega))$.

######  Proof 4.4.

Let us assume that $\ell_{n}\rightharpoonup\ell$ in
$H^{1}_{0}(\mathcal{X}_{c})\subset H^{1}(\mathcal{X}_{c})$. Since
$\mathcal{X}_{c}$ is compactly embedded into $\mathcal{X}$,
$H^{1}(\mathcal{X}_{c})$ is compactly embedded into $C(\mathcal{X})$ and
hence, $\mathcal{G}\ell_{n}\rightarrow\mathcal{G}\ell$ in
$L^{2}(\mathbf{H}^{1}(\Omega))$ and
$(\mathcal{G}\ell_{n})(t)\rightarrow(\mathcal{G}\ell)(t)$ in
$\mathbf{H}^{1}(\Omega)$ for all $t\in[0,T]$, in particular for $t=T$. We
conclude that the sequence $u_{D,n}:=\mathcal{G}\ell_{n}+\mathfrak{a}$
fulfills (ii) in 3.3 with $u_{D}:=\mathcal{G}\ell+\mathfrak{a}$. The claim
then follows from Proposition 3.14.

Given the (weak) continuity properties of $\SS$, one readily deduces the
following

######  Theorem 4.5 (Existence of optimal solutions).

There exists at least one global solution of Eq. P.

######  Proof 4.6.

The assertion follows from the standard direct method of the calculus of
variations using the coercivity of the Tikhonov term in the objective with
respect to $\ell$, the weakly lower semicontinuity of $J$, and the weak
continuity of $\SS$. Note that $H^{1}_{0}(\mathcal{X}_{c})$ is weakly closed
due to the continuous embedding $H^{1}(\mathcal{X}_{c})\hookrightarrow
C(\mathcal{X}_{c})$.

######  Remark 4.7.

Corollary 4.3 and Theorem 4.5 also hold when $H^{1}_{0}(\mathcal{X}_{c})$ is
replaced by any other weakly closed subset of $H^{1}(\mathcal{X}_{c})$. The
set $H^{1}_{0}(\mathcal{X}_{c})$ is motivated by practical applications (as
explained in the introduction) and will be used in our numerical experiments
in Section 6.

###  4.2 Convergence of Global Minimizers

Let us proceed with the approximation of global solutions to Eq. 2.
Additionally to 4.1 we impose the following assumption for the rest of this
section.

######  Assumption 4.8 (Regularization parameters).

Let
$\\{(\varepsilon_{n},\lambda_{n})\\}_{n\in\mathbb{N}}\subset\mathbb{R}^{2}\setminus\\{0\\}$
be a sequence such that $\varepsilon_{n},\lambda_{n}\geq 0$,
$(\varepsilon_{n},\lambda_{n})\rightarrow 0$ and
$(\sigma_{0}+\varepsilon_{n}\mathbb{B}(\mathbb{A}\sigma_{0}-\mathbb{C}\nabla^{s}\mathcal{T}(0,\mathfrak{a})))\in\mathcal{K}(\Omega)$,
whenever $\lambda_{n}=0$.

######  Definition 4.9 (Solution operator for the regularized state equation).

According to Corollary 3.23, for every $(\varepsilon_{n},\lambda_{n})$, there
exists a unique solution $(u_{n},\sigma_{n},z_{n})\in
H^{1}(\mathbf{H}^{1}(\Omega)\times\mathbb{L}^{2}(\Omega)\times\mathbb{L}^{2}(\Omega))$
of Eq. 5 with respect to $u_{D}=\mathcal{G}\ell+\mathfrak{a}\in
H^{1}(\mathbf{H}^{1}(\Omega))$ for a given $\ell\in
H^{1}_{0}(\mathcal{X}_{c})$. We may thus define the solution operator

$\displaystyle\mathcal{S}_{n}:H^{1}_{0}(\mathcal{X}_{c})\rightarrow
H^{1}(\mathbb{L}^{2}(\Omega)),\text{ \ \ \ \ \ \ }\ell\mapsto\sigma_{n}.$

With the regularized solution operator at hand, we define the following
regularized version of Eq. P for a given tuple $(\varepsilon_{n},\lambda_{n})$
of regularization parameters:

(Pn) $\min_{\ell\in H^{1}_{0}(\mathcal{X}_{c})}\;J(\SS_{n}(\ell),\ell).$

######  Definition 4.10.

Given the operator
$\mathcal{G}\in\mathcal{L}(\mathcal{X},\mathbf{H}^{1}(\Omega))$ and the
solution mapping $\mathcal{T}$ from Eq. 7, we define the linear and continuous
operator

$\displaystyle
R\in\mathcal{L}(\mathcal{X};\mathbb{L}^{2}(\Omega)),\quad\ell\mapsto\mathbb{C}\nabla^{s}\mathcal{T}(0,\mathcal{G}\ell).$

We denote the restriction of this operator to $\mathcal{X}_{c}$ with the same
symbol. Moreover, we set
$\mathfrak{A}:=\mathbb{C}\nabla^{s}\mathcal{T}(0,\mathfrak{a})\in
H^{1}(\mathbb{L}^{2}(\Omega))$.

######  Proposition 4.11 (Existence of optimal solutions of the regularized
problems).

For every $n\in\mathbb{N}$, there exists a global solution of Eq. Pn.

######  Proof 4.12.

Using Proposition 3.21 and the definition of $R$ one obtains that
$(u_{n},\sigma_{n},z_{n})\in
H^{1}(\mathbf{H}^{1}(\Omega)\times\mathbb{L}^{2}(\Omega)\times\mathbb{L}^{2}(\Omega))$
is a solution of Eq. 5 with respect to $u_{D}=\mathcal{G}\ell+\mathfrak{a}$
with $\ell\in H^{1}_{0}(\mathcal{X}_{c})$, if and only if $z_{n}$ is a
solution of

(13) $\displaystyle\overset{\boldsymbol{.}}{z}_{n}\in\partial
I_{n}(R\ell+\mathfrak{A}-Q_{n}z_{n}),\text{ \ \ \ \
}z_{n}(0)=\nabla^{s}\mathfrak{a}(0)-\mathbb{A}\sigma_{0}$

(where $Q_{n}$ is as defined in Definition 3.20) and $u_{n}$ and $\sigma_{n}$
are determined through $z_{n}$ via

(14)
$u_{n}=\mathcal{T}(-\operatorname{div}(\mathbb{C}z_{n}),\mathcal{G}\ell+\mathfrak{a})\quad\text{and}\quad\sigma_{n}=\mathbb{C}(\nabla^{s}u_{n}-z_{n}).$

Note that $\ell\in H^{1}_{0}(\mathcal{X}_{c})$ implies $\ell(0)=0$, which
leads to the initial condition in Eq. 13, and that
$R\ell(0)+\mathfrak{A}(0)-Q_{n}z_{n}(0)=\sigma_{0}+\varepsilon_{n}\mathbb{B}(\mathbb{A}\sigma_{0}-\mathfrak{A}(0))\in\mathcal{K}(\Omega)$,
according to 4.8. We next show the weak continuity of the solution operator of
Eq. 13, denoted by $\SS^{(z)}_{n}$, as a mapping from $H^{1}(\mathcal{X}_{c})$
to $H^{1}(\mathbb{L}^{2}(\Omega))$. In case of $\lambda_{n}=0$ (and thus
$\varepsilon_{n}>0$), Eq. 13 corresponds to an evolution variational
inequality with a maximal monotone operator as for instance discussed in [16,
section 3]. The continuity properties thereof are stated in [16, Theorem
3.10]. Since in particular $Q_{n}$ is coercive when $\varepsilon_{n}>0$ as
shown in Proposition 3.21, all assumptions of this theorem are fulfilled
except for the offset $\mathfrak{A}$, which is zero in [16]. It is however
easily seen that this does not affect the underlying analysis such that this
continuity result together with the compact embedding of
$H^{1}(\mathcal{X}_{c})$ in $L^{1}(\mathcal{X})$ yields the desired weak
continuity of $\SS^{(z)}_{n}$.

If $\lambda_{n}>0$, then $\partial I_{n}$ is a Lipschitz continuous mapping
from $\mathbb{L}^{2}(\Omega)$ to $\mathbb{L}^{2}(\Omega)$, which, together
with Gronwall’s inequality, gives the Lipschitz continuity of the solution
mapping of Eq. 13 from $L^{2}(\mathcal{X})$ to
$H^{1}(\mathbb{L}^{2}(\Omega))$, cf. [16, proof of Proposition 4.4]. Together
with the compactness of $H^{1}(\mathcal{X}_{c})\hookrightarrow
L^{2}(\mathcal{X})$, this yields the weak continuity of $\SS^{(z)}_{n}$ in
this case.

Since all operators in Eq. 14 are linear (resp. affine) and continuous in
their respective spaces, the weak continuity of $\SS_{n}^{(z)}$ carries over
to solution mapping $\SS_{n}$ from Definition 4.9. Now the assertion can be
proven analogously to the proof of Theorem 4.5 by means of the standard direct
method of the calculus of variations.

######  Proposition 4.13 (Approximation properties of the solution operators).

The following two properties hold:

  * (i)

$\ell_{n}\rightharpoonup\ell$ in $H^{1}_{0}(\mathcal{X}_{c})$
$\Longrightarrow$ $\mathcal{S}_{n}(\ell_{n})\rightharpoonup\mathcal{S}(\ell)$
in $H^{1}(\mathbb{L}^{2}(\Omega))$,

  * (ii)

$\ell_{n}\rightarrow\ell$ in $H^{1}_{0}(\mathcal{X}_{c})$ $\Longrightarrow$
$\mathcal{S}_{n}(\ell_{n})\rightarrow\mathcal{S}(\ell)$ in
$H^{1}(\mathbb{L}^{2}(\Omega))$.

######  Proof 4.14.

The proof is the same as the proof of Corollary 4.3, except that we employ
Theorem 3.31 instead of Proposition 3.14.

######  Theorem 4.15 (Approximation of global minimizers).

Let $\\{\ell_{n}\\}_{n\in\mathbb{N}}$ be a sequence of global minimizers of
Eq. Pn. Then every weak accumulation point of
$\\{\ell_{n}\\}_{n\in\mathbb{N}}$ is a strong accumulation point and a global
minimizer of Eq. P. Moreover, there exists an accumulation point.

######  Proof 4.16.

The proof follows standard arguments using the continuity properties in
Proposition 4.13. Let us nonetheless shortly sketch the proof for convenience
of the reader. Since $\Psi$ is bounded from below by our standing assumptions,
the Tikhonov term in the objective together with the constraints in
$H^{1}_{0}(\mathcal{X}_{c})$ imply that the sequence $\\{\ell_{n}\\}$ is
bounded in $H^{1}_{0}(\mathcal{X}_{c})$. Since $\mathcal{X}_{c}$ is assumed to
be a Hilbert space, there exists a weakly converging subsequence with weak
limit $\overline{\ell}\in H^{1}_{0}(\mathcal{X}_{c})$. Due to Proposition
4.13(i), the associated states $\SS_{n}(\ell_{n})$ converge weakly to the
reduced solution $\overline{\sigma}:=\SS(\overline{\ell})$, and the weak lower
semicontinuity of the objective ensures the global optimality of
$(\overline{\sigma},\overline{\ell})$.

From Proposition 4.13(ii), we moreover deduce that
$\SS_{n}(\overline{\ell})\to\overline{\sigma}$ in
$H^{1}(\mathbb{L}^{2}(\Omega))$ such that the continuity of $\Psi$ implies

$J(\overline{\sigma},\overline{\ell})\leq\liminf_{n\to\infty}J(\SS_{n}(\ell_{n}),\ell_{n})\leq\limsup_{n\to\infty}J(\SS_{n}(\ell_{n}),\ell_{n})\leq\limsup_{n\to\infty}J(\SS_{n}(\overline{\ell}),\overline{\ell})=J(\overline{\sigma},\overline{\ell}),$

i.e., the convergence of the objective. Since both components of the objective
are weakly lower semicontinuous, we obtain
$\|\overset{\boldsymbol{.}}{\ell_{n}}\|_{L^{2}(\mathcal{X}_{c})}\to\|\overset{\boldsymbol{.}}{\overline{\ell}}\|_{L^{2}(\mathcal{X}_{c})}$,
which in turn implies strong convergence.

As the above reasoning applies to every weakly convergent subsequence, we
deduce that every weak accumulation point is actually a strong one and a
global minimizer of Eq. P, which completes the proof.

##  5 Optimality Conditions

Unfortunately, the Yosida regularization does in general not yield a Gâteaux-
differentiable control-to-state mapping. We will demonstrate this for a
particular case of the set of admissible stresses below. Therefore, in order
to derive an optimality system by the standard adjoint calculus, a further
smoothing is necessary, which will be addressed next.

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

We consider now the regularized system Eq. 5 for a fixed $n\in\mathbb{N}$ and
set $(\varepsilon,\lambda):=(\varepsilon_{n},\lambda_{n})$. Accordingly, we
also abbreviate $Q:=Q_{n}$ (see Definition 3.20).

For the construction of the smoothing of the Yosida regularization and its
differentiability properties, we impose the following assumption for the rest
of this section:

######  Assumption 5.1 (Smoothing of the Yosida regularization).

  * (i)

We fix $p\in(2,\overline{p}]$ in Lemma 3.17.

  * (ii)

The operator $\mathcal{G}$ is linear and continuous from $\mathcal{X}_{c}$ to
$\mathbf{W}^{1,p}(\Omega)$ and the Dirichlet-offset satisfies $\mathfrak{a}\in
H^{1}(\mathbf{W}^{1,p}(\Omega))$.

  * (iii)

We assume $\lambda>0$ (note that $\varepsilon=0$ is possible).

  * (iv)

The set $K$ from Definition 3.2 is given in terms of the von Mises yield
condition, i.e.,

(15) $K:=\\{\tau\in\mathbb{R}^{n\times n}_{s}:|\tau^{D}|_{F}\leq\gamma\\},$

where $\tau^{D}:=\tau-\frac{1}{n}\operatorname{tr}(\tau)I$ is the deviator of
$\tau\in\mathbb{R}^{n\times n}_{s}$, $\gamma>0$ denotes the initial uniaxial
yield stress, and $|\cdot|_{F}$ is the Frobenius norm.

A straightforward calculations shows that, in case of the von Mises yield
condition, the Yosida-approximation of $\partial I_{\mathcal{K}(\Omega)}$ is
given by

$\displaystyle\partial
I_{\lambda}(\tau)=\frac{1}{\lambda}\operatorname{max}\Big{\\{}0,1-\frac{\gamma}{|\tau^{D}|_{F}}\Big{\\}}\tau^{D},$

cf. e.g. [9]. Herein, with a slight abuse of notation, we denote the Nemyzki
operator in $L^{\infty}(\Omega)$ associated with the pointwise maximum, i.e.,
$\mathbb{R}\ni r\mapsto\max\\{0,r\\}\in\mathbb{R}$, by the same symbol. In
addition, we set $\operatorname{max}\\{0,1-\gamma/r\\}:=0$, if $r=0$. As
indicated above, we indeed observe that $\partial I_{\lambda}$ is still a non-
smooth mapping, giving in turn that the associated solution operator of the
regularized state equation is not Gâteaux-differentiable. We therefore
additionally smoothen the Yosida-approximation to obtain a differentiable
mapping:

(16)
$A_{\delta}:\mathbb{L}^{2}(\Omega)\rightarrow\mathbb{L}^{2}(\Omega),\text{ \ \
\ \ \ \
}\tau\mapsto\frac{1}{\lambda}\operatorname{max}_{\delta}\Big{(}1-\frac{\gamma}{|\tau^{D}|_{F}}\Big{)}\tau^{D},$

where

$\operatorname{max}_{\delta}:\mathbb{R}\rightarrow\mathbb{R}\text{ \ \ \ \ \ \
}r\mapsto\begin{cases}\max\\{0,r\\},&|r|\geq\delta,\\\
\tfrac{1}{4\delta}(r+\delta)^{2},&|r|<\delta.\end{cases}$

for a fixed $\delta\in(0,1)$. Again, we denote the Nemyzki operator associated
with $\operatorname{max}_{\delta}$ by the same symbol. One easily checks that
$\operatorname{max}_{\delta}\in C^{1}(\mathbb{R})$ and that

(17) $\displaystyle\|A_{\delta}(\tau)-\partial
I_{\lambda}(\tau)\|_{\mathbb{L}^{2}(\Omega)}\leq\frac{|\Omega|\gamma\delta}{4\lambda(1-\delta)}$

for all $\tau\in\mathbb{L}^{2}(\Omega)$. Furthermore, we denote the
restriction of $A_{\delta}$ to $\mathbb{L}^{p}(\Omega)$ by the same symbol.

Let us now turn to the smoothed state equation and the associated optimization
problem. The smoothed state equation reads

(18a) $\displaystyle-\operatorname{div}\sigma(t)$ $\displaystyle=0$
$\displaystyle\text{in }\mathbf{W}^{-1,p}_{D}(\Omega),$ (18b)
$\displaystyle\sigma(t)$ $\displaystyle=\mathbb{C}(\nabla^{s}u(t)-z(t))$
$\displaystyle\text{in }\mathbb{L}^{p}(\Omega),$ (18c)
$\displaystyle\overset{\boldsymbol{.}}{z}(t)$
$\displaystyle=A_{\delta}(\sigma(t)-\varepsilon\mathbb{B}z(t))$
$\displaystyle\text{in }\mathbb{L}^{p}(\Omega),$ (18d) $\displaystyle
u(t)-u_{D}(t)$ $\displaystyle\in\mathbf{W}^{1,p}_{D}(\Omega),$ (18e)
$\displaystyle(u,\sigma)(0)$ $\displaystyle=(u_{D}(0),\sigma_{0})$
$\displaystyle\text{in }\mathbf{W}^{1,p}(\Omega)\times\mathbb{L}^{p}(\Omega).$

As in the proofs of Proposition 4.11 resp. Proposition 3.21, in the case
$u_{D}=\mathcal{G}\ell+\mathfrak{a}$, this system can equivalently be
transformed into

(19a) $\displaystyle\overset{\boldsymbol{.}}{z}$
$\displaystyle=A_{\delta}(R\ell+\mathfrak{A}-Qz),$ $\displaystyle\text{ \ \ \
\ }z(0)$ $\displaystyle=\nabla^{s}\mathfrak{a}(0)-\mathbb{A}\sigma_{0},$ (19b)
$\displaystyle u$
$\displaystyle=\mathcal{T}(-\operatorname{div}(\mathbb{C}z),\mathcal{G}\ell+\mathfrak{a}),$
$\displaystyle\sigma$ $\displaystyle=\mathbb{C}(\nabla^{s}u-z),$

where $Q$, $R$, and $\mathfrak{A}$ are defined as in Definition Definition
3.20 and Definition 4.10. Again, we used $\ell\in H^{1}_{0}(\mathcal{X}_{c})$
implying $\ell(0)=0$ for the initial condition in Eq. 19a. As in case of the
Yosida regularization in Corollary 3.23, the existence of solutions to Eq. 19
can again be deduced from Banach’s fixed point theorem owing to the global
Lipschitz continuity of $A_{\delta}$. This time, we consider the fixed point
mapping associated with the integral equation corresponding to Eq. 19a as a
mapping in $L^{2}(0,T;\mathbb{L}^{p}(\Omega))$. Note in this context that, by
virtue of 5.1(ii) and Lemma 3.17, $Q$ and $R$ are mappings from
$\mathbb{L}^{p}(\Omega)$ and $\mathcal{X}_{c}$, respectively, to
$\mathbb{L}^{p}(\Omega)$ and $\mathfrak{A}\in H^{1}(\mathbb{L}^{p}(\Omega))$.
This gives rise to the following

######  Definition 5.2 (Smoothed solution operator).

For $\ell\in H^{1}_{0}(\mathcal{X}_{c})$ there exists a unique solution
$(u,\sigma,z)$ of Eq. 18 with respect to $u_{D}=\mathcal{G}\ell+\mathfrak{a}$.
We denote the associated solution operator by

$\displaystyle\mathcal{S}_{\delta}:H^{1}_{0}(\mathcal{X}_{c})\rightarrow
H^{1}(\mathbb{L}^{p}(\Omega))\text{ \ \ \ \ \ \ }\ell\mapsto\sigma.$

Of course, this operator also depends on $\lambda$ and $\varepsilon$, but we
suppress this dependency to ease notation.

Given $\SS_{\delta}$, the smoothed optimal control problem reads as follows:

(Pδ) $\min_{\ell\in H^{1}_{0}(\mathcal{X}_{c})}\;J(\SS_{\delta}(\ell),\ell).$

The existence of optimal solution to Eq. Pδ follows form standard arguments
completely analogous to Proposition 4.11. Let us shortly interrupt the
derivation of optimality conditions for Eq. Pδ in order to briefly address the
convergence of global minimizers.

######  Proposition 5.3.

Let $\\{\lambda_{n}\\}\subset\mathbb{R}^{+}\setminus\\{0\\}$ be a sequence
converging to zero and assume for simplicity that $\varepsilon_{n}=0$ for all
$n\in\mathbb{N}$. Suppose moreover that the smoothing parameter $\delta_{n}$
is chosen such that

(20)
$\delta_{n}=\delta(\lambda_{n})=o\Big{(}\lambda_{n}^{2}\exp\big{(}-\tfrac{T\|Q\|_{\mathcal{L}(\mathbb{L}^{2}(\Omega))}}{\lambda_{n}}\big{)}\Big{)}.$

Let $\\{\ell_{n}\\}$ denote a sequence of solutions of Eq. Pδ with
$\lambda=\lambda_{n}$ and $\delta=\delta_{n}$. Then every weak accumulation
point is actually a strong one and a minimizer of Eq. P. In addition, there is
an accumulation point.

######  Proof 5.4.

In principle, we only need to estimate the difference in the solution of Eq. 5
and Eq. 18. For this purpose, we use the equivalent formulations in Eq. 9 and
Eq. 19 to see that Eq. 17 gives

$\displaystyle\|\overset{\boldsymbol{.}}{z}_{\lambda}(t)-\overset{\boldsymbol{.}}{z}_{\delta}(t)\|_{\mathbb{L}^{2}(\Omega)}$
$\displaystyle\leq\|\partial
I_{\lambda}(R\ell(t)+\mathfrak{A}-Q(z_{\delta}(t)))-A_{\delta}(R\ell(t)+\mathfrak{A}-Q(z_{\delta}(t)))\|_{\mathbb{L}^{2}(\Omega)}$
$\displaystyle\quad+\|\partial
I_{\lambda}(R\ell(t)+\mathfrak{A}-Q(z_{\delta}(t)))-\partial
I_{\lambda}(R\ell(t)+\mathfrak{A}-Q(z_{\lambda}(t)))\|_{\mathbb{L}^{2}(\Omega)}$
$\displaystyle\leq\frac{|\Omega|\gamma\delta}{4\lambda(1-\delta)}+\frac{1}{\lambda}\,\|Q\|_{\mathcal{L}(\mathbb{L}^{2}(\Omega))}\|z_{\delta}(t)-z_{\lambda}(t)\|_{\mathbb{L}^{2}(\Omega)}$

such that Gronwall’s inequality in turn implies

(21)
$\|\overset{\boldsymbol{.}}{z}_{\lambda}(t)-\overset{\boldsymbol{.}}{z}_{\delta}(t)\|_{\mathbb{L}^{2}(\Omega)}\leq\Bigg{(}\frac{\|Q\|_{\mathcal{L}(\mathbb{L}^{2}(\Omega))}}{\lambda}T\exp\Big{(}\frac{\|Q\|_{\mathcal{L}(\mathbb{L}^{2}(\Omega))}}{\lambda}\,T\Big{)}+1\Bigg{)}\frac{|\Omega|\gamma\delta}{4\lambda(1-\delta)}.$

We observe that the error induced by the additional smoothing is independent
of the control $\ell$. Therefore, if $\lambda$ and $\delta$ are coupled as
indicated in Eq. 20, then the convergence results from Proposition 4.13
readily carry over to the solution operator with additional smoothing and we
can use exactly the same arguments as in the proof of Theorem 4.15 to
establish the claim.

######  Remark 5.5.

The above proof is completely along the lines of [16, Sections 4.2 and 7.4],
but we have briefly presented it for convenience of the reader. We underline
that we do not claim that the coupling of $\lambda$ and $\delta$ in Eq. 20 is
optimal.

The next lemma covers the differentiability of $A_{\delta}$. Although the
function $\operatorname{max}_{\delta}$ slightly differs from the one in [16,
Section 7.4], it is straight forward to transfer the analysis thereof to our
setting giving the following

######  Lemma 5.6 (Differentiability of $A_{s}$, [16, Lemma 7.24 & Corollary
7.25]).

The operator $A_{\delta}$ is continuously Fréchet differentiable from
$\mathbb{L}^{p}(\Omega)$ to $\mathbb{L}^{2}(\Omega)$ and its directional
derivative at $\tau\in\mathbb{L}^{p}(\Omega)$ in direction
$h\in\mathbb{L}^{2}(\Omega)$ is given by

$A_{\delta}^{\prime}(\tau)h=\frac{1}{\lambda}\operatorname{max}_{\delta}^{\prime}\Big{(}1-\frac{\gamma}{|\tau^{D}|_{F}}\Big{)}\frac{\gamma}{|\tau^{D}|_{F}^{3}}(\tau^{D}\colon
h^{D})\tau^{D}+\frac{1}{\lambda}\operatorname{max}_{\delta}\Big{(}1-\frac{\gamma}{|\tau^{D}|_{F}}\Big{)}h^{D}.$

Moreover, for every $\tau\in\mathbb{L}^{p}(\Omega)$,
$A_{\delta}^{\prime}(\tau)$ can be extended to an operator in
$\mathcal{L}(\mathbb{L}^{2}(\Omega);\mathbb{L}^{2}(\Omega))$, which is self-
adjoint and satisfies
$\|A_{\delta}^{\prime}(\tau)\|_{\mathcal{L}(\mathbb{L}^{2}(\Omega))}\leq C$
with a constant independent of $\tau$.

######  Proposition 5.7 (Differentiability of the smoothed solution operator).

The solution operator $\mathcal{S}_{\delta}$ is Fréchet differentiable from
$H^{1}_{0}(\mathcal{X}_{c})$ to $H^{1}(\mathbb{L}^{2}(\Omega))$. Its
directional derivative at $\ell\in H^{1}_{0}(\mathcal{X}_{c})$ in direction
$h\in H^{1}_{0}(\mathcal{X}_{c})$, denoted by
$\tau=\SS_{\delta}^{\prime}(\ell)h$, is the second component of the unique
solution $(v,\tau,\eta)\in
H^{1}(\mathbf{H}^{1}(\Omega)\times\mathbb{L}^{2}(\Omega)\times\mathbb{L}^{2}(\Omega))$
of

(22a) $\displaystyle-\operatorname{div}\tau(t)$ $\displaystyle=0$
$\displaystyle\text{in }\mathbf{H}^{-1}_{D}(\Omega),$ (22b)
$\displaystyle\tau(t)$ $\displaystyle=\mathbb{C}(\nabla^{s}v(t)-\eta(t))$
$\displaystyle\text{in }\mathbb{L}^{2}(\Omega),$ (22c)
$\displaystyle\overset{\boldsymbol{.}}{\eta}(t)$
$\displaystyle=A_{\delta}^{\prime}(\sigma(t)-\varepsilon\mathbb{B}z(t))(\tau(t)-\varepsilon\mathbb{B}\eta(t))$
$\displaystyle\text{in }\mathbb{L}^{2}(\Omega),$ (22d) $\displaystyle
v(t)-(\mathcal{G}h)(t)$ $\displaystyle\in\mathbf{H}^{1}_{D}(\Omega),$ (22e)
$\displaystyle(v,\tau)(0)$ $\displaystyle=(0,0)$ $\displaystyle\text{in
}\mathbf{H}^{1}(\Omega)\times\mathbb{L}^{2}(\Omega).$

where $(u,\sigma,z)$ is the solution of Eq. 18 associated with
$u_{D}=\mathcal{G}\ell+\mathfrak{a}$.

######  Proof 5.8.

We again employ the equivalent formulation in Eq. 19. The operator
differential equation in Eq. 19a has exactly the form as the one investigated
in [16, Section 5], except that there is an additional offset $\mathfrak{A}$
and $Q$ is not coercive, if $\varepsilon=0$. It is however easily seen that
these differences have no influence on the sensitivity analysis in [16,
Section 5]. While it is rather evident that the constant offset does not play
any role in this context, the coercivity of $Q$ is only needed in [16] to
verify the existence of solutions, if $A_{\delta}$ is replaced by $\partial
I_{\mathcal{K}(\Omega)}$, and is not used for the sensitivity analysis of the
smoothed equation. All in all, we see that, thanks to Lemma 5.6, [16, Theorem
5.5] is applicable giving that the solution mapping of Eq. 19a is Fréchet-
differentiable from $H^{1}_{0}(\mathcal{X}_{c})$ to
$H^{1}(\mathbb{L}^{2}(\Omega))$ and its derivative at $\ell$ in direction $h$
is the unique solution of

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

Since all mappings in Eq. 19b are linear and affine, respectively, they are
trivially Fréchet-differentiable in their respective spaces and the respective
derivatives are given by
$v=\mathcal{T}(-\operatorname{div}(\mathbb{C}\eta),\mathcal{G}h)$ and
$\tau=\mathbb{C}(\nabla^{s}v-\eta)$. In view of the definition of
$\mathcal{T}$, $R$, and $Q$, we finally end up with Eq. 22.

###  5.2 Adjoint Equation

We now choose a concrete objective function, namely

(23) $\displaystyle J:H^{1}(\mathbb{L}^{2}(\Omega))\times
H^{1}_{0}(\mathcal{X}_{c})\rightarrow\mathbb{R},\quad(\sigma,\ell)\mapsto\frac{1}{2}\|\sigma(T)-\sigma_{d}\|_{\mathbb{L}^{2}(\Omega)}^{2}+\frac{\alpha}{2}\|\overset{\boldsymbol{.}}{\ell}\|_{L^{2}(\mathcal{X}_{c})},$

where $\alpha>0$ is a Tikhonov paramenter and
$\sigma_{d}\in\mathbb{L}^{2}(\Omega)$ a given desired stress. The transfer of
the upcoming analysis to other Fréchet-differentiable objectives is
straightforward, but, in order to keep the discussion concise and since the
objective in Eq. 23 is certainly of practical interest, we restrict ourselves
to this particular setting. The smoothed optimization problem then reads

(Pδ) $\displaystyle\min_{\ell\in H^{1}_{0}(\mathcal{X}_{c})}\text{ \ \ }$
$\displaystyle J(\SS_{\delta}(\ell),\ell).$

In the following, we will derive first-order necessary optimality conditions
for this problem involving an adjoint equation.

######  Definition 5.9 (Adjoint equation).

Let $(\sigma,z)\in H^{1}(\mathbb{L}^{2}(\Omega)\times\mathbb{L}^{2}(\Omega))$
be given. Then the _adjoint equation_ is given by

(24a) $\displaystyle-\operatorname{div}\mathbb{C}\nabla^{s}w_{\varphi}(t)$
$\displaystyle=-\operatorname{div}\mathbb{C}A_{\delta}^{\prime}(\sigma(t)-\varepsilon\mathbb{B}z(t))\varphi(t)$
$\displaystyle\text{in }\mathbf{H}^{-1}_{D}(\Omega),$ (24b) $\displaystyle
w_{\varphi}(t)$ $\displaystyle\in\mathbf{H}^{1}_{D}(\Omega),$ (24c)
$\displaystyle\overset{\boldsymbol{.}}{\varphi}(t)$
$\displaystyle=(\mathbb{C}+\varepsilon\mathbb{B})A_{\delta}^{\prime}(\sigma(t)-\varepsilon\mathbb{B}z(t))\varphi(t)-\mathbb{C}\nabla^{s}w_{\varphi}(t)$
$\displaystyle\text{in }\mathbb{L}^{2}(\Omega),$ (24d)
$\displaystyle\varphi(T)$
$\displaystyle=\mathbb{C}(\sigma(T)-\sigma_{d}-\nabla^{s}w_{T})$
$\displaystyle\text{in }\mathbb{L}^{2}(\Omega),$ (24e)
$\displaystyle-\operatorname{div}\mathbb{C}\nabla^{s}w_{T}$
$\displaystyle=-\operatorname{div}\mathbb{C}(\sigma(T)-\sigma_{d})$
$\displaystyle\text{in }\mathbf{H}^{-1}_{D}(\Omega),$ (24f) $\displaystyle
w_{T}$ $\displaystyle\in\mathbf{H}^{1}_{D}(\Omega).$

A triple $(w_{\varphi},\varphi,w_{T})\in
H^{1}(\mathbf{H}^{1}_{D}(\Omega))\times
H^{1}(\mathbb{L}^{2}(\Omega))\times\mathbf{H}^{1}_{D}(\Omega)$ is called
_adjoint state_ , if it fulfills Eq. 24 for almost all $t\in(0,T)$.

######  Lemma 5.10.

For every $(\sigma,z)\in
H^{1}(\mathbb{L}^{2}(\Omega)\times\mathbb{L}^{2}(\Omega))$, there exists a
unique adjoint state.

######  Proof 5.11.

Thanks to the definition of $Q$ and $\mathcal{T}$ in Definition 3.20 and Lemma
3.17, the adjoint equation is equivalent to

(25)
$\overset{\boldsymbol{.}}{\varphi}=QA_{\delta}^{\prime}(\sigma-\varepsilon\mathbb{B}z)\varphi,\quad\varphi(T)=\mathbb{C}\big{[}\sigma(T)-\sigma_{d}-\nabla^{s}\mathcal{T}(-\operatorname{div}(\mathbb{C}(\sigma(T)-\sigma_{d})),0)\big{]}.$

This is an operator equation backward in time, whose existence again follows
from Banach’s contraction principle thanks to the boundedness of
$A_{\delta}^{\prime}(\sigma-\varepsilon\mathbb{B}z)$ as an operator from
$\mathbb{L}^{2}(\Omega)$ to $\mathbb{L}^{2}(\Omega)$ by Lemma 5.6.
Alternatively, the existence of solutions to Eq. 25 can be deduced via
duality, cf. [16, Lemma 5.11].

With the help of the adjoint state we can express the derivative of the so-
called reduced objective, defined by

$F_{\delta}:H^{1}_{0}(\mathcal{X}_{c})\to\mathbb{R},\quad\ell\mapsto
J(S_{\delta}(\ell),\ell),$

in a compact form, as the following result shows:

######  Proposition 5.12 (Differentiability of the reduced objective
function).

The reduced objective $F_{\delta}$ is Fréchet differentiable from
$H^{1}_{0}(\mathcal{X}_{c})$ to $\mathbb{R}$. Its directional derivative at
$\ell\in H^{1}_{0}(\mathcal{X}_{c})$ in direction $h\in
H^{1}_{0}(\mathcal{X}_{c})$ is given by

(26)
$F_{\delta}^{\prime}(\ell)h=\partial_{\sigma}J(\sigma,\ell)S_{\delta}^{\prime}(\ell)h+\partial_{\ell}J(\sigma,\ell)h=(\mathfrak{q},h)_{L^{2}(\mathcal{X}_{c})}+\alpha\big{(}\overset{\boldsymbol{.}}{\ell},\overset{\boldsymbol{.}}{h}\big{)}_{L^{2}(\mathcal{X}_{c})},$

where $\mathfrak{q}\in L^{2}(\mathcal{X}_{c})$ is defined by

(27)
$\mathfrak{q}:=\mathcal{G}^{*}\big{[}-\operatorname{div}\mathbb{C}\big{(}A_{\delta}^{\prime}(\sigma-\varepsilon\mathbb{B}z)\varphi-\nabla^{s}w_{\varphi}\big{)}\big{]}$

and $(u,\sigma,z)$ is the solution of Eq. 18 associated with $\ell$ and
$(w_{\varphi},\varphi,w_{T})$ is the corresponding adjoint state.

######  Proof 5.13.

We define
$\Psi:H^{1}_{0}(\mathcal{X}_{c})\ni\ell\mapsto\frac{1}{2}\|\SS_{\delta}(\ell)(T)-\sigma_{d}\|_{\mathbb{L}^{2}(\Omega)}^{2}\in\mathbb{R}$.
According to Proposition 5.7 and the chain rule, $\Psi$ is Fréchet-
differentiable. If we denote by $(u,\sigma,z)$ and $(v,\tau,\eta)$ the
solutions of Eq. 18 and Eq. 22, respectively, and the adjoint state by
$(w_{\varphi},\varphi,w_{T})$, then we obtain for its directional derivative

$\displaystyle\Psi^{\prime}(\ell)h$
$\displaystyle=\left(\sigma(T)-\sigma_{d},\tau(T)\right)_{\mathbb{L}^{2}(\Omega)}$
$\displaystyle=\left(\mathbb{C}(\sigma(T)-\sigma_{d}-\nabla^{s}w_{T}),\nabla^{s}v(T)-\eta(T)\right)_{\mathbb{L}^{2}(\Omega)}$
(by Eq. 22a, Eq. 24f, and Eq. 22b)
$\displaystyle=(\mathbb{C}(\sigma(T)-\sigma_{d}-\nabla^{s}w_{T}),\nabla^{s}\mathcal{G}h(T))_{\mathbb{L}^{2}(\Omega)}$
$\displaystyle\hskip
113.81102pt-\left(\varphi(T),\eta(T)\right)_{\mathbb{L}^{2}(\Omega)}$ (by Eq.
24e, Eq. 22d, and Eq. 24d)
$\displaystyle=-(\varphi(T),\eta(T))_{\mathbb{L}^{2}(\Omega)}$ (since $h\in
H^{1}_{0}(\mathcal{X}_{c})$).

For the last term we find

$\displaystyle(\varphi(T),\eta(T))_{\mathbb{L}^{2}(\Omega)}$
$\displaystyle=(\varphi(T),\eta(T))_{\mathbb{L}^{2}(\Omega)}-(\varphi(0),\eta(0))_{\mathbb{L}^{2}(\Omega)}$
(by Eq. 22e and Eq. 22b)
$\displaystyle=(\overset{\boldsymbol{.}}{\varphi},\eta)_{L^{2}(\mathbb{L}^{2}(\Omega))}+(\varphi,\overset{\boldsymbol{.}}{\eta})_{L^{2}(\mathbb{L}^{2}(\Omega))}$
$\displaystyle=\big{(}(\mathbb{C}+\varepsilon\mathbb{B})A_{\delta}^{\prime}(\sigma-\varepsilon\mathbb{B}z)\varphi-\mathbb{C}\nabla^{s}w_{\varphi},\eta\big{)}_{L^{2}(\mathbb{L}^{2}(\Omega))}$
$\displaystyle\qquad+\big{(}\varphi,A_{\delta}^{\prime}(\sigma-\varepsilon\mathbb{B}z)(\tau-\varepsilon\mathbb{B}\eta)\big{)}_{L^{2}(\mathbb{L}^{2}(\Omega))}$
(by Eq. 24c and Eq. 22c)
$\displaystyle=-(\mathbb{C}\nabla^{s}w_{\varphi},\eta)_{L^{2}(\mathbb{L}^{2}(\Omega))}+(\mathbb{C}A_{\delta}^{\prime}(\sigma-\varepsilon\mathbb{B}z)\varphi,\nabla^{s}v)_{L^{2}(\mathbb{L}^{2}(\Omega))}$
(by Eq. 22b)
$\displaystyle=-(\mathbb{C}\nabla^{s}w_{\varphi},\eta-\nabla^{s}v+\nabla^{s}\mathcal{G}h)_{L^{2}(\mathbb{L}^{2}(\Omega))}$
$\displaystyle\qquad+(\mathbb{C}A_{\delta}^{\prime}(\sigma-\mathbb{B}z)\varphi,\nabla^{s}\mathcal{G}h)_{L^{2}(\mathbb{L}^{2}(\Omega))}$
(by Eq. 24a and Eq. 22d)
$\displaystyle=(\nabla^{s}w_{\varphi},\tau)_{L^{2}(\mathbb{L}^{2}(\Omega))}$
$\displaystyle\qquad+\big{(}\mathbb{C}(\nabla^{s}w_{\varphi}-A_{\delta}^{\prime}(\sigma-\varepsilon\mathbb{B}z)\varphi),\nabla^{s}\mathcal{G}h\big{)}_{L^{2}(\mathbb{L}^{2}(\Omega))}$
(by Eq. 22b) $\displaystyle=-(\mathfrak{q},h)_{L^{2}(\mathcal{X}_{c})}$
$\displaystyle\text{(by \lx@cref{creftype~refnum}{eq:momtau} and
\lx@cref{creftype~refnum}{eq:qdef})}.$

Note that $A_{\delta}^{\prime}(\sigma-\varepsilon\mathbb{B}z)\in
L^{\infty}(\mathcal{L}(\mathbb{L}^{2}(\Omega)))$ by Lemma 5.6 and
$\mathcal{G}^{*}$ maps $\mathbf{H}^{-1}(\Omega)$ to
$\mathcal{X}_{c}^{*}\cong\mathcal{X}_{c}$, which give the asserted regularity
of $\mathfrak{q}$.

######  Theorem 5.14 (KKT-Conditions for Eq. Pδ).

Let $\ell\in H^{1}_{0}(\mathcal{X}_{c})$ be locally optimal for Eq. Pδ with
associated state $(u,\sigma,z)\in
H^{1}(\mathbf{W}^{1,p}(\Omega)\times\mathbb{L}^{p}(\Omega)\times\mathbb{L}^{p}(\Omega))$.
Then there exists an adjoint state $(w_{\varphi},\varphi,w_{T})\in
H^{1}(\mathbf{H}^{1}_{D}(\Omega))\times
H^{1}(\mathbb{L}^{2}(\Omega))\times\mathbf{H}^{1}_{D}(\Omega)$ such that
$\ell$ satisfies for almost all $t\in(0,T)$ the boundary value problem

(28) $\alpha\,\partial_{tt}^{2}\ell(t)=\mathfrak{q}(t)\quad\text{in
}\mathcal{X}_{c},\quad\ell(0)=\ell(T)=0$

with $\mathfrak{q}$ as defined in Eq. 27. This in particular implies that
$\ell\in H^{2}(\mathcal{X}_{c})$.

######  Proof 5.15.

If $\ell\in H^{1}_{0}(\mathcal{X}_{c})$ is a local minimizer of Eq. Pδ, then
Proposition 5.12 implies

$\alpha(\overset{\boldsymbol{.}}{\ell},\overset{\boldsymbol{.}}{h})_{L^{2}(\mathcal{X}_{c})}+(\mathfrak{q},h)_{L^{2}(\mathcal{X}_{c})}=0\quad\forall\,h\in
H^{1}_{0}(\mathcal{X}_{c}).$

Thus the second distributional time derivative of $\ell$ is a regular
distribution in $L^{2}(\mathcal{X}_{c})$, namely $\mathfrak{q}$, which is just
Eq. 28.

######  Remark 5.16.

An optimality condition for the original non-smooth optimal control problem
(P) could be derived by passing to the limit $\lambda,\delta\searrow 0$ in the
regularized optimality system (24) and (28). This has been done for the case
with hardening in [26] and for a scalar rate-independent system with uniformly
convex energy in [20]. The optimality systems obtained in the limit are
comparatively weak compared to what can be derived by regularization in the
static case, see [11] for the latter. We expect that results similar to [26]
can also be obtained in case of (P). This would however go beyond the scope of
this paper and is subject to future research.

##  6 Numerical Experiments

The last section is devoted to the numerical solution of the smoothed problem
Eq. Pδ. We start with a concrete realization of the operator $\mathcal{G}$
mapping our control variable in form of the pseudo-force $\ell$ to the
Dirichlet data. Given the precise form of the operator $\mathcal{G}$, we can
use Proposition 5.12 to obtain an implementable characterization of the
gradient of the reduced objective, see Algorithm 1 below. We moreover describe
the discretization of the involved PDEs and report on numerical results.

###  6.1 A Realization of the Operator $\mathcal{G}$

Let us recall the assumptions imposed on $\mathcal{G}$ throughout the paper:
$\mathcal{G}$ is a linear and continuous operator from $\mathcal{X}$ to
$\mathbf{H}^{1}(\Omega)$ and from $\mathcal{X}_{c}$ to
$\mathbf{W}^{1,p}(\Omega)$ with some $p\in(2,\overline{p}]$ and a Hilbert
space $\mathcal{X}_{c}$, which is compactly embedded in $\mathcal{X}$. In
principle, there are various ways to realize such an operator, for instance by
means of convolution. As we are dealing with a problem in computational
mechanics anyway, we choose $\mathcal{G}$ to be the solution operator of a
particular linear elasticity problem. For this purpose, we split
$\partial\Omega$ into two disjoint measurable parts $\Lambda_{D}$ and
$\Lambda_{N}$, called _pseudo Dirichlet boundary_ and _pseudo Neumann
Boundary_. As for $\Gamma_{D}$ and $\Gamma_{N}$, we require that $\Lambda_{N}$
is relatively open in $\partial\Omega$, while $\Lambda_{D}$ is relatively
closed and has positive measure. Moreover, we assume that
$\Omega\cup\Lambda_{N}$ is regular in the sense of Gröger. Therefore,
according to [10], there is an index $\overline{p}$ such that, for every
$p\in[\overline{p}^{\prime},\overline{p}]$, the linear elasticity equation

(29)
$(\mathbb{C}\nabla^{s}\upsilon,\nabla^{s}\zeta)_{\mathbb{L}^{2}(\Omega)}=\langle
b,\zeta\rangle\quad\forall\,\zeta\in\mathbf{W}^{1,p^{\prime}}_{\Lambda}(\Omega),\quad\upsilon\in\mathbf{W}^{1,p}_{\Lambda}(\Omega)$

admits a unique solution in $\mathbf{W}^{1,p}_{\Lambda}(\Omega)$ for every
right hand side $b\in\mathbf{W}^{-1,p}_{\Lambda}(\Omega)$. Herein,
$\mathbf{W}^{1,p}_{\Lambda}(\Omega)$ is defined as
$\mathbf{W}^{1,p}_{D}(\Omega)$ in Eq. 1 with $\Lambda_{D}$ instead of
$\Gamma_{D}$. Depending on the precise geometrical structure, the index
$\overline{p}$ may well differ from the one in Lemma 3.17, but, in order to
ease the notation, we assume that both are equal (just take the minimum of
both, which is still greater two). As in Section 5, we fix
$p\in(2,\overline{p}]$ in what follows and assume in addition that
$p<2n/(n-1)$. Furthermore, we require that $\Gamma_{D}\subset\Lambda_{N}$ and
that $\Gamma_{D}$ and $\Lambda_{D}$ have positive distance to each other,
i.e.,

(30)
$\operatorname{dist}(\Gamma_{D},\Lambda_{D})=\inf_{x\in\Lambda_{D},\,\xi\in\Gamma_{D}}|x-\xi|>0.$

Similarly to Eq. 7, we denote the linear and continuous solution operator of
Eq. 29 by
$\mathcal{T}_{\Lambda}:\mathbf{W}^{-1,p}_{\Lambda}(\Omega)\to\mathbf{W}^{1,p}_{\Lambda}(\Omega)$.
This operator will also be considered as a mapping from
$\mathbf{H}^{-1}_{\Lambda}(\Omega)$ to $\mathbf{H}^{1}_{\Lambda}(\Omega)$,
which we denote by the same symbol. Since $p<2n/(n-1)$ by assumption, Sobolev
embeddings and trace theorems give that the embedding and trace operator

$E:\mathbf{W}^{1,p^{\prime}}_{\Lambda}(\Omega)\to\mathbf{L}^{2}(\Omega),\quad\operatorname{tr}:\mathbf{W}^{1,p^{\prime}}_{\Lambda}(\Omega)\to\mathbf{L}^{2}(\Lambda_{N})$

are compact. With these definitions at hand, we define $\mathcal{X}$ and
$\mathcal{X}_{c}$ by

(31)
$\mathcal{X}:=\mathbf{H}^{-1}_{\Lambda}(\Omega)\quad\text{and}\quad\mathcal{X}_{c}:=\mathbf{L}^{2}(\Omega)\times\mathbf{L}^{2}(\Lambda_{N})$

so that, due to the compactness of $E$ and $\operatorname{tr}$, we indeed have
that $\mathcal{X}_{c}$ is compactly embedded in
$\mathbf{W}^{-1,p}_{\Lambda}(\Omega)\hookrightarrow\mathcal{X}$. Moreover,
considered as an operator from $\mathcal{X}=\mathbf{H}^{-1}_{\Lambda}(\Omega)$
to $\mathbf{H}^{1}(\Omega)$, we simply set
$\mathcal{G}:=\mathcal{T}_{\Lambda}$, while, with a slight abuse of notation,
we define $\mathcal{G}$ as an operator from $\mathcal{X}_{c}$ to
$\mathbf{W}^{1,p}_{\Lambda}(\Omega)$ by

(32) $\mathcal{G}:=\mathcal{T}_{\Lambda}\circ(E^{*},\operatorname{tr}^{*}),$

i.e., given $(f,g)\in\mathcal{X}_{c}$, $\mathcal{G}$ is the solution operator
of Eq. 29 with $\langle
b,\zeta\rangle=(f,\zeta)_{\mathbf{L}^{2}(\Omega)}+(g,\zeta)_{\mathbf{L}^{2}(\Lambda_{N})}$.
Note that, since
$\mathcal{X}_{c}\hookrightarrow\mathbf{W}^{-1,p}_{\Lambda}(\Omega)$, this
equation indeed admits a solution in $\mathbf{W}^{1,p}_{\Lambda}(\Omega)$.
Moreover, the following result shows that our control space $\mathcal{X}_{c}$
is “large enough”:

######  Lemma 6.1.

There holds
$\mathcal{T}(0,\mathbf{H}^{2}(\Omega))\subset\mathcal{T}(0,\mathcal{G}(\mathcal{X}_{c}))$,
where $\mathcal{T}$ is the solution operator from Eq. 7.

######  Proof 6.2.

Due to Eq. 30, there is a function $\phi\in
C^{\infty}(\mathbb{R}^{n};\mathbb{R})$ such that $0\leq\phi\leq 1$,
$\phi\equiv 1$ on $\Gamma_{D}$ and $\phi\equiv 0$ on $\Lambda_{D}$. Let
$u_{D}\in\mathbf{H}^{2}(\Omega)$ be arbitrary and define $\tilde{u}_{D}:=\phi
u_{D}\in\mathbf{H}^{2}(\Omega)\cap\mathbf{W}^{1,p}_{\Lambda}(\Omega)$. From
construction of $\phi$ it follows that such that
$\mathcal{T}(0,u_{D})=\mathcal{T}(0,\tilde{u}_{D})$ holds. Moreover, if we
define
$f:=-\operatorname{div}\mathbb{C}\nabla^{s}\tilde{u}_{D}\in\mathbf{L}^{2}(\Omega)$
and
$g:=\operatorname{tr}\mathbb{C}\nabla^{s}\tilde{u}_{D}\in\mathbf{L}^{2}(\Lambda_{N})$,
then $\mathcal{G}(f,g)=\tilde{u}_{D}$ and hence,
$\mathcal{T}(0,\mathcal{G}(f,g))=\mathcal{T}(0,u_{D})$, which proves the
assertion.

Let us now investigate the precise structure of the gradient of the reduced
objective for this particular realization of $\mathcal{G}$.

######  Lemma 6.3.

Let $\ell,h\in H^{1}_{0}(\mathcal{X}_{c})$ be arbitrary and denote the
components of $\ell$ and $h$ by $\ell_{\Omega},h_{\Omega}\in
H^{1}_{0}(\mathbf{L}^{2}(\Omega))$ and $\ell_{N},h_{N}\in
H^{1}_{0}(\mathbf{L}^{2}(\Lambda_{N}))$. Then

(33)
$F_{\delta}^{\prime}(\ell)h=\int_{0}^{T}\int_{\Omega}(\overset{\boldsymbol{.}}{\psi}+\alpha\,\overset{\boldsymbol{.}}{\ell}_{\Omega})\cdot\overset{\boldsymbol{.}}{h}_{\Omega}\,dx+\int_{0}^{T}\int_{\Lambda_{N}}(\overset{\boldsymbol{.}}{\psi}+\alpha\,\overset{\boldsymbol{.}}{\ell}_{N})\cdot\overset{\boldsymbol{.}}{h}_{N}\,ds,$

with $\psi\in H^{2}(\mathbf{H}^{1}_{\Lambda}(\Omega))\cap
H^{1}_{0}(\mathbf{H}^{1}_{\Lambda}(\Omega))$ defined by

(34)
$\psi(t):=\int_{0}^{t}\int_{0}^{s}q(r)\,dr\,ds-\frac{t}{T}\int_{0}^{T}\int_{0}^{s}q(r)\,dr\,ds,$

where $q\in L^{2}(\mathbf{H}^{1}_{\Lambda}(\Omega))$ denotes the solution of

(35)
$\displaystyle(\mathbb{C}\nabla^{s}q(t),\nabla^{s}\zeta)_{\mathbb{L}^{2}(\Omega)}=$
$\displaystyle\qquad\Big{(}\mathbb{C}\big{(}A_{\delta}^{\prime}(\sigma(t)-\varepsilon\mathbb{B}z(t))\varphi(t)-\nabla^{s}w_{\varphi}(t)\big{)},\nabla^{s}\zeta\Big{)}_{\mathbb{L}^{2}(\Omega)}\quad\forall\,\zeta\in\mathbf{H}^{1}_{\Lambda}(\Omega).$

Thus the Riesz representation of $F_{\delta}^{\prime}(\ell)$ w.r.t. the
$H^{1}_{0}(\mathcal{X}_{c})$-scalar product is
$(E\psi,\operatorname{tr}\psi)+\alpha\ell$.

######  Proof 6.4.

The definition of $\mathcal{G}$ in Eq. 32 yields for $\mathfrak{q}$ as defined
in Eq. 27

(36)
$\mathfrak{q}=(E,\operatorname{tr})\mathcal{T}_{\Lambda}^{*}\big{[}-\operatorname{div}\mathbb{C}\big{(}A_{\delta}^{\prime}(\sigma-\varepsilon\mathbb{B}z)\varphi-\nabla^{s}w_{\varphi}\big{)}\big{]}.$

Now, since $\varphi,w_{\varphi}\in
C([0,T];\mathbb{L}^{2}(\Omega)\times\mathbf{H}^{1}_{D}(\Omega))$ by Lemma
5.10, we have
$[-\operatorname{div}\mathbb{C}(A_{\delta}^{\prime}(\sigma-\varepsilon\mathbb{B}z)\varphi-\nabla^{s}w_{\varphi})](t)\in\mathbf{H}^{-1}_{\Lambda}(\Omega)$
for all $t\in[0,T]$. As
$\mathcal{T}_{\Lambda}:\mathbf{H}^{-1}_{\Lambda}(\Omega)\to\mathbf{H}^{1}_{\Lambda}(\Omega)$
is self adjoint due to the symmetry of $\mathbb{C}$, the definition of $q$ via
Eq. 35 thus implies $\mathfrak{q}=(Eq,\operatorname{tr}q)$ and hence, Eq. 26
becomes

$F_{\delta}^{\prime}(\ell)h=\alpha(\overset{\boldsymbol{.}}{\ell},\overset{\boldsymbol{.}}{h})_{L^{2}(\mathcal{X}_{c})}+\int_{0}^{T}\int_{\Omega}q\cdot
h_{\Omega}\,dx\,dt+\int_{0}^{T}\int_{\Lambda_{N}}q\cdot h_{N}\,ds\,dt.$

Since $\partial_{tt}^{2}\psi=q$ by construction, integration by parts in time
implies the assertion.

The precise structure of $\mathfrak{q}$ in Eq. 36 together with the gradient
equation in Eq. 28 immediately gives the following regularity result:

######  Corollary 6.5.

If $\mathcal{G}$ is chosen as in Eq. 32, then the set of local minimizers of
Eq. Pδ is a subset of $H^{2}(\mathbf{H}^{1}_{\Lambda}(\Omega))\cap
H^{1}_{0}(\mathbf{H}^{1}_{\Lambda}(\Omega))$.

The characterization of the Riesz representation of the gradient of the
reduced objective in Lemma 6.3 is of course crucial for the construction of
gradient based optimization methods. We observe that, if we start with an
initial guess for the control of the form
$(E\ell_{0},\operatorname{tr}\ell_{0})$ with a function $\ell_{0}\in
H^{2}(\mathbf{H}^{1}_{\Lambda}(\Omega))\cap
H^{1}_{0}(\mathbf{H}^{1}_{\Lambda}(\Omega))$, then the gradient update will
preserve this structure, i.e., the next iterate
$\ell_{1}:=\ell_{0}-\sigma_{0}(\psi_{0}+\alpha\ell_{0})$ with a suitable step
size $\sigma_{0}>0$ will again be an element of
$H^{2}(\mathbf{H}^{1}_{\Lambda}(\Omega))\cap
H^{1}_{0}(\mathbf{H}^{1}_{\Lambda}(\Omega))$. Note moreover that, due to the
additional regularity of locally optimal controls in Corollary 6.5, it makes
perfectly sense to restrict to control functions in
$H^{2}(\mathbf{H}^{1}_{\Lambda}(\Omega))\cap
H^{1}_{0}(\mathbf{H}^{1}_{\Lambda}(\Omega))$. The overall computation of the
reduced gradient by means of the adjoint approach is given as a pseudo-code in
Algorithm 1.

Algorithm 1 Computation of the Reduced Gradient

1:control function $\ell\in H^{2}(\mathbf{H}^{1}_{\Lambda}(\Omega))\cap
H^{1}_{0}(\mathbf{H}^{1}_{\Lambda}(\Omega))$

2: Compute the Dirichlet data $u_{D}$ by solving for all $t\in[0,T]$
$(\mathbb{C}\nabla^{s}\upsilon(t),\nabla^{s}\zeta)_{\mathbb{L}^{2}(\Omega)}=\int_{\Omega}\ell(t)\cdot\zeta\,dx+\int_{\Lambda_{N}}\ell(t)\cdot\zeta\,ds\quad\forall\,\zeta\in\mathbf{W}^{1,p^{\prime}}_{\Lambda}(\Omega).$

3:Compute the state $(u,\sigma,z)$ as solution of Eq. 18 with $u_{D}$ from
step 2.

4:Solve the adjoint equation in Eq. 24 with solution
$(w_{\varphi},\varphi,w_{T})$.

5:Compute $q$ as solution of Eq. 35.

6:Integrate $q$ according to Eq. 34 to obtain $\psi$.

7:return $\psi+\alpha\ell$ as Riesz representative of
$F_{\delta}^{\prime}(\ell)$.

Based on Algorithm 1, gradient-based first-order optimization algorithm like
the classical gradient descent method or nonlinear CG methods can be used to
solve the smoothed problem Eq. Pδ. For the computations in Section 6.4 below,
we used a standard gradient method with an Armijo line search. As termination
criterion, we require that the norm of the gradient is smaller than the
tolerance TOL = 5e-04. If this criterion is not met, the algorithm will stop
after $100$ iterations. Note that the natural scalar product (and associated
norm) for the termination criterion as well as for the step size control is

$(\overset{\boldsymbol{.}}{g},\overset{\boldsymbol{.}}{\ell})_{L^{2}(\mathcal{X}_{c})}=(\overset{\boldsymbol{.}}{g},\overset{\boldsymbol{.}}{\ell})_{L^{2}(\mathbf{L}^{2}(\Omega))}+(\overset{\boldsymbol{.}}{g},\overset{\boldsymbol{.}}{\ell})_{L^{2}(\mathbf{L}^{2}(\Gamma_{N}))}.$

###  6.2 Discretization

In order to obtain an implementable algorithm, we need to discretize the PDEs
in Algorithm 1. We follow the “ _first optimize, then discretize_ ”-approach,
i.e., we discretize the continuous gradient as given in Algorithm 1, see
Remark 6.6 below.

Let us begin with the discretization in space. The computational domain is
discretized by means of a regular triangulation, which exactly fits the
boundary (which does not cause any trouble in our test scenarios, since our
computational domain is polygonally bounded). For the displacement-like
variables $u$, $w_{\varphi}$, $w_{T}$, and $q$, we use standard continuous and
piecewise linear finite elements, whereas the stress- and strain-like
variables $\sigma$, $z$, and $\varphi$ are discretized by means of piecewise
constant ansatz functions. The state system is reduced to displacement and
plastic strain only by eliminating the stress field by means of Eq. 18b. We
are aware that this type of discretization will in general lead to locking
effects, but we assume that these can be neglected, as we do not consider
“thin” computational domains. A suitable discretization of state and adjoint
equation accounting for locking is however essential, especially in case of
stress tracking, and therefore subject to future research.

Concerning the time discretization, we apply an implicit Euler scheme to Eq.
18c and Eq. 24c. The numerical integration for the computation of $\psi$ and
the evaluation of the objective is performed by an exact integration of the
linear interpolant built upon the iterates of the implicit Euler scheme.

To solve the discretized equations in every iteration of the implicit Euler
scheme, we use the finite element toolbox FEniCS (version 2018.1.0). The
nonlinear state equation is solved by the FEniCS’s inbuilt Newton-solver with
a relative and absolute tolerance of $10^{-10}$.

######  Remark 6.6.

Let us emphasize that our “first optimize, then discretize”-approach leads to
a mismatch between the discretization of the derivative of the reduced
objective in function space and the derivative of the discretized objective.
Thus, the “gradient” computed by means of a discretization of Algorithm 1 does
not coincide with the true discrete gradient. In our numerical experiments, it
however turned out that, as expected, this mismatch only plays a role for
large time step sizes (as expected) and small values of $\lambda$, see Table 2
below.

###  6.3 The Test Setting

For our numerical test, we choose the following data:

#### Domain

The two-dimensional computational domain is set to
$\Omega:=(0,4)\times(0,1)\subset\mathbb{R}^{2}$ with the boundaries
$\Gamma_{D}:=[\\{0\\}\cup\\{4\\}]\times[0,1]$,
$\Lambda_{D}:=[1,3]\times[\\{0\\}\cup\\{1\\}]$ and
$\Gamma_{N}:=\partial\Omega\setminus\Gamma_{D}$,
$\Lambda_{N}:=\partial\Omega\setminus\Lambda_{D}$.

#### Elasticity tensor, hardening and smoothing parameters

We choose typical material parameters of steel:

$\displaystyle E=210\;\big{[}\textup{kN/mm}^{2}\big{]}$
$\displaystyle\text{(Young's modulus)},$ $\displaystyle\nu=0.3$
$\displaystyle\text{(Poisson's ratio)},$ $\displaystyle\begin{aligned}
\lambda&=\frac{E\nu}{(1+\nu)(1-2\nu)}\thickapprox
121.1538\;\big{[}\textup{kN/mm}^{2}\big{]}\\\\[-2.15277pt]
\mu&=\frac{E}{2+2\nu}\thickapprox
80.7692\;\big{[}\textup{kN/mm}^{2}\big{]}\end{aligned}$
$\displaystyle\text{(Lam\'{e} parameters)},$
$\displaystyle\gamma=0.45\;\big{[}\textup{kN/mm}^{2}\big{]}$ (uniaxial yield
stress)

and define the elasticity tensor by
$\mathbb{C}\boldsymbol{\epsilon}:=\lambda\operatorname{tr}(\boldsymbol{\epsilon})I+2\mu\,\boldsymbol{\epsilon}$
for all $\boldsymbol{\epsilon}\in\mathbb{R}^{n\times n}_{s}$.

In our numerical tests, we set $\varepsilon=0$ such that there is no
hardening. We again underline that this case is covered by our analysis, see
4.8 and 5.1(iii).

The smoothing parameter $\delta$ of the $\operatorname{max}$-function in (16)
is set to $10^{-8}$. During the numerical experiments, it turned out that this
parameter appears to have only little influence on the results and the
performance of the algorithm so that we simply fix it to this value.

#### End time and initial condition

We set $T=1$ and $\sigma_{0}\equiv 0$.

#### Desired Dirichlet displacement

The offset in the Dirichlet condition is chosen to be
$\mathfrak{a}(t):=t\,\mathfrak{a}_{e}$, where
$\mathfrak{a}_{e}(x,y):=\frac{1}{200}(x-2,0)$ for $(x,y)\in\Omega$.

#### Optimization problem

We set the desired stress to zero, i.e., $\sigma_{d}\equiv 0$, and the
Tikhonov parameter $\alpha$ to $10^{-4}$.

The above setting is motivated by the following application-driven
optimization problem: The aim of the optimization is to reach a desired
displacement of the Dirichlet boundary (given by $\mathfrak{a}_{e}$) and, at
the same time, to minimize the overall stress distribution at end time. For
this reason, the left and right boundary of the body occupying $\Omega$ is
pulled apart constantly in time. The control $\ell$ (respectively $u_{D}$) can
alter this process for $t\in(0,T)$, but at the end (and also the beginning)
the control is zero, hence, the position of the Dirichlet boundary at $t=T$ is
predefined, namely by the desired $\mathfrak{a}_{e}$. The minimization of the
stress at end time is reflected by setting $\sigma_{d}\equiv 0$ and choosing a
comparatively small Tikhonov parameter.

###  6.4 Numerical Results

Let us finally present the numerical results. In order to assess the impact of
the Yosida regularization, we vary the parameter $\lambda$ and consider the
distance of the stress field to the feasible set $\mathcal{K}(\Omega)$ at the
end of the iteration as an indicator for the effect of the regularization. To
be more precise, given the feasible set of the von Mises yield condition in
(15) and a discrete solution $\sigma_{h}$, we compute

$\operatorname{dist}_{\mathcal{K}}:=\operatorname*{ess\,sup}_{(t,x)\in(0,T)\times\Omega}\frac{|\sigma_{h}^{D}(t,x)|_{F}-\gamma}{\gamma}.$

Furthermore, we evaluate the error induced by the inexact computation of the
reduced gradient caused by the first-optimize-then-discretize approach. It
turned out that this error is entirely induced by the time discretization
while the spatial discretization had no effect here (which is to be expected,
as we used a Galerkin scheme). Therefore, we vary the time step size and use
the difference between in the (inexact) directional derivative and a
difference quotient as error indicator. To describe this in detail, let
$\ell_{h}$ denote the (discrete) control variable in the last iteration and
denote the inexact reduced gradient computed by the discretized counterpart of
Algorithm 1 by $g_{h}$. Then we compute

$\textup{err}=\Bigg{|}\frac{\langle
g_{h},-g_{h}\rangle_{H^{1}_{0}(\mathcal{X}_{c})}-\tau^{-1}\big{(}F_{\delta}(\ell_{h}-\tau\,g_{h})-F_{\delta}(\ell_{h})\big{)}}{\tau^{-1}\big{(}F_{\delta}(\ell_{h}-\tau\,g_{h})-F_{\delta}(\ell_{h})\big{)}}\Bigg{|},$

i.e., we compute the relative error of the directional derivative in the anti-
gradient direction (which is also our search direction). The step size in the
difference quotient is set to $\tau=10^{-8}$.

Table 1 shows the numerical results for different values of $\lambda$. For the
computations, we chose an equidistant time step size by dividing $[0,T]$ in
$n_{t}=128$ intervals of the same length. The spatial mesh is equidistant,
too, with $n_{x}=64$ elements in horizontal and $n_{y}=16$ in vertical
direction. Recall that we focus on the last iteration of the gradient method,
that is, either the norm of the gradient was smaller than TOL = 5e-04 (i.e.,
$\langle
g_{h},-g_{h}\rangle_{H^{1}_{0}(\mathcal{X}_{c})}\geq-\texttt{TOL}^{2}=-2.5\cdot
10^{-7}$) or the 100th iteration was reached.

$\lambda$ | iteration | $\langle g_{h},-g_{h}\rangle_{H^{1}_{0}(\mathcal{X}_{c})}$ | $\frac{F_{\delta}(\ell_{h}-\tau\,g_{h})-F_{\delta}(\ell_{h})}{\tau}$ | err | $\operatorname{dist}_{\mathcal{K}}$
---|---|---|---|---|---
0.001 | 100 | -4.7174e-07 | -4.8520e-07 | 0.027751 | 0.00048
0.01 | 25 | -2.0089e-07 | -2.0869e-07 | 0.037369 | 0.00192
0.1 | 33 | -2.4687e-07 | -2.5552e-07 | 0.033854 | 0.01781
1 | 58 | -2.1643e-07 | -2.1790e-07 | 0.006773 | 0.13652
10 | 100 | -2.0106e-06 | -2.0122e-06 | 0.000833 | 0.62584
100 | 62 | -2.4884e-07 | -2.4876e-07 | 0.000338 | 5.31148
Table 1:  Comparison of the numerical results for different values of
$\lambda$.

We observe that the adjoint approach becomes less accurate for small values of
$\lambda$ reflecting the non-smoothness of the limit problem. Furthermore, the
relative distance of $|\sigma_{h}^{D}|_{F}$ to the yield stress $\gamma$
decreases when $\lambda$ decreases, illustrating the efficiency Yosida-
regularization.

In Table 2, we analyze the impact of the number of time steps on the last
iteration of the gradient method. The spatial mesh is again equidistant with
$n_{x}=64$ and $n_{y}=16$ and we set $\lambda=1$.

$n_{t}$ | iteration | $\langle g_{h},-g_{h}\rangle_{H^{1}_{0}(\mathcal{X}_{c})}$ | $\frac{F_{\delta}(\ell_{h}-\tau\,g_{h})-F_{\delta}(\ell_{h})}{\tau}$ | err | $\operatorname{dist}_{\mathcal{K}}$
---|---|---|---|---|---
4 | 55 | -2.4601e-07 | -3.1816e-07 | 0.226817 | 0.0502
8 | 51 | -2.3590e-07 | -2.8903e-07 | 0.183828 | 0.0478
16 | 52 | -2.4577e-07 | -2.6541e-07 | 0.074012 | 0.0497
32 | 45 | -2.4318e-07 | -2.5225e-07 | 0.035941 | 0.1066
64 | 77 | -2.4627e-07 | -2.5056e-07 | 0.017121 | 0.1017
128 | 58 | -2.1643e-07 | -2.1790e-07 | 0.006773 | 0.1365
256 | 34 | -2.4476e-07 | -2.4562e-07 | 0.003481 | 0.1417
512 | 48 | -2.2542e-07 | -2.2541e-07 | 0.000045 | 0.1318
1024 | 43 | -1.9258e-07 | -1.9225e-07 | 0.001736 | 0.1339
2048 | 41 | -2.3150e-07 | -2.3165e-07 | 0.000662 | 0.1339
Table 2:  Comparison of the numerical results for different numbers of time
steps.

We observe that, as expected, the relative error of the directional derivative
decreases when the number of time steps increases such that the error caused
by the first-optimize-then-discretize approach disappears if the time step
size goes to zero. Moreover, for larger number of time steps, the time
discretization has no effect on the feasibility of the stress (which is of
course mainly influenced by the Yosida parameter as seen before).

We end the description of our numerical results with the time evolution of the
stress field after optimization. For these computations, we set $\lambda=1$,
$n_{t}=256$, $n_{x}=128$, and $n_{y}=32$. The result of the optimization after
150 iterations in form of the stress field at selected time points is shown in
Fig. 2. Therein, and also in Fig. 3, the displacement was scaled by a factor
20.

0.00.20.40.60.81.0

Figure 1: Legend; values in $\big{[}\textup{kN/mm}^{2}\big{]}$.

(a) $n_{t}=0$, $t=0$

(b) $n_{t}=21$, $t\approx 0.0820$

(c) $n_{t}=42$, $t\approx 0.1641$

(d) $n_{t}=63$, $t\approx 0.2461$

(e) $n_{t}=84$, $t\approx 0.3281$

(f) $n_{t}=136$, $t\approx 0.5312$

(g) $n_{t}=188$, $t\approx 0.7343$

(h) $n_{t}=240$, $t\approx 0.9375$

(i) $n_{t}=244$, $t\approx 0.9531$

(j) $n_{t}=248$, $t\approx 0.9688$

(k) $n_{t}=252$, $t\approx 0.9844$

(l) $n_{t}=256$, $t=1$

Figure 2:  Evolution of $|\sigma(x,t)|_{F}$.

We observe that until $n_{t}=84$ the norm of the stress increases constantly
in time. Afterwards, between $n_{t}=84$ and $n_{t}=240$, the yield surface is
reached and the norm of the stress stays almost constant. Moreover, until
$n_{t}=240$ the beam is slowly but constantly pulled apart. From $n_{t}=240$
on, the beam is fast pressed together and the norm of the stress shrinks to
almost zero as desired.

(a) $n_{t}=188$, $t\approx 0.7343$

(b) $n_{t}=240$, $t\approx 0.9375$

(c) $n_{t}=252$, $t\approx 0.9844$

(d) $n_{t}=256$, $t=1$

Figure 3:  Zoom to the left part of the beam from Fig. 2.

Fig. 3 shows a zoom to the left Dirichlet boundary. We observe that the
optimal displacement of the Dirichlet boundary is not constant in vertical
direction. Instead there is a slight curvature of the Dirichlet boundary,
i.e., the optimal Dirichlet displacement pulling the beam in horizontal
direction slightly varies in vertical direction during the evolution.

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