# Converse Lyapunov theorems for control systems with unbounded controls

Anna Chiara Lai  A. C, Lai, Dipartimento di Scienze di Base e Applicate per
l’Ingegneria, Sapienza Università di Roma, Via Scarpa 16, 00181, Roma, Italy
Telefax (39)(06)442401836, Telephone (39)(06) 49766555
<EMAIL_ADDRESS>and  Monica Motta  M. Motta, Dipartimento di
Matematica, Università di Padova
Via Trieste, 63, Padova 35121, Italy
Telefax (39)(49) 827 1499, Telephone (39)(49) 827 1368<EMAIL_ADDRESS>
###### Abstract.

In this paper we extend well-known relationships between global asymptotic
controllability, sample stabilizability, and the existence of a control
Lyapunov function to a wide class of control systems with unbounded controls,
which includes control-polynomial systems. In particular, we consider open
loop controls and discontinuous stabilizing feedbacks, which may be unbounded
approaching the target, so that the corresponding trajectories may present a
chattering behaviour. A key point of our results is to prove that global
asymptotic controllability, sample stabilizability, and existence of a control
Lyapunov function for these systems or for an impulsive extension of them are
equivalent.

###### Key words and phrases:

Converse Lyapunov theorem, Asymptotic controllability, Asymptotic
stabilizability, Discontinuous feedback law, Impulsive control systems.

###### 2020 Mathematics Subject Classification:

93B05, 93D15,93D20, 93C10, 93C27

This research is partially supported by the INdAM-GNAMPA Project 2020
“Extended control problems: gap, higher order conditions and Lyapunov
functions”.

##  1\. Introduction

In this paper we extend classic equivalence results between global asymptotic
controllability to a set $\mathcal{C}$, sample stabilizability to
$\mathcal{C}$, and the existence of a control Lyapunov function to a control
system of the form

(1) $\dot{x}(t)=f(x(t),u(t))\ \text{a.e.},\qquad u(t)\in U\ \text{a.e.},$

where the (unbounded) control set $U\subseteq{\mathbb{R}}^{m}$ is a closed
cone, the target set $\mathcal{C}\subset{\mathbb{R}}^{n}$ is closed with
compact boundary, and the function $f:{\mathbb{R}}^{n}\times
U\to{\mathbb{R}}^{n}$ satisfies suitable growth assumptions in the control
variable, which include control-polynomial dependence (see hypothesis (Hg)
below). The extension lies in the fact that, following [LMR16, LM20, LM21], we
consider notions of global asymptotic controllability and sample
stabilizability which involve open loop controls $u\in L^{\infty}_{loc}$ and
locally bounded feedback laws $K:{\mathbb{R}}^{n}\setminus\mathcal{C}\to U$
with possibly $\limsup_{x\to\bar{x}\in\partial\mathcal{C}}|K(x)|=+\infty$,
respectively.

The problems of asymptotic controllability and feedback stabilization to a
point or to a set of control systems that are nonlinear (and for which
linearization fails), and their relationships with the existence of control
Lyapunov functions have been central topics in control theory since the 1980s.
It is now well-known that a smooth control Lyapunov function, which guarantees
the asymptotic controllability of the system, may not exist and a continuous
stabilizing feedback fails in general to exist either (see [Bro83, SS80,
Art83, Son83, Rya94, CR94, SS95]). Under suitable assumptions on the dynamics
function $f$ and on the control set $U$, these problems were solved in
[CLSS97] by the introduction of nonsmooth control Lyapunov functions,
discontinuous feedback laws $K=K(x)$, and a “sample and hold” solution
concept, similar to that used in differential games in [KSK87].

In this context, converse Lyapunov theorems have been established, showing how
global asymptotic controllability (GAC), which is equivalent to sample
stabilizability by a result in [CLSS97], implies the existence of continuous
(see [Son83]), and actually locally Lipschitz and even semiconcave control
Lyapunov functions. The latter property plays a a fundamental role in the the
explicit construction of stabilizing feedback strategies (see [CLRS00, Rif00,
Rif03, MRS04, KT00, KT04]). We have limited ourselves to mentioning only a few
key articles and those most related to the present work. For a broader
overview we refer e.g. to [Cla10, DES11, Kel15, Tsi12, TT16] and references
therein.

Two are the key hypotheses in the above results: (i) $f$ is continuous in
$(x,u)$ and Lipschitz continuous in $x$ on compact subsets of
${\mathbb{R}}^{n}$ (or ‘uniform in distance to the set $\mathcal{C}$’, as in
[KT00]), uniformly with respect to $U$; (ii) the vector field $f(x,u)$
associated to $u=u(t)$ or $u=K(x)$ and steering trajectories of (1) to
$\mathcal{C}$ in a uniform way, is bounded in any neighborhood of the target.
Actually, it is usually assumed that these open-loop and feedback controls are
themselves bounded for states close to $\mathcal{C}$. Our aim is to extend
these results to a wider class of control systems where conditions (i), (ii)
above do not hold. This extension is not achieved by refining the techniques
used in the case of classic assumptions on $f$. Rather, following an approach
commonly adopted in optimal impulsive control, as generalized in [RS00, MS14,
KDOPS14], we embed the original control system into an extended control system
with bounded controls, to which the known results apply. Our main result is
that GAC, sample stabilizability, and existence of a control Lyapunov function
for the extended system or for the original system are all equivalent
properties (see the Converse Lyapunov Theorem 4.3). These relationships
between the original control system and its impulsive extension are relevant
in themselves. Indeed, on the one hand, it is not obvious a priori that
asymptotic controllability or stabilizability to the target by means of
impulsive inputs guarantees controllability and stabilizability of (1) to
$\mathcal{C}$ over unbounded controls, since trajectories of the impulsive
extension may not be approximated by trajectories of (1) with the same
endpoint. On the other hand, the explicit construction of control strategies
for the (impulsive) extended system, which is usually simpler, can be used to
build a stabilizing feedback also for the original control system (1), as
described in Section 4. More in detail, the main hypotheses we will consider
on $f$, are: (Hg) there exists a strictly increasing, bijective function
$\nu:[0,+\infty)\to[0,+\infty)$, to which we refer to as growth rate, such
that

(i) the function ${\bar{f}}:({\mathbb{R}}^{n}\setminus\mathcal{C})\times
U\to{\mathbb{R}}^{n}$, defined by
$\displaystyle\bar{f}(x,u):=\frac{{f}(x,u)}{1+\nu(|u|)},$ is uniformly
continuous on ${\mathcal{K}}\times U$ for any compact set
${\mathcal{K}}\subset{\mathbb{R}}^{n}\setminus\mathcal{C}$ and bounded on
$(B_{R}(\mathcal{C})\setminus\mathcal{C})\times U$ for any $R>0$;

(ii) the function
$F:\overline{({\mathbb{R}}^{n}\setminus\mathcal{C})}\times{[0,+\infty)\times
U}\to{\mathbb{R}}^{n}$, defined as

$F(x,w_{0},w):=\lim_{r\to
w_{0}^{+}}{\bar{f}}\left(x,\frac{w}{|w|}\,\nu^{-1}\left(\frac{|w|}{r}\right)\right)=\lim_{r\to
w_{0}^{+}}f\left(x,\frac{w}{|w|}\,\nu^{-1}\left(\frac{|w|}{r}\right)\right)r\
$

is not identically zero, it is continuous, and locally bounded on
$\overline{({\mathbb{R}}^{n}\setminus\mathcal{C})}\times\overline{{\mathbb{U}}}$,
where ${\mathbb{U}}:=\\{(w_{0},w)\in(0,+\infty)\times U:\ \ w_{0}+|w|=1\\}.$
The function $\nu$ represents the maximal growth of $f$ in the control $u$. We
will refer to $\bar{f}$ and $F$ as rescaled dynamics and extended dynamics
function, respectively. This extension consists essentially in a control-
compactification, obtained by adding the scalar control $w_{0}$, so that the
pairs $(w_{0},w)=(0,w)$ with $|w|=1$ represent the points of $U$ ‘at
infinity’. Observe that control-polynomial systems of degree $d$ with
continuous coefficients satisfy hypothesis (Hg) with growth rate
$\nu(r)=r^{d}$. Under assumption (Hg) and some standard Lipschitz continuity
hypotheses on the extended dynamics $F$ specified in Section 4, we establish
our Converse Lyapunov Theorem. The proof of this theorem relies on three key
results: Theorem 2.5, dealing with two equivalent notions of GAC for unbounded
dynamics, which is crucial in order to prove that GAC of $\dot{x}=f(x,u)$ and
GAC of the rescaled control system $\dot{x}={\bar{f}}(x,u)$ are equivalent
(see Theorems 3.1, 3.2); the interplay between the rescaled and the extended
control system (see Propositions 3.1, 3.1); and the fact that sample
stabilizability implies GAC also for dynamics which are merely continuous on
${\mathbb{R}}^{n}\setminus\mathcal{C}$ (see Theorem 2.9).

Let us point out that there are interesting situations in which considering
bounded controls (and dynamics) the system is not asymptotically controllable
and not stabilizable, whereas it becomes so if dynamics that can become
unbounded when approaching the target are allowed (see the example in Section
5). This is the case, for instance, of some applications to Lagrangian
mechanics where part of the coordinates act as controls. The evolution of the
remaining coordinates is then described by an “impulsive” control system,
where the dynamics function is linearly or quadratically dependent on the
derivatives of the controlled coordinates, derivatives which are identifiable
with unbounded controls (see [Bre90, Mar91], and [BR10] with references
therein). In particular, in [BR10] the authors exhibit mechanical examples for
which stabilization can only be achieved by “vibrating controls”, namely
allowing unbounded inputs. The paper is organized as follows. In Section 2 we
introduce the notions of GAC and GAC with $U\cap\sigma$ controls, prove that
they are equivalent, and show that sample stabilizability implies GAC. Section
3 is devoted to establish some relationships between the rescaled and the
extended system. In Section 4 we prove the converse Lyapunov theorem and
describe how related explicit feedback constructions for the original and for
the extended control system can be implemented. In Section 5, an example
concludes the paper.

###  1.1. Notations

For $a,b\in{\mathbb{R}}$, we set $a\vee b:=\max\\{a,b\\}$, $a\land
b:=\min\\{a,b\\}$. Let $\Omega\subseteq{\mathbb{R}}^{N}$ for some integer
$N\geq 1$ be a nonempty set. For every $r\geq 0$, we set
$B_{r}(\Omega):=\\{x\in{\mathbb{R}}^{n}:\ d(x,\Omega)\leq r\\}$, where $d$ is
the usual Euclidean distance. We use $\overline{\Omega}$, $\partial\Omega$,
and $\mathring{\Omega}$ to denote the closure, the boundary, and the interior
of $\Omega$, respectively. For any interval $I\subseteq{\mathbb{R}}$,
$L^{\infty}(I,\Omega)$, $AC(I,\Omega)$ are the sets of functions
$x:I\to\Omega$, which are essentially bounded or absolutely continuous,
respectively, on $I$. We use $L^{\infty}_{loc}(I,\Omega)$,
$AC_{loc}(I,\Omega)$ to denote the sets of functions $x:I\to\Omega$, which are
essentially bounded or absolutely continuous on any compact subset $J\subset
I$. When no confusion may arise, we simply write $L^{\infty}(I)$, $AC(I)$,
$L^{\infty}_{loc}(I)$, $AC_{loc}(I)$.

##  2\. Global asymptotic controllability and sample stabilizability

We introduce two concepts of global asymptotic controllability and prove that
they are equivalent. Furthermore, we show that sample stabilizability implies
global asymptotic controllability, as in the case of bounded controls. Unless
otherwise specified, we assume $f:({\mathbb{R}}^{n}\setminus\mathcal{C})\times
U\to{\mathbb{R}}^{n}$ continuous. Let us set
$\mathbf{d}(x):=d(x,\mathcal{C})$.

###  2.1. Equivalent concepts of global asymptotic controllability

######  Definition 2.1 (Admissible trajectory-control pair).

A couple $(x,u)$ is called an admissible trajectory-control pair for $(f,U)$
if there exists $T_{x}\leq+\infty$ such that: the control $u$ belongs to
$L^{\infty}_{loc}([0,T_{x}),U)$; the trajectory $x\in
AC_{loc}([0,T_{x}),{\mathbb{R}}^{n}\setminus\mathcal{C})$ verifies

(2) $\dot{x}(t)=f(x(t),u(t))\qquad\text{a.e. $t\in[0,T_{x})$;}$

and, if $T_{x}<+\infty$, one has $\lim_{t\to T_{x}^{-}}\mathbf{d}(x(t))=0$. If
$(x,u)$ is an admissible trajectory-control pair for $(f,U)$ and
$T_{x}<+\infty$, we extend $x$ to $[0,+\infty)$ by setting $x(t):=\bar{z}$ for
any $t\geq T_{x}$, where $\bar{z}$ is an arbitrary point of the set

$\mathcal{C}_{x}:=\big{\\{}\zeta\in\partial\mathcal{C}\text{ :}\
\exists\tau_{i}\uparrow T_{x}\,\,\,\text{as $i\to+\infty$ and
}\lim_{i\to+\infty}x(\tau_{i})=\xi\big{\\}}.$

When no confusion may arise, we will simply call admissible trajectory-control
pair any admissible trajectory-control pair for $(f,U)$.

As customary, we use ${\mathcal{KL}}$ to denote the set of all continuous
functions $\beta:[0,+\infty)\times[0,+\infty)\to[0,+\infty)$ such that: (1)
$\beta(0,t)=0$ and $\beta(\cdot,t)$ is strictly increasing and unbounded for
each $t\geq 0$; (2) $\beta(r,\cdot)$ is strictly decreasing for each $r\geq
0$; (3) $\beta(r,t)\to 0$ as $t\to+\infty$ for each $r\geq 0$. We refer to any
function $\beta\in{\mathcal{KL}}$ as a descent rate.

######  Definition 2.2 (GAC).

The system (2) is called Globally Asymptotically Controllable (GAC) to
$\mathcal{C}$ if there exists a function $\beta\in\mathcal{K}\mathcal{L}$ such
that for any initial point $z\in{\mathbb{R}}^{n}\setminus\mathcal{C}$ there is
an admissible trajectory-control pair $(x,u)$ for $(f,U)$ with $x(0)=z$, such
that

(3) $\mathbf{d}(x(t))\leq\beta(\mathbf{d}(z),t)\quad\forall t\geq 0.$

######  Definition 2.3 (GAC with $U\cap\sigma$ controls).

Let $\sigma:(0,+\infty)\to(0,+\infty)$ be a continuous function. We say that
the system (2) is Globally Asymptotically Controllable (GAC) to $\mathcal{C}$
with $U\cap\sigma$ controls if there exists a descent rate
$\beta\in\mathcal{K}\mathcal{L}$ such that for any initial point
$z\in{\mathbb{R}}^{n}\setminus\mathcal{C}$ there is an admissible trajectory-
control pair $(x,u)$ for $(f,U)$ with $x(0)=z$, such that

$\mathbf{d}(x(t))\leq\beta(\mathbf{d}(z),t)\quad\forall t\geq 0$

and

(4) $|u(t)|\leq\sigma(\mathbf{d}(x(t))\quad\text{for a.e. $t\in[0,T_{x})$}.$

######  Remark 2.4.

The concept of GAC with $U\cap\sigma$ controls introduced above only
apparently coincides with the definition considered, e.g., in the survey
papers [Son99, KT00]. In fact, in the previous literature the function
$\sigma$ was supposed to be increasing, in order to prevent unbounded inputs
around the target. On the contrary, in Definition 2.3 it may happen that
$\lim_{r\to 0^{+}}\sigma(r)=+\infty$, thus allowing for controls with
$L^{\infty}$ norm diverging to $+\infty$ as the trajectory approaches
$\mathcal{C}$.

The concept of GAC with $U\cap\sigma$ controls, although apparently stronger
than GAC, when $f$ is locally Lipschitz continuous in $x$, is equivalent to
GAC. Precisley, let us consider the following hypothesis: (Hl) the function
$f:({\mathbb{R}}^{n}\setminus\mathcal{C})\times U\to{\mathbb{R}}^{n}$ is
continuous and, for every pair of compact sets
${\mathcal{K}}\subset{\mathbb{R}}^{n}\setminus\mathcal{C}$, $U_{1}\subset U$,
there is some constant $L>0$ such that

$|f(x,u)-f(y,u)|\leq L\,|x-y|\qquad\forall x,y\in{\mathcal{K}},\ \forall u\in
U_{1}.$

######  Theorem 2.5.

Let $f$ satisfy (Hl). Then, system (2) is GAC to $\mathcal{C}$ if and only if
it is GAC to $\mathcal{C}$ with $U\cap\sigma$ controls.

###### Proof.

If system (2) is GAC to $\mathcal{C}$ with $U\cap\sigma$, it is trivially GAC
to $\mathcal{C}$. So, let us assume that (2) is GAC to $\mathcal{C}$ and prove
that it is GAC with $U\cap\sigma$ controls by building a continuous positive
function $\sigma$ and a descent rate $\bar{\beta}$. Step 1.
($\beta$-admissible trajectory-control pairs) Without loss of generality, in
the definition of GAC we can assume the inequality in (3) strict, namely, that
there exists a descent rate $\beta\in\mathcal{KL}$ such that for all
$z\in{\mathbb{R}}^{n}\setminus\mathcal{C}$ there is some admissible
trajectory-control pair $(x,u)$ with $x(0)=z$, such that

(5) $\mathbf{d}(x(t))<\beta(\mathbf{d}(z),t)\qquad\forall t\geq 0.$

We refer to such $(x,u)$ as a $\beta$-_admissible trajectory-control pair from
$z$_. Let us define the set $\mathcal{A}_{\beta}$, given by the triplets
$(x,u,z)$, where $z\in{\mathbb{R}}^{n}\setminus\mathcal{C}$ and, for any $z$,
we select one $(x,u)$ among the admissible trajectory-control pairs from $z$.
Note that, by (5), one has

(6) $\beta(R,0)>R\quad\forall R>0.$

Step 2. ($\beta$-strips) Let $r_{0}:=1$ and recursively define
$(r_{i})_{i\in{\mathbb{Z}}}$ by

$r_{i-1}=\beta(r_{i},0)\qquad i\in{\mathbb{Z}},$

so that, for instance, $r_{1}$ is the solution of $\beta(r_{1},0)=1=r_{0}$ and
$r_{-1}=\beta(r_{0},0)$. By (6) and by the definition of $\mathcal{KL}$
functions, we have that $(r_{i})_{i\in{\mathbb{Z}}}$ is strictly decreasing,
positive and

$\lim_{i\to+\infty}r_{i}=0\quad\text{ and
}\quad\lim_{i\to-\infty}r_{i}=+\infty.$

For every $i\in{\mathbb{Z}}$, set
$\mathcal{B}_{i}:=\\{z\in{\mathbb{R}}^{n}\setminus\mathcal{C}:\
\mathbf{d}(z)\in[r_{i},r_{i-1}]\\}$. We define the _$i$ -th $\beta$-strip_ as
the set $\mathcal{A}^{i}_{\beta}:=\\{(x,u,z)\in\mathcal{A}_{\beta}:\ \
z\in\mathcal{B}_{i}\\}$. Note that for all $i\in{\mathbb{Z}}$,
$r_{i-2}=\beta(r_{i-1},0)$, therefore, for every
$(x,u,z)\in\mathcal{A}^{i}_{\beta}$ one has $\mathbf{d}(x(t))<r_{i-2}$ for all
$t\geq 0$.

Fix $i\in{\mathbb{Z}}$ and consider a triplet
$(x,u,z)\in\mathcal{A}^{i}_{\beta}$. Define

(7) $T_{i,z}:=\inf\left\\{t\geq 0:\
\mathbf{d}(x(t))=\frac{r_{i}+r_{i+1}}{2}\right\\}.$

Clearly, $0<T_{i,z}<T_{x}$ and the fact that $u\in
L_{loc}^{\infty}([0,T_{x}))$ implies

(8) $\|u\|_{L^{\infty}([0,T_{i,z}])}<+\infty.$

Set

$\tilde{\varepsilon}_{i,z}:=\inf\left\\{\frac{1}{2}(\beta(\mathbf{d}(z),t)-\mathbf{d}(x(t)):\
t\in[0,T_{i,z}]\right\\}.$

Note that, by the continuity of $\beta$ and $x$, $\tilde{\varepsilon}_{i,z}$
is actually a minimum. Furthermore, $\tilde{\varepsilon}$ is positive in view
of (5). Define

$\bar{\varepsilon}_{i}:=\frac{r_{i}-r_{i+1}}{4},\qquad\varepsilon_{i,z}:=\min\\{\tilde{\varepsilon}_{i,z},\bar{\varepsilon}_{i}\\}.$

In view of the Lipschitzianity hypothesis (Hl), there exists $\delta_{i,z}>0$
such that, for all $\bar{z}\in{\mathbb{R}}^{n}\setminus\mathcal{C}$ verifying
$|z-\bar{z}|<\delta_{i,z}$, the Cauchy problem $\dot{x}=f(x,u)$,
$x(0)=\bar{z}$, admits a unique solution, denoted in the following by
$x(\cdot\,;u,\bar{z})$, which is defined on the whole interval $[0,T_{i,z}]$
and verifies

(9) $|x(t)-x(t;u,\bar{z})|<\varepsilon_{i,z}\quad\forall t\in[0,T_{i,z}].$

From the definition of $\varepsilon_{i,z}$, it follows that

$\begin{array}[]{l}\displaystyle\mathbf{d}(x(t;u,\bar{z}))<\mathbf{d}(x(t))+\frac{1}{2}(\beta(\mathbf{d}(z),t)-\mathbf{d}(x(t))<\beta(\mathbf{d}(z),t)\\\
\ \quad\qquad\qquad\leq\beta(r_{i-1},0)=r_{i-2}\quad\forall
t\in[0,T_{i,z}],\end{array}$

while the definition of $T_{i,z}$ yields

$\mathbf{d}(x(t;u,\bar{z}))\geq\mathbf{d}(x(T_{i,z}))-\tilde{\varepsilon}_{i}>r_{i+1}\quad\forall
t\in[0,T_{i,z}].$

In conclusion, for any $(x,u,z)\in\mathcal{A}^{i}_{\beta}$ there exists some
$\delta_{i,z}>0$ such that, for all
$\bar{z}\in{\mathbb{R}}^{n}\setminus\mathcal{C}$ verifying
$|z-\bar{z}|<\delta_{i,z}$, one obtains

(10) $\begin{cases}\mathbf{d}(x(t;u,\bar{z}))\in(r_{i+1},r_{i-2})&\forall
t\in[0,T_{i,z}],\\\ \bar{x}(T_{i,z})\in{\mathcal{B}}_{i+1}.\end{cases}$

Step 3. (Construction of $\sigma$ and $\bar{\beta}$) Fix $i\in{\mathbb{Z}}$
and consider the cover of $\mathcal{B}_{i}$ given by the collection of open
balls $\mathring{B}_{\delta_{i,z}}(\\{z\\})$, with $z\in\mathcal{B}_{i}$.
Since $\partial\mathcal{C}$ is compact, then the $i$-th strip
$\mathcal{B}_{i}$ is compact, as well, and consequently it admits a finite
subcover $\\{\mathring{B}_{\delta_{i,z}}(\\{z\\})\\}_{z\in\mathcal{Z}_{i}}$,
where $\mathcal{Z}_{i}$ is a finite subset of $\mathcal{B}_{i}$. Now, define

(11) $\bar{\sigma}(i):=\max\\{\|u\|_{L^{\infty}([0,T_{i,z}])}:\ \
(x,u,z)\in\mathcal{A}^{i}_{\beta},\ \ z\in\mathcal{Z}_{i}\\},$

where $T_{i,z}$ is as in (7), and $\sigma:(0,+\infty)\to(0,+\infty)$, given by

$\sigma(r):=\max\\{\bar{\sigma}(i-i),\bar{\sigma}(i),\bar{\sigma}(i+1)\\}\qquad\forall
r\in[r_{i},r_{i-1}).$

Note that, for every $i\in{\mathbb{Z}}$, one has

(12) $\sigma(r)\geq\bar{\sigma}(i)\qquad\forall r\in[r_{i+1},r_{i-2}).$

To build a new descent rate function, (to be associated to controls $u$ such
that $|u(t)|\leq\sigma(\mathbf{d}(x(t)))$ for every $t\geq 0$), set

(13) $T_{i}:=\max\\{T_{i,z}:\ z\in\mathcal{Z}_{i}\\}.$

Replacing any time $T_{i}$ with a larger value if necessary, we can assume
that, for every $i\in{\mathbb{Z}}$, one has
$\sum_{j=0}^{+\infty}T_{i+j}=+\infty$. Then, for every $i\in{\mathbb{Z}}$ and
$N\in{\mathbb{N}}$, define

(14) $\bar{T}_{i,-1}:=0,\qquad\bar{T}_{i,N}:=\sum_{j=0}^{N}T_{i+j}.$

Note that $\bar{T}_{i,N}\to+\infty$ as $N\to\infty$ for every fixed
$i\in{\mathbb{Z}}$. Consider the piecewise constant function
$b:[0,+\infty)\times[0,+\infty)\to[0,+\infty)$, given by

$\left\\{\begin{split}&b(R,t):=r_{i+N-2}\quad\text{if
}R\in[r_{i},r_{i-1})\text{ and }t\in[\bar{T}_{i,N-1},\bar{T}_{i,N}),\\\
&b(0,t)=0\qquad\qquad\forall t\geq 0,\end{split}\right.$

for all $i\in{\mathbb{Z}}$ and $N\in{\mathbb{N}}$. To make notation more
compact, we introduce the decreasing, integer valued function
$i(R):=i\in{\mathbb{Z}}$, such that $R\in[r_{i},r_{i-1})$. Note that
$i(R)\to+\infty$ as $R\to 0$ and $i(R)\to-\infty$ as $R\to+\infty$. We then
rewrite the definition of $b$ as follows

$b(R,t):=r_{i(R)+N-2}\quad\text{ if
}t\in[\bar{T}_{i(R),N-1},\bar{T}_{i(R),N}),$

for $N\in{\mathbb{N}}$. Since $(r_{i})$ vanishes as $i\to+\infty$, for all
$R\in(0,+\infty)$, $b(R,t)\to 0$ as $t\to+\infty$. Moreover, since
$r_{i(R)}\to+\infty$ as $R\to+\infty$, for every $t\geq 0$, we have that
$b(R,t)\to+\infty$ as $R\to+\infty$. Then $b$ can be dominated by some
$\mathcal{KL}$ function, (say, a larger, continuous linear interpolation) that
we call $\bar{\beta}$.

Step 4. (GAC with $U\cap\sigma$ controls) To conclude, for every initial datum
$\bar{z}\in{\mathbb{R}}^{n}\setminus\mathcal{C}$ we need to provide a
$\bar{\beta}$-admissible trajectory control pair $(\bar{x},\bar{u})$ from
$\bar{z}$, satisfying $|\bar{u}(t)|\leq\sigma(\mathbf{d}(\bar{x}(t))$ for all
$t\geq 0$.

To this aim, fix $\bar{z}\in{\mathbb{R}}^{n}\setminus\mathcal{C}$ and, for
brevity, set $i:=i(\mathbf{d}(\bar{z}))$, so that $\bar{z}\in\mathcal{B}_{i}$.
By Step 3 it follows that
$\bar{z}\in\mathring{B}_{\delta_{i,z_{0}}}(\\{z_{0}\\})$ for some
$z_{0}\in\mathcal{Z}_{i}$ and, if
$(x_{0},u_{0},z_{0})\in\mathcal{A}^{i}_{\beta}$ (taking into account also the
inequality (12)), there exists $\hat{t}_{0}:=T_{i,z_{0}}\leq T_{i}$ such that
the trajectory $\bar{x}:=x(t;u_{0},\bar{z})$, satisfies

(15) $\displaystyle\mathbf{d}(\bar{x}(t))\in(r_{i+1},r_{i-2})\quad\forall
t\in[0,\hat{t}_{0}],$ (16)
$\displaystyle\bar{x}(\hat{t}_{0}))\in\mathcal{B}_{i+1},$ (17)
$\displaystyle|u_{0}(t)|\leq\bar{\sigma}(i)\leq\sigma(\mathbf{d}(\bar{x}(t)))\quad\forall
t\in[0,\hat{t}_{0}].$

By repeating the same argument starting from the point
$\bar{z}_{1}:=\bar{x}(\hat{t}_{0})\in\mathcal{B}_{i+1}$, one obtains a time
$\hat{t}_{1}\leq T_{i+1}$ and a control $u_{1}\in
L^{\infty}([0,\hat{t}_{1}],U)$ which satisfy an updated version of (15)-(17),
with $i$ replaced by $i+1$. Iterating this procedure, one gets a sequence of
times $(\hat{t}_{N})_{N}$ and a sequence of controls $(u_{N})$, such that
$\hat{t}_{N}\leq T_{i+N}$ and $u_{N}\in L^{\infty}([0,\hat{t}_{N}],U)$ for
every $N$. Therefore, setting for every $N\in{\mathbb{N}}$,

$\begin{array}[]{l}\hat{T}_{-1}:=0,\quad\hat{T}_{N}:=\sum_{j=0}^{N}\,\hat{t}_{j},\quad\hat{T}_{\infty}:=\sum_{j=0}^{+\infty}\,\hat{t}_{j},\\\
\hat{u}(t):=u_{N}(t-{\hat{T}_{N-1}})\quad\forall{t\in(\hat{T}_{N-1},\hat{T}_{N}]},\end{array}$

one obtains a control $\hat{u}\in L_{loc}^{\infty}([0,\hat{T}_{\infty}),U)$
such that the corresponding trajectory $\hat{x}:=x(t;\hat{u},\bar{z})$ is
defined on the whole interval $[0,\hat{T}_{\infty})$ and enjoys the following
properties:

(18) $\displaystyle\mathbf{d}(\hat{x}(t))\in(r_{i+N+1},r_{i+N-2})\quad\forall
t\in[\hat{T}_{N-1},\hat{T}_{N}],$ (19)
$\displaystyle\hat{x}(\hat{T}_{N})\in\mathcal{B}_{i+N+1},$ (20)
$\displaystyle|\hat{u}(t)|\leq\sigma(\mathbf{d}(\hat{x}(t)))\quad\forall
t\in[0,\hat{T}_{N}].$

In particular, $\mathbf{d}(\hat{x}(t))\to 0$ as $t\to\hat{T}_{\infty}^{-}$.

At this point, a simple inductive argument shows that

$\mathbf{d}(\hat{x}(t))<r_{i+N-1}\quad\forall t\geq\hat{T}_{N}.$

Furthermore, for every fixed $N$, $\hat{T}_{N}$, which depends on $\bar{z}$,
is bounded above by a constant which depends only on $\mathbf{d}(\bar{z})$.
Indeed, by construction, $\hat{T}_{N}\leq\bar{T}_{i,N}$, for all $N\geq 0$.
Hence, one finally obtains that

$\mathbf{d}(\hat{x}(t))<r_{i(\mathbf{d}(z))+N-2}=b(\mathbf{d}(\bar{z}),t)\leq\bar{\beta}(\mathbf{d}(\bar{z}),t),\
t\in[\bar{T}_{i(\mathbf{d}(z)),N-1},\bar{T}_{i(\mathbf{d}(z)),N}).$

Since $\bar{T}_{i(\mathbf{d}(\bar{z})),N}\to+\infty$ as $N\to\infty$, this
concludes the proof. ∎

###  2.2. Sample stabilizability

The feedback counterpart of a notion of GAC which involves admissible
trajectory-control pairs $(x,u)$ with controls $u$ in
$L^{\infty}_{loc}([0,T_{x}),U)$, requires necessarily to consider locally
bounded feedback functions $K:{\mathbb{R}}^{n}\setminus\mathcal{C}\to U$,
which may have
$\displaystyle\limsup_{x\to\bar{x}\in\partial\mathcal{C}}|K(x)|=+\infty.$

In correspondence of such feedbacks, following [LM20] we adopt the notion of
sample stabilizability below. A partition (of $[0,+\infty)$) is a sequence
$\pi=(t_{k})$ such that $t_{0}=0,\quad t_{k-1}<t_{k}$ $\forall k\geq 1$, and
$\lim_{k\to+\infty}t_{k}=+\infty$. The value diam$(\pi):=\sup_{k\geq
1}(t_{k}-t_{k-1})$ will be called the diameter or the sampling time of the
partition $\pi$.

######  Definition 2.6 (Sampling trajectory-control pair).

Given a locally bounded feedback ${K}:{\mathbb{R}}^{n}\setminus\mathcal{C}\to
U$, a partition $\pi=(t_{k})$, and a point
$z\in{\mathbb{R}}^{n}\setminus\mathcal{C}$, we call $\pi$-sampling trajectory-
control pair for $\dot{x}=f(x,u)$ from $z$, a pair $(x,u)$, where the sampling
trajectory $x$ is a continuous function defined by recursively solving

$\dot{x}={f}(x(t),{K}(x(t_{k-1})))\qquad\text{a.e.
}t\in[t_{k-1},t_{k}],~{}(x(t)\in{\mathbb{R}}^{n}\setminus\mathcal{C})$

from the initial time $t_{k-1}$ up to time

$\tau_{k}:=t_{k-1}\vee\sup\\{\tau\in[t_{k-1},t_{k}]:\ x\mathrm{\
is\,defined\,on}\,[t_{k-1},\tau)\\},$

such that $x(t_{0})=x(0)=z$. In this case, the trajectory $x$ is defined on
the right-open interval from time zero up to time $T^{-}:=\inf\\{\tau_{k}:\
\tau_{k}<t_{k}\\}$. Accordingly, for every $k\geq 1$ and for all
$t\in[t_{k-1},t_{k})\cap[0,T^{-})$, the sampling control is defined as

(21) $u(t):={K}(x(t_{k-1}))\quad\forall t\in[t_{k-1},t_{k})\cap[0,T^{-}),\quad
k\geq 1.$

If $T^{-}=T_{x}<+\infty$ such that $\lim_{t\to T^{-}_{x}}\mathbf{d}(x(t))\to
0$, we extend $x$ to $[0,+\infty)$ as described in Definition 2.1.

######  Definition 2.7 (Sample stabilizability).

A locally bounded feedback ${K}:{\mathbb{R}}^{n}\setminus\mathcal{C}\to U$ is
said to sample stabilize the control system $\dot{x}={f}(x,u)$ to
$\mathcal{C}$ if there is a descent rate $\beta\in{\mathcal{KL}}$ satisfying
the following: for each pair $0<r<R$ there exists $\delta=\delta(R,r)>0$, such
that, for every partition $\pi$ with $\text{diam}(\pi)\leq\delta$ and for any
$z\in{\mathbb{R}}^{n}\setminus\mathcal{C}$ such that $\mathbf{d}(z)\leq R$,
any $\pi$-sampling trajectory-control pair $(x,u)$ with $x(0)=z$ is admissible
and verifies:

(22) $\mathbf{d}(x(t))\leq\max\\{\beta(\mathbf{d}(z),t),r\\}\qquad\forall
t\in[0,+\infty).$

We call $\dot{x}={f}(x,u)$ sample stabilizable to $\mathcal{C}$ if there is a
feedback ${K}$ as above.

######  Remark 2.8.

Given a discontinuous feedback $K$, in this paper we only consider sampling
trajectories, which are classical solutions corresponding to piecewise
constant controls. We just point out that, because of the mere continuity of
$f$ and the unboundedness of $K$, sampling trajectories can have a finite
blow-up time and chattering phenomena may occur. As a consequence, classical
Euler solutions –defined in [CLRS00] as uniform limits of sampling solutions–
may not exist. For this reason, in [LM20] (see also [LM21]) we proposed a
notion of _weak_ Euler solution, given by the pointwise limit of a sequence of
suitably truncated sampling trajectories. In particular, in [LM20] it has been
shown that sample stabilizability implies weak Euler stabilizability.

The main result of this subsection is:

######  Theorem 2.9.

Let $f:({\mathbb{R}}^{n}\setminus\mathcal{C})\times U\to{\mathbb{R}}^{n}$ be
continuous. Then, if the control system $\dot{x}=f(x,u)$ is sample
stabilizable to $\mathcal{C}$, it is GAC to $\mathcal{C}$.

###### Proof.

Assume that $\dot{x}={f}(x,u)$ is sample stabilizable to $\mathcal{C}$. Let
$K:{\mathbb{R}}^{n}\setminus\mathcal{C}\to U$ be a locally bounded, sample
stabilizing feedback, let $\beta\in{\mathcal{KL}}$ be an associated descent
rate, and, for any $r,R\in(0,+\infty)$ with $r<R$, let $\delta(R,r)>0$ be as
in Definition 2.7.

Consider the sequence $(r_{i})_{i\in{\mathbb{Z}}}$ introduced in Step 2 of the
proof of Theorem 2.5, defined in a recursive way by setting

$r_{0}:=1,\qquad r_{i-1}=\beta(r_{i},0)\quad\forall i\in{\mathbb{Z}}.$

As already observed, this sequence is positive, strictly decreasing, and
satisfies $\displaystyle\lim_{i\to-\infty}r_{i}=+\infty$,
$\displaystyle\lim_{i\to+\infty}r_{i}=0$. For every $i\in{\mathbb{Z}}$, let
$\mathcal{B}_{i}:=\\{z\in{\mathbb{R}}^{n}\setminus\mathcal{C}:\
\mathbf{d}(z)\in(r_{i},r_{i-1}]\\}$ and choose a positive sequence
$(\hat{t}_{i})_{i\in\mathbb{Z}}$ such that $\beta(r_{i-1},\hat{t}_{i})\leq
r_{i}$ for all $i\in\mathbb{Z}$, and also satisfying

$\sum_{n=0}^{+\infty}\hat{t}_{i+n}=+\infty.$

For each $i\in{\mathbb{Z}}$, set $\hat{T}_{i,-1}:=0$ and
$\hat{T}_{i,N}:=\sum_{n=0}^{N}\hat{t}_{i+n}$ for any $N\in{\mathbb{N}}$.
Hence, define the piecewise constant function
$b:[0,+\infty)\times[0,+\infty)\to[0,+\infty)$, given by

$b(R,t):=\begin{cases}r_{i-2+N}&\text{if
}R\in[r_{i},r_{i-1}),t\in[\hat{T}_{i,N-1},\hat{T}_{i,N})\\\ 0&\text{if
}R=0,t\geq 0,\end{cases}$

for all $i\in{\mathbb{Z}}$ and $N\in{\mathbb{N}}$. Note that if
$t\in[0,\hat{t}_{i})\subseteq[\hat{T}_{i,-1},\hat{T}_{i,0})$ then

(23) $b(r_{i},t)=r_{i-2}=\beta(r_{i-1},0)\geq\beta(r_{i-1},t),$

for all $i\in{\mathbb{Z}}$. As observed in the proof of Theorem 2.5, we can
approximate this function $b$ with a $\mathcal{K}\mathcal{L}$ function
$\bar{\beta}$ such that $\bar{\beta}(R,t)\geq b(R,t)$ for all
$(R,t)\in[0,+\infty)\times[0,+\infty)$.

Now, fixed $i\in{\mathbb{Z}}$, define $\delta_{i}:=\delta(r_{i},r_{i-1})$ and
consider the partition $\pi_{i}:=(t_{i,k})_{k\geq 0}$ where
$t_{i,k}:=k\delta_{i}$. For any $z\in\mathcal{B}_{i}$, select a $\pi_{i}$
sampling-trajectory $x_{i}(t;z)$ from $z$ associated to the sample stabilizing
feedback $K$. Then, by the above definitions,

$\mathbf{d}(x_{i}(t;z))\leq\min\\{\beta(\mathbf{d}(z),t),r_{i}\\}\quad\text{for
}t\geq 0.$

In particular, also in view of (23), we have

(24) $\mathbf{d}(x_{i}(t;z))\leq\beta(r_{i-1},t)\leq
b(r_{i},t)\leq\bar{\beta}(r_{i},t)\leq\bar{\beta}(\mathbf{d}(z),t)\ \text{for
}t\in[0,\hat{t}_{i}].$

and, since $r_{i}\geq\beta(r_{i-1},t)$ for all $t\geq\hat{t}_{i}$,

(25) $\mathbf{d}(x_{i}(t;z))\leq r_{i}\quad\text{for }t\geq\hat{t}_{i}.$

Let $u_{i}(t)$ be the sampling control associated to $x_{i}(t;z)$.

Consider the map $i:{\mathbb{R}}^{n}\setminus\mathcal{C}\to{\mathbb{Z}}$,
defined as $i(z):=i$ whenever $z\in\mathcal{B}_{i}$. Fix
$z\in{\mathbb{R}}^{n}\setminus\mathcal{C}$. Let us build in a recursive way an
increasing sequence of times $\\{T_{N}\\}_{N\geq 0}$ such that
$T_{N}\leq\hat{T}_{i{(z)},N}$ for all $N\geq 0$, and a trajectory-control pair
$(x,u)$ defined in $[0,T_{N}]$ such that

$\mathbf{d}(x(t))\leq\bar{\beta}(\mathbf{d}(z),t)\qquad\forall
t\in[0,T_{N}],\qquad\mathbf{d}(x(T_{N}))=r_{i(z)+N}.$

Precisely, for $N=0$, we define $T_{0}:=\inf\\{t>0:\
x_{i(z)}(t;z)\in\mathcal{B}_{i(z)+1}\\}$. Note that, in view of (25),
$T_{0}\leq\hat{t}_{i(z)}=\hat{T}_{i(z),0}$. Set

$x(t):=x_{i(z)}(t;z),\qquad u(t):=u_{i(z)}(t)\qquad t\in[0,{T_{0}}].$

From (24) and from the definition of $T_{0}$ we derive that

$\mathbf{d}(x(t))\leq\bar{\beta}(\mathbf{d}(z),t)\qquad\forall
t\in[0,T_{0}];\qquad\mathbf{d}(x(T_{0}))=r_{i(z)}.$

Let now $N>0$ and let be defined $T_{0},\dots,T_{N-1}$ (satisfying
$T_{n}\leq\hat{T}_{i(z),n}$ for all $n=0,\dots,N-1$) and a trajectory-control
pair $(x,u)$ on $[0,T_{N-1}]$ satisfying

$\mathbf{d}(x(t))\leq\bar{\beta}(\mathbf{d}(z),t),\qquad\forall
t\in[0,T_{N-1}],\qquad\mathbf{d}(x(T_{N-1}))=r_{i(z)+N-1}.$

Set $z_{N}:=x(T_{N-1})$ and observe that $z_{N}\in\mathcal{B}_{i(z)+N}$.
Define ${t_{N}}:=\inf\\{t>0:\
x_{i(z)+N}(t;z_{N})\in\mathcal{B}_{i_{0}+N+1}\\}$. Then, in view of (25),
${t_{N}}\leq\hat{t}_{i(z)+N}$. Set $T_{N}:=T_{N-1}+t_{N}$ and note that
$T_{N}\leq\hat{T}_{i(z),N}$. We extend the definition of $(x,u)$ to
$(T_{N-1},T_{N}]$ as follows

$x(t)=x_{i(z)+N}(t-T_{N-1};z_{N}),\quad u(t)=u_{i(z)+N}(t-T_{N-1}),\quad
t\in(T_{N-1},T_{N}].$

From (24) and (23), we deduce that

(26) $\mathbf{d}(x(t))=\mathbf{d}(x_{i(z)+N}(t-T_{N-1};z_{N}))\leq
r_{i(z)+N-1},\qquad t\in{(T_{N-1},T_{N}]}.$

On the other hand, one has

$r_{i(z)+N-1}=b(\mathbf{d}(z),t)\leq\bar{\beta}(\mathbf{d}(z),t),\qquad
t\in[\hat{T}_{i(z),N-1},\hat{T}_{i(z),N}].$

Since $\bar{\beta}(\mathbf{d}(z),\cdot)$ is decreasing, then
$r_{i(z)+N-1}\leq\bar{\beta}(\mathbf{d}(z),t)$ for all
$t\in[0,\hat{T}_{i(z),N}]$. In particular,
$r_{i(z)+N-1}\leq\bar{\beta}(\mathbf{d}(z),t)$ for all $t\in[0,T_{N}]$,
because $T_{N}\leq\hat{T}_{i(z),N}$. This, together with (26), implies that

(27) $\mathbf{d}(x(t))\leq\bar{\beta}(\mathbf{d}(z),t)\qquad\forall
t\in{(T_{N-1},T_{N}]}.$

Moreover, we have by construction $\mathbf{d}(x(T_{N}))=r_{i(z)+N}$.

So far, we iteratively constructed an admissible trajectory-control pair
$(x,u)$ from $z$, which is defined in $[0,\tilde{T})$, where
$\tilde{T}:=\lim_{N\to\infty}T_{N}$ ($\leq+\infty$),
$\mathbf{d}(x(T_{N}))=r_{i(z)+N}$ for all $N\geq 0$, and such that

$\mathbf{d}(x(t))\leq\bar{\beta}(\mathbf{d}(z),t)\qquad\forall
t\in[0,\tilde{T}).$

Therefore, $\mathbf{d}(x(t))\to 0$ as $t\to\tilde{T}^{-}$ and, extending $x$
to $[0,+\infty)$ as in Definition 2.1 when $\tilde{T}<+\infty$, this yields

$\mathbf{d}(x(t))\leq\bar{\beta}(\mathbf{d}(z),t)\qquad\forall t\geq 0,$

so concluding the proof.

∎

It is worth noting that the results of Theorems 2.5, 2.9 are constructive, in
the sense that, given a decrease rate $\beta$ associated with the GAC with
$U\cap\sigma$ controls or the sample stabilizability, respectively, we
explicitly indicate how to obtain a decrease rate for the GAC. In addition,
thanks to Theorem 2.9, sufficient conditions for GAC (in the case of unbounded
control systems) obtained in [MR13, LMR16], follow now as corollaries by the
sample stabilizability results in [LM19, LM20].

##  3\. The rescaled system and the impulsive extension

In this section we establish some relationships between the global asymptotic
controllability to $\mathcal{C}$ of a rescaled control system and the
associated impulsive extension. These results will be crucial to obtain the
Converse Lyapunov Theorem of Section 4.

###  3.1. GAC of the rescaled problem

Throughout this subsection, the function
$f:({\mathbb{R}}^{n}\setminus\mathcal{C})\times U\to{\mathbb{R}}^{n}$ is
continuous and satisfies the growth assumption (Hg), (i), for some growth rate
$\nu$. Let ${\bar{f}}$ denote the associated rescaled dynamics. For the
purpose of distinguishing the admissible trajectory-control pairs of $(f,U)$
from those of $({\bar{f}},U)$, we will denote by $(x,u)$ the former and by
$(y,v)$ the latter. Precisely, we simply say that $(x,u)$ is an admissible
trajectory-control pair when $u\in L^{\infty}_{loc}([0,T_{x}),U)$, $x\in
AC_{loc}([0,T_{x}),{\mathbb{R}}^{n}\setminus\mathcal{C})$, and $x$ solves the
original control system

(28) $\dot{x}(t)=f(x(t),u(t))\quad\text{a.e. }t\in(0,T_{x}),$

where $\lim_{t\to T^{-}_{x}}\mathbf{d}(x(t))=0$ whenever $T_{x}<+\infty$. We
call $(y,v)$ an admissible rescaled trajectory-control pair when $v\in
L^{\infty}_{loc}([0,S_{y}),U)$, $y\in
AC_{loc}([0,S_{y}),{\mathbb{R}}^{n}\setminus\mathcal{C})$, and $y$ solves the
rescaled control system

(29) $y^{\prime}(s)={\bar{f}}(y(s),v(s))\quad\text{a.e. }s\in(0,S_{y}),\ $

where $\lim_{t\to S^{-}_{y}}\mathbf{d}(y(s))=0$ whenever $S_{x}<+\infty$. When
$T_{x}$ [$S_{y}$] is finite, we mean that $x$ [$y$] is extended to
$[0,+\infty)$ as described in Definition 2.1. As an easy consequence of the
chain rule, admissible rescaled trajectory-control pairs $(y,v)$ are in one-
to-one correspondence with admissible trajectory-control pairs $(x,u)$ through
a time-change.

######  Lemma 3.1.

Assume $f$ continuous and satisfying (Hg), (i). Fix
$z\in{\mathbb{R}}^{n}\setminus\mathcal{C}$.

(i) Given an admissible process $(x,u)$ from $z$, set

$\displaystyle s(t):=\int_{0}^{t}\left(1+\nu(|u(\tau)|)\right)d\tau\,\ \forall
t\in[0,T_{x}),\ \ S_{y}:=\lim_{t\to T_{x}^{-}}s(t),\quad
t(\cdot):=s^{-1}(\cdot).$

Then $(y,v)(s):=(x,u)\circ t(s)$, $s\in[0,S_{y})$, is an admissible rescaled
trajectory-control pair from $z$.

(ii) Vice-versa, let $(y,v)$ be an admissible rescaled trajectory-control pair
from $z$ and set

$t(s):=\int_{0}^{s}(1+\nu(|v(\sigma)|)^{-1}d\sigma\ \ \forall s\in[0,S_{y}),\
\ T_{x}:=\lim_{s\to S_{y}^{-}}t(s),\quad s(\cdot):=t^{-1}(\cdot).$

Then, $(x,u)(t):=(y,v)\circ s(t)$, $t\in[0,T_{x})$, is an admissible
trajectory-control pair from $z$.

The following theorem establishes the equivalence between the GAC to
$\mathcal{C}$ with $U\cap\sigma$ controls of the original and the rescaled
control system.

######  Theorem 3.1.

Assume $f$ continuous and satisfying (Hg), (i). Then, the original control
system (28) is GAC to $\mathcal{C}$ with $U\cap\sigma$ controls if and only if
the rescaled control system (29) is GAC to $\mathcal{C}$ with $U\cap\sigma$
controls (for the same $\sigma$).

###### Proof.

Suppose first that the rescaled control system (29) is GAC to $\mathcal{C}$
with $U\cap\sigma$ controls, for some continuous function
$\sigma:(0,+\infty)\to(0,+\infty)$ and some descent rate
$\beta\in\mathcal{K}\mathcal{L}$. Hence, for every
$z\in{\mathbb{R}}^{n}\setminus\mathcal{C}$ there is an admissible rescaled
trajectory-control pair $(y,v)$ such that $y(0)=z$ and

$\mathbf{d}(y(s))\leq\beta(\mathbf{d}(z),s),\quad|v(s)|\leq\sigma(\mathbf{d}(y(s))\quad\forall
s\geq 0.$

Consider now $(x,u)(t):=(y,v)\circ s(t)$ where $s(t)$ is the time-change
introduced in Lemma (3.1),(ii). Since $s(t)\geq t$ for all $t\geq 0$, from the
monotonicity properties of $\beta(r,\cdot)$ it follows that

$\mathbf{d}(x(t))=\mathbf{d}(y(s(t)))\leq\beta(\mathbf{d}(z),s(t))\leq\beta(\mathbf{d}(z),t)\quad\forall
t\geq 0.$

Thus, $\beta$ is a descent rate also for the original control system (28).
Moreover,

$|u(t)|=|v(s(t))|\leq\sigma(\mathbf{d}(y(s(t)))=\sigma(\mathbf{d}(x(t))\quad\forall
t\in[0,T_{x}),$

where $T_{x}=\lim_{s\to S_{y}^{-}}t(s)$, as in Lemma (3.1),(ii).

To prove the converse implication, let us assume that the original control
system (28) is GAC to $\mathcal{C}$ with $U\cap\sigma$ controls, for some
continuous function $\sigma:(0,+\infty)\to(0,+\infty)$ and some descent rate
$\beta\in\mathcal{K}\mathcal{L}$.

For any couple $(R,r)$ of real numbers such that $0<r<R$, let us set

$N(R,r):=\nu\big{(}\max\sigma([r,\beta(R,0)]\big{)}.$

By the continuity of $\nu$, $\beta$, and $\sigma$ and by the monotonicity
properties of $\nu$ and $\beta$, one deduces immediately that the function
$N:\\{(R,r):\ 0<r<R\\}\to(0,+\infty)$ is continuous, for any $R$, $r\mapsto
N(R,r)$ is decreasing, and, for any $r$, $R\mapsto N(R,r)$ is increasing. By
eventually enlarging $N$, we can assume without loss of generality that $N$ is
strictly monotone with respect to both $r$ and $R$.

Now, let $S(R,r)>0$ be the value of $s$ implicitly defined by the equation

$\beta\left(R,\frac{s}{1+N(R,r)}\right)=r.$

From the monotonicity and continuity properties of $\beta$ and $N$, it follows
that $S$ is a continuous function on $\\{(R,r):\ 0<r<R\\}$, such that
$r\mapsto S(R,r)$ is strictly decreasing and $R\mapsto S(R,r)$ is strictly
increasing. As a consequence, if, for any $R$, we denote by $\rho=\rho(R,s)$
the inverse of the map $\rho\mapsto S(R,\rho)$, one easily obtains that $\rho$
is a ${\mathcal{KL}}$ function. Moreover, we have the identity

(30) $\beta\left(R,\frac{s}{1+N(R,\rho(R,s))}\right)=\rho(R,s).$

Let us show that $\rho$ is a descent rate for the rescaled control system
(29). To this aim, fix $z\in{\mathbb{R}}^{n}\setminus\mathcal{C}$ and let
$(x,u)$ be an admissible trajectory-control pair of (28) with $x(0)=z$ and
satisfying

(31)
$\mathbf{d}(x(t))\leq\beta(\mathbf{d}(z),t),\qquad|u(t)|\leq\sigma(\mathbf{d}(x(t))\quad\forall
t\geq 0.$

Now, define $(y,v)(s):=(x,u)\circ t(s)$ for all $s\in[0,S_{y})$, where

$\displaystyle s(t):=\int_{0}^{t}\left(1+\nu(|u(\tau)|)\right)d\tau\,\ \forall
t\in[0,T_{x}),\quad S_{y}:=\lim_{t\to T_{x}^{-}}s(t),\quad
t(\cdot):=s^{-1}(\cdot).$

By Lemma 3.1 $(y,v)$ is an admissible trajectory-control pair for
$(\bar{f},U)$. Furthermore, the rescaled control $v$ verifies

$|v(s)|=|u(t(s))|\leq\sigma(\mathbf{d}(x(t(s)))=\sigma(\mathbf{d}(y(s))\quad\forall
s\in[0,S_{y}).$

Let us show that $\mathbf{d}(y(s))\leq\rho(\mathbf{d}(z),s)$ for every
$s\in[0,S_{y})$. By contradiction, suppose that there exists some
$s\in[0,S_{y})$ such that

(32) $\mathbf{d}(y(s))>\rho(\mathbf{d}(z),s).$

Since $y(s)=x(t(s))$, we deduce that

$\rho(\mathbf{d}(z),s)<\mathbf{d}(x(t(s))\leq\beta(\mathbf{d}(z),0).$

By the definitions of $N$, $t(s)$, and the monotonicity properties of $\beta$,
we get

(33) $t(s)\geq\frac{s}{1+N(\mathbf{d}(z),\rho(\mathbf{d}(z),s))}.$

Using (31) and (30) we obtain the required contradiction with (32). Indeed,
(33) and the monotonicity of $\beta$, imply that

$\begin{array}[]{l}\mathbf{d}(y(s))=\mathbf{d}(x(t(s)))\leq\beta(\mathbf{d}(z),t(s))\leq\beta\left(\mathbf{d}(z),\frac{s}{1+N(\mathbf{d}(z),\rho(\mathbf{d}(z),s))}\right)\\\
\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad=\rho(\mathbf{d}(z),s).\end{array}$

At this point, if $S_{y}=+\infty$ the proof is concluded. If instead
$S_{y}<+\infty$, by definition we extend the trajectory $y$ to $[0,+\infty)$
as described in Definition 2.1, so $\mathbf{d}(y(s))=0<\rho(\mathbf{d}(z),s)$
for every $s\geq S_{y}$ trivially. ∎

From Theorems 2.5, 3.1 one derives that, under the Lipschitz continuity
assumption (Hl), the original system is GAC to $\mathcal{C}$ if and only if
the rescaled system is GAC to $\mathcal{C}$. Precisely, one has:

######  Theorem 3.2.

Assume that $f$ satisfies (Hl) and (Hg), (i). Then, the original system (28)
is GAC to $\mathcal{C}$ if and only if the rescaled system (29) is GAC to
$\mathcal{C}$. In addition, for both systems being GAC to $\mathcal{C}$ is
equivalent to being GAC to $\mathcal{C}$ with $U\cap\sigma$ controls.

###### Proof.

Theorem 3.1 establishes that the original system (28) is GAC to $\mathcal{C}$
with $U\cap\sigma$ controls if and only if the rescaled system (29) is GAC to
$\mathcal{C}$ with $U\cap\sigma$ controls. To conclude the proof, it is enough
to observe that, when $f$ verifies (Hl) and (Hg), (i), the rescaled dynamics
${\bar{f}}$ verifies (Hl) too. Hence, Theorem 2.5 applied both to $f$ and
${\bar{f}}$ implies the equivalence of the notions of GAC to $\mathcal{C}$ and
GAC to $\mathcal{C}$ with $U\cap\sigma$ controls both for the original system
and the rescaled system. ∎

###  3.2. GAC of the impulsive extension

Let $f$ be a continuous function satisfying (Hg), and let $\nu$, ${\bar{f}}$,
$F$, $\mathbb{U}$ be as in (Hg). We now embed the original system into an
extended control system and show that GAC to $\mathcal{C}$ of the rescaled
control system implies GAC to $\mathcal{C}$ of the extended system.

######  Definition 3.3 (Admissible extended trajectory-control pairs).

A triple $(y,w_{0},w)$ is called an admissible extended trajectory-control
pair if there exists $S_{y}\leq+\infty$ such that: the control $(w_{0},w)\in
L^{\infty}([0,S_{y}),\overline{\mathbb{U}})$; the trajectory $y\in
AC([0,S_{y}),{\mathbb{R}}^{n}\setminus\mathcal{C})$ is a solution of the
extended control system

(34) $y^{\prime}(s)=F(y(s),w_{0}(s),w(s))\qquad\text{a.e. $s\in[0,S_{y})$;}$

and, if $S_{y}<+\infty$, one has $\lim_{s\to S_{y}^{-}}\mathbf{d}(y(s))=0$. If
$(y,w_{0},w)$ is an admissible extended trajectory-control pair and
$S_{y}<+\infty$, we extend $y$ to $[0,+\infty[$ by setting
$y(s):=\lim_{\sigma\to S_{y}^{-}}y(\sigma)$ for any $s\geq S_{y}$.444In this
case, the limit at $S_{y}$ always exists, since $F$ is bounded on any
neighborhood of the target, which has compact boundary.

The original and the rescaled system can be embedded in the extended system as
follows.

######  Lemma 3.2.

Assume that $f$ is a continuous function satisfying (Hg) for some growth rate
$\nu$. Let $(x,u)$ be an admissible trajectory-control pair for $(f,U)$. Set

$\displaystyle s(t)$
$\displaystyle:=\int_{0}^{t}\left(1+\nu(|u(\tau)|)\right)d\tau\,\ \forall
t\in[0,T_{x}),\ \ S_{y}:=\lim_{t\to T_{x}^{-}}s(t),\quad t:=s^{-1},$
$\displaystyle y(s)$ $\displaystyle:=x\circ t(s)\ \ \forall s\in[0,S_{y}),$
$\displaystyle(w_{0},w)(s)$
$\displaystyle:=\left(1,\frac{u\,\nu(|u|)}{|u|}\circ
t(s)\right)\,t^{\prime}(s)=\left(\frac{1}{1+\nu(|u|)}\,,\,\frac{u\,\nu(|u|)}{|u|(1+\nu(|u|))}\right)\circ
t(s)$

for a.e. $s\in[0,S_{y})$. Then $(y,v)$, where $v(s):=u\circ t(s)$ for a.e.
$s\in[0,S_{y})$, is an admissible rescaled trajectory-control pair, while
$(y,w_{0},w)$ is an admissible extended trajectory-control pair, with
$w_{0}>0$ a.e. on $[0,S_{y})$.

###### Proof.

The rescaled trajectory-control pair $(y,v)$ is admissible by Lemma 3.1, (i).
In view of the definition of $(w_{0},w)$ and of $F$, straightforward
calculations yield that

(35) $y^{\prime}(s)={\bar{f}}(y(s),v(s))=F(y(s),w_{0}(s),w(s)),\quad\text{a.e.
$s\in[0,S_{y})$.}$

This shows that $(y,w_{0},w)$ is an admissible extended trajectory-control
pair with $w^{0}>0$ a.e.. ∎

######  Remark 3.4.

When $\nu(r)=r^{\,\bar{d}}$ for some integer $\bar{d}\geq 1$, the extended
dynamics $F$ is equivalent to the extended dynamics introduced in [RS00],
whose definition is based on the notion of recession function. Indeed, if for
any
$(x,w_{0},w)\in\overline{({\mathbb{R}}^{n}\setminus\mathcal{C})}\times\overline{{\mathbb{U}}}$
we set
$(\tilde{w}_{0},\tilde{w}):=\left(w_{0}^{1/\,\bar{d}},\frac{w}{|w|}\,|w|^{1/\,\bar{d}}\right)$,
then $\tilde{w}_{0}^{\,\bar{d}}+|\tilde{w}|^{\,\bar{d}}=1$ and we obtain that

$\begin{array}[]{l}\displaystyle
F(x,\tilde{w}_{0},\tilde{w})=\lim_{r\to\tilde{w}_{0}^{+}}{\bar{f}}\left(x,\frac{\tilde{w}}{r}\right)=\lim_{r\to\tilde{w}_{0}^{+}}f\left(x,\frac{\tilde{w}}{r}\right)\,r^{\,\bar{d}}.\end{array}$

In particular, in the case of control-affine $f$ one has $\bar{d}=1$ and
$(\tilde{w}_{0},\tilde{w})\equiv(w_{0},w)$, so that the classical impulsive
extension of the graph-completion approach considered in [LM21], coincides
with the present one.

The extension consists in considering $(y,w_{0},w)$, where $w_{0}$ may be zero
on nondegenerate subintervals of $[0,S_{y})$. On these intervals, the time
variable $t=\int_{0}^{s}w_{0}(\sigma)\,d\sigma$ is constant –i.e. the time
stops–, while the state variable $y$ evolves, according to the equation
$y^{\prime}=F(t,y,0,w)$, sometimes called the fast dynamics. For this reason,
system (34) is often referred to as the impulsive extension of the original
control system (28), despite the fact that it is an ordinary control system,
as the extended controls $(w_{0},w)$ take values in the compact set
$\overline{\mathbb{U}}$ and the trajectory $y$ is absolutely continuous. A
detailed discussion of this topic goes beyond the purposes of the paper. We
just mention that an equivalent, $t$-based description of this extension,
where $u$ is no more a function and the trajectory $x$ is a discontinuous map
whose total variation is bounded on $[0,T]$ for every $T<T_{x}$, but possibly
unbounded on $[0,T_{x})$, in short $x\in BV_{loc}[0,T_{x})$, could be given
(see [KDOPS14] and also [KDOPS15, AR15, MS18, MS20]).

######  Proposition 3.1.

Assume $f$ continuous and satisfying (Hg). If the rescaled system (29) is GAC
to $\mathcal{C}$, then the extended system (34) is GAC to $\mathcal{C}$.

###### Proof.

Since the rescaled control system (29) is GAC to $\mathcal{C}$, there exists a
descent rate $\beta\in\mathcal{KL}$ such that for all
$z\in{\mathbb{R}}^{n}\setminus\mathcal{C}$ there is an admissible rescaled
trajectory-control pair $(y,v)$, such that $y(0)=x$ and

(36) $\mathbf{d}(y(s))\leq\beta(\mathbf{d}(z),s)\quad\forall s\geq 0.$

Define for a.e. $s\in[0,S_{y})$ the extended control

$(w_{0},w)(s)=\left(\frac{1}{1+\nu(|v(s)|)}\,,\,\frac{v(s)\,\nu(|v(s)|)}{|v(s)|(1+\nu(|v(s)|))}\right).$

Since $(w_{0},w)(s)\in\mathbb{U}$ for a.e. $s\in[0,S_{y})$, from Lemma 3.2 it
follows that $(y,w_{0},w)$ is an admissible extended trajectory-control pair
for (34) and this, together with (36) and the arbitrariness of $z$, implies
that the extended system (34) is GAC to $\mathcal{C}$, with the same descent
rate $\beta$ as the rescaled system (29). ∎

##  4\. A Converse Lyapunov Theorem

In this section we state our main result, which extends to control systems
with unbounded controls and their impulsive extensions well-known
relationships between GAC, sample stabilizability, and existence of control
Lyapunov functions. Furthermore, we relate explicit stabilizing feedback
constructions for the extended and for the original system, which are based on
the existence of a semiconcave control Lyapunov function.

###  4.1. Main result

To begin with, let us introduce the notion of (nonsmooth) control Lyapunov
function. In the following, given an open set
$\Omega\subseteq{\mathbb{R}}^{N}$, a continuous function
$W:\overline{\Omega}\to[0,+\infty)$ is said positive definite on $\Omega$ if
$W(x)>0$ $\forall x\in\Omega$ and $W(x)=0$ $\forall x\in\partial\Omega$. The
function $W$ is called proper on $\Omega$ if the pre-image $W^{-1}(K)$ of any
compact set $K\subset[0,+\infty)$ is compact. As customary, $\partial_{P}W(x)$
refers to the proximal subdifferential of $W$ at $x$ (which may very well be
empty). We recall that $p$ belongs to $\partial_{P}W(x)$ if and only if there
exist $\sigma$ and $\eta>0$ such that

$W(y)-W(x)+\sigma|y-x|^{2}\geq\langle p\,,\,y-x\rangle\qquad\forall y\in
B_{\eta}(\\{x\\}).\mathcal{C}$

The limiting subdifferential $\partial_{L}W(x)$ of $W$ at $x\in\Omega$, is
defined as

$\displaystyle\partial_{L}W(x):=\Big{\\{}\lim_{i\to+\infty}\,p_{i}:\
p_{i}\in\partial_{P}W(x_{i}),\ \lim_{i\to+\infty}x_{i}=x\Big{\\}}.$

When the function $W$ is locally Lipschitz continuous on $\Omega$, the
limiting subdifferential $\partial_{L}W(x)$ is nonempty at every point, the
set-valued map $x\rightsquigarrow\partial_{L}W(x)$ is upper semicontinuous,
and the Clarke generalized gradient at $x$ can be derived as co
$\partial_{L}W(x)$. As sources for nonsmooth analysis we refer e.g. to [CS04,
CLSW08, Vin00]. For any nonempty closed set
$\mathbf{U}\subseteq{\mathbb{R}}^{M}$ for some integer $M>0$ and any
continuous function
$\mathbf{f}:({\mathbb{R}}^{n}\setminus\mathcal{C})\times\mathbf{U}\to{\mathbb{R}}^{n}$,
let consider the control system

(37) $\dot{x}=\mathbf{f}(x,u),\qquad u\in\mathbf{U},$

and the Hamiltonian
$H_{\mathbf{f},\mathbf{U}}:({\mathbb{R}}^{n}\setminus\mathcal{C})\times
U\to[-\infty,+\infty)$, given by

(38) $\displaystyle
H_{\mathbf{f},\mathbf{U}}(x,p):=\inf_{u\in\mathbf{U}}\left\\{\langle
p\,,\,\mathbf{f}(x,u)\rangle\right\\}.$

Notice that $H_{\mathbf{f},\mathbf{U}}$ may be discontinuous and equal to
$-\infty$ at some points.

######  Definition 4.1 (Control Lyapunov Function).

Let $W:\overline{{\mathbb{R}}^{n}\setminus{\mathcal{C}}}\to[0,+\infty)$ be a
locally Lipschitz continuous function on
$\overline{{\mathbb{R}}^{n}\setminus{\mathcal{C}}}$, which is positive
definite and proper on ${\mathbb{R}}^{n}\setminus\mathcal{C}$. We say that $W$
is a _Control Lyapunov Function,_ (CLF), for the system (37) if there exists
some continuous, strictly increasing function
$\gamma:(0,+\infty)\to(0,+\infty)$, that we call a decrease rate, such that
the following _(infinitesimal) decrease condition_ is satisfied:

(39) $H_{\mathbf{f},\mathbf{U}}(x,\partial_{L}W(x))<-\gamma(W(x))\quad\forall
x\in{{{\mathbb{R}}^{n}}\setminus\mathcal{C}}.\,$

######  Remark 4.2.

If the continuous function
$\mathbf{f}:({\mathbb{R}}^{n}\setminus\mathcal{C})\times\mathbf{U}\to{\mathbb{R}}^{n}$
is bounded in $(B_{R}(\mathcal{C})\setminus\mathcal{C})\times\mathbf{U}$ for
some $R>0$ and continuous in $x$ uniformly with respect to $\mathbf{U}$ –as it
is for the rescaled dynamics ${\bar{f}}$ and for the extended dynamics $F$–,
then the Hamiltonian $H_{\mathbf{f},\mathbf{U}}$ is continuous and the
decrease condition (39) is equivalent to the usual condition

(40) $H_{\mathbf{f},\mathbf{U}}(x,\partial_{P}W(x))<-V(x)\quad\forall
x\in{{{\mathbb{R}}^{n}}\setminus\mathcal{C}},$

expressed in terms of the proximal subdifferential, for some continuous
function $V:\overline{{\mathbb{R}}^{n}\setminus{\mathcal{C}}}\to[0,+\infty)$,
which is positive definite and proper on
${\mathbb{R}}^{n}\setminus\mathcal{C}$, used e.g. in [CLSS97, CLRS00, KT04]
(see [LM19, Prop. 4.2]). Incidentally, in this case (39) has also an
equivalent formulation, which involves the Dini derivative.

The rescaled system and the extended system share the same CLFs.

######  Proposition 4.1.

Assume $f:({\mathbb{R}}^{n}\setminus\mathcal{C})\times U\to{\mathbb{R}}^{n}$
continuous and satisfying assumption (Hg). A map
$W:{\mathbb{R}}^{n}\setminus\mathcal{C}\to{\mathbb{R}}$ is a CLF for the
rescaled problem (29) if and only if it is a CLF for the extended problem
(34).

###### Proof.

Consider the maps

(41) $\displaystyle u$
$\displaystyle\mapsto(w_{0},w)(u):=\left(\frac{1}{1+\nu(|u|)}\,,\,\frac{u\,\nu(|u|)}{|u|(1+\nu(|u|))}\right)\quad\forall
u\in U,$ $\displaystyle(w_{0},w)$ $\displaystyle\mapsto
u(w_{0},w):=\frac{w}{|w|}\,\nu^{-1}\left(\frac{|w|}{w_{0}}\right)\quad\forall(w_{0},w)\in\mathbb{U},$

where $\mathbb{U}$ and $\nu$ are as in (Hg). The definitions of ${\bar{f}}$
and $F$ imply that

$\begin{array}[]{l}\displaystyle
F(x,(w_{0},w)(u))={\bar{f}}(x,u)\qquad\forall(x,u)\in({\mathbb{R}}^{n}\setminus\mathcal{C})\times
U,\\\ \displaystyle{\bar{f}}(x,u(w_{0},w))=F(x,w_{0},w)\ \
\forall(x,w_{0},w)\in({\mathbb{R}}^{n}\setminus\mathcal{C})\times\mathbb{U}.\end{array}$

Thus, for any $x\in{\mathbb{R}}^{n}\setminus\mathcal{C}$ and
$p\in{\mathbb{R}}^{n}$, one has

$\begin{array}[]{l}\displaystyle\inf_{u\in U}\left\\{\langle
p\,,\,{\bar{f}}(x,u)\rangle\right\\}=\inf_{u\in U}\left\\{\langle
p\,,\,F(x,(w_{0},w)(u))\rangle\right\\}\\\ \displaystyle\qquad\
\qquad\quad\qquad\qquad\geq\inf_{(w_{0},w)\in\overline{\mathbb{U}}}\left\\{\langle
p\,,\,F(x,w_{0},w)\rangle\right\\}\\\
\displaystyle\quad\qquad\qquad\qquad\qquad=\inf_{(w_{0},w)\in\mathbb{U}}\left\\{\langle
p\,,\,F(x,w_{0},w)\rangle\right\\}\\\
\displaystyle\quad\qquad\qquad\qquad\qquad=\inf_{(w_{0},w)\in\mathbb{U}}\left\\{\langle
p\,,\,{\bar{f}}(x,u(w_{0},w))\rangle\right\\}\\\ \displaystyle\qquad\
\qquad\quad\qquad\qquad\geq\inf_{u\in U}\left\\{\langle
p\,,\,{\bar{f}}(x,u)\rangle\right\\}.\end{array}$

Therefore $H_{{\bar{f}},U}\equiv H_{F,\,\overline{\mathbb{U}}}$. As a
consequence, a map $W$ is a CLF for (28) if and only if it is a CLF for (34).
∎

We are ready to state the main result of the paper. To this aim, we introduce
the following stronger assumptions. (Hg)∗ The function
$f:{\mathbb{R}}^{n}\times U\to{\mathbb{R}}^{n}$ is continuous and a stronger
version of hypothesis (Hg) is valid, where ${\mathbb{R}}^{n}$ replaces
${\mathbb{R}}^{n}\setminus\mathcal{C}$. Furthermore, for any compact set
$\mathcal{K}\subset{\mathbb{R}}^{n}$ there is some constant $\bar{L}>0$ such
that

$|F(x,w_{0},w)-F(y,w_{0},w)|\leq L|x-y|\qquad\forall x,y\in\mathcal{K},\
\forall(w_{0},w)\in\bar{\mathbb{U}}.$

######  Theorem 4.3 (Converse Lyapunov Theorem).

Assume hypothesis (Hg)∗. Then the following properties are equivalent:

  * (i)

the original control system $\dot{x}=f(x,u)$ is GAC to $\mathcal{C}$;

  * (ii)

the extended control system $y^{\prime}=F(y,w_{0},w)$ is GAC to $\mathcal{C}$;

  * (iii)

there exists a CLF for the extended control system $y^{\prime}=F(y,w_{0},w)$;

  * (iv)

there exists a CLF for the original control system $\dot{x}=f(x,u)$;

  * (v)

the system $\dot{x}=f(x,u)$ is sample stabilizable to $\mathcal{C}$;

  * (vi)

the system $y^{\prime}=F(y,w_{0},w)$ is sample stabilizable to $\mathcal{C}$.

###### Proof.

Let us preliminarily observe that the Lipschitz continuity hypothesis on $F$
in (Hg)∗ implies both that $f$ satisfies hypothesis (Hl) and that ${\bar{f}}$
is locally Lipschitz continuous in $x$ (on
$\overline{{\mathbb{R}}^{n}\setminus\mathcal{C}}$), uniformly w.r.t. $u\in U$.
${\rm(i)}\Longrightarrow{\rm(ii)}$. From Theorem 3.2 it follows that the
original control system is GAC to $\mathcal{C}$ if and only if the rescaled
control system $y^{\prime}={\bar{f}}(y,v)$ is GAC to $\mathcal{C}$. This
implies that the extended control system is GAC to $\mathcal{C}$, in view of
Proposition 3.1. ${\rm(ii)}\Longleftrightarrow{\rm(iii)}$. The fact that the
extended control system is GAC to $\mathcal{C}$ if and only if there exists a
CLF for it follows from [KT00, Theorem 1] (see also [KT04, Thm 3.2, Rmk. 4]).
Observe that this result is applicable to the impulsive extension essentially
because the set $\overline{\mathbb{U}}$ of extended control values is bounded.
${\rm(iii)}\Longrightarrow{\rm(iv)}$. Let $W$ be a CLF for the extended
control system $y^{\prime}=F(y,w_{0},w)$, for some decrease rate $\gamma$.
Hence, by Proposition 4.1 $W$ is also a CLF for the rescaled system
$y^{\prime}={\bar{f}}(y,v)$, with the same $\gamma$. Since the definition of
$\bar{f}$ implies that

(42) $H_{f,U}(x,p)\leq H_{\bar{f},U}(x,p)<-\gamma(W(x))\quad\forall
x\in{\mathbb{R}}^{n}\setminus\mathcal{C},\ \ \forall p\in\partial_{L}W(x),$

we can finally conclude that $W$ is a CLF also for the original control
system. ${\rm(iv)}\Longrightarrow{\rm(v)}$. Let $W$ be a CLF for (28). Then,
[LM20, Theorem 4.6] implies that $\dot{x}=f(x,u)$ is sample stabilizable to
$\mathcal{C}$. ${\rm(v)}\Longrightarrow{\rm(i)}$. The sample stabilizability
of $\dot{x}=f(x,u)$ to $\mathcal{C}$ implies that it is GAC to $\mathcal{C}$,
by Theorem 2.9. With this, we have shown that (i),(ii),(iii),(iv), and (v) are
equivalent. To conclude the proof it suffices to observe that, by the same
arguments as above, ${\rm(iii)}\Longrightarrow{\rm(vi)}$ and
${\rm(vi)}\Longrightarrow{\rm(ii)}$, namely, the existence of a CLF for
$y^{\prime}=F(y,w_{0},w)$ implies sample stabilizability of the extended
system, which in turn implies that $y^{\prime}=F(y,w_{0},w)$ is GAC to
$\mathcal{C}$. ∎

###  4.2. Semiconcave CLFs and stabilizing feedback construction

Given an open set $\Omega\subseteq{\mathbb{R}}^{N}$, a function
$W:\Omega\to{\mathbb{R}}$ is called locally semiconcave if for every compact
subset $\mathcal{K}\subset\Omega$ there exists $\rho>0$ such that, for all
$x$, $\hat{x}\in\mathcal{K}$ with $[x,\hat{x}]\subset\mathcal{K}$, one has

$W(x)+W(\hat{x})-2W\left(\frac{x+\hat{x}}{2}\right)\leq\rho|x-\hat{x}|^{2}.$

Locally semiconcave functions are locally Lipschitz continuous and twice
differentiable almost everywhere (see e.g. [CS04]).

Since the works by Rifford [Rif00, Rif03], semiconcave control Lyapunov
functions have proven to play a crucial role for the explicit construction of
sample stabilizing feedback strategies. It is therefore worth noting that in
Theorem 4.3 we can assume without loss of generality that CLFs are locally
semiconcave on ${\mathbb{R}}^{n}\setminus\mathcal{C}$. Precisely, one has:

######  Proposition 4.2.

Under the assumptions of Theorem 4.3, 666Actually, from the results in [LM19,
LM20] this statement is valid even if $f$, ${\bar{f}}$, and $F$ satisfy the
assumptions in (Hg)∗ only for
$x\in\overline{{\mathbb{R}}^{n}\setminus\mathcal{C}}$. if there exists a CLF
either for the extended control system (34) or for the original control system
(28), then there exists a CLF for (34) or (28), respectively, which is locally
semiconcave on ${\mathbb{R}}^{n}\setminus\mathcal{C}$.

###### Proof.

If there exists a locally Lipschitz continuous CLF for the extended system
(34), the dynamics of which meet classical Lipschitz continuity and
boundedness assumptions, then from [LM19, Theorem 4.3] there is also a locally
semiconcave CLF for (34). On the other hand, when there exists a locally
Lipschitz continuous CLF for the original system (28), [LM20, Theorem 4.3]
guarantees the existence of a locally Lipschitz continuous CLF for the
rescaled system (29). At this point, [LM19, Theorem 4.3] again implies the
existence of a locally semiconcave CLF for (29), which, as one deduces by
(42), is also a locally semiconcave CLF for (28). The proof of the claim is
thus complete. ∎

Thanks to Proposition 4.2, we can explicitly build a stabilizing feedback for
the original control system from a stabilizing feedback for the impulsive
extension. To this aim, we need the following preliminary result.

######  Proposition 4.3.

Assume that
$F:({\mathbb{R}}^{n}\setminus\mathcal{C})\times\overline{\mathbb{U}}\to{\mathbb{R}}$
is a continuous function. Let $W$ be a CLF for $y^{\prime}=F(y,w_{0},w)$, with
decrease rate $\gamma$. Then, there exists a continuous function
$N:(0,+\infty)\to(0,1]$ such that

(43) $H_{F,\mathbb{U}_{N(W(x))}}(x,\partial_{L}W(x))<-\gamma(W(x))\quad\forall
x\in{\mathbb{R}}^{n}\setminus\mathcal{C},$

where, for every $\rho\in(0,1]$,

$\mathbb{U}_{\rho}:=\\{(w_{0},w)\in\mathbb{U}:\ w_{0}\geq\rho\\}.$

###### Proof.

To prove the statement, we show that there is some continuous function
$N:(0,+\infty)\to(0,1]$, which is increasing in $(0,1]$, decreasing in
$[1,+\infty)$, and such that, for any $r>0$, one has

(44) $H_{F,\mathbb{U}_{N(r)}}(x,\partial_{L}W(x))<-\gamma(W(x))\quad\forall
x\in W^{-1}([r\land 1,r\vee 1]).$

In fact, proven this, from the monotonicity properties of $N$ it immediately
follows that in (44) we can replace $r$ with $W(x)$, and that implies (43) by
the arbitrariness of $r>0$.

To prove (44), fix $r>0$ and set

$\Gamma_{r}:=\\{(x,p):\ \ x\in W^{-1}([r\land 1,r\vee 1]),\
p\in\partial_{L}W(x)\\}.$

Notice that the properties of $W$ –in particular, the properness of $W$ and
the upper semicontinuity of the set-valued map
$x\rightsquigarrow\partial_{L}W(x)$– imply that $\Gamma_{r}$ is a compact set.
For every $(x,p)\in\Gamma_{r}$, define

$w_{0}(x,p):=\sup\\{w_{0}:\ (w_{0},w)\in\mathbb{U}\ \text{and}\ \langle
p,F(x,w_{0},w)\rangle<-\gamma(W(x))\\}.$

This set is nonempty because $W$ is a CLF, and $w_{0}(x,p)\in[0,1]$. At this
point, set

$\hat{N}(r):=\inf\\{w_{0}(x,p):\ (x,p)\in\Gamma_{r}\\}.$

By construction, $\hat{N}$ is nonnegative, increasing in $(0,1]$ and
decreasing in $[1,+\infty)$, and $\hat{N}([0,+\infty))\subseteq[0,1]$. If
$\hat{N}(r)>0$ for all $r>0$, then the required $N$ is given by any
continuous, positive approximation from below of $\hat{N}$, which is
increasing in $(0,1]$ and decreasing in $[1,+\infty)$. To conclude, it only
remains to prove that $\hat{N}(r)>0$ for all $r>0$. To this end, assume by
contradiction that $\hat{N}(r)=0$ for some $r>0$. Then, there is some sequence
$((x_{k},p_{k}))_{k\geq 1}\subset\Gamma_{r}$, such that
$w_{0}(x_{k},p_{k})<1/k$ for all $k$. Hence, the definition of
$w_{0}(x_{k},p_{k})$ yields

(45) $\langle p_{k},F(x_{k},w_{0},w)\rangle<-\gamma(W(x_{k}))\ \text{ for }\
(w_{0},w)\in\overline{\mathbb{U}}\Longrightarrow w_{0}<\frac{1}{k}.$

Since $\Gamma_{r}$ is compact, there exists a subsequence, that we still
denote $((x_{k},p_{k}))_{k}$, converging to some
$(\bar{x},\bar{p})\in\Gamma_{r}$. Since $W$ is a CLF with decrease rate
$\gamma$, there exists some $(\bar{w}_{0},\bar{w})\in\mathbb{U}$ (with
$\bar{w}_{0}>0$) such that

$\langle\bar{p},F(\bar{x},\bar{w}_{0},\bar{w})\rangle<-\gamma(W(\bar{x})).$

Therefore, by the continuity of $F,W$ and $\gamma$, for a sufficiently large
$k$ one has $1/k<\bar{\omega}_{0}$ and

$\langle p_{k},F(x_{k},\bar{w}_{0},\bar{w})\rangle<-\gamma(W(x_{k})),$

in contradiction with (45), so that the proof is complete. ∎

From Proposition 4.3 it follows that, given a semiconcave control Lyapunov
function for the extended control system, we can always select a stabilizing
feedback $\hat{K}(x)=(\hat{w}_{0}(x),\hat{w}(x))$ which is not impulsive,
namely such that $\hat{w}_{0}(x)>0$ for every
$x\in{\mathbb{R}}^{n}\setminus\mathcal{C}$.

######  Proposition 4.4.

Consider the same assumptions as in Theorem 4.3. Let the extended system (34)
be sample stabilizable to $\mathcal{C}$. Then, there exist a continuous
function $N:(0,+\infty)\to(0,1)$ and a stabilizing feedback
$\hat{K}:{\mathbb{R}}^{n}\setminus\mathcal{C}\to\mathbb{U}$,
$\hat{K}(x)=(\hat{w}_{0}(x),\hat{w}(x))$ for (34), satisfying

(46) $w_{0}(x)\geq N(W(x))\qquad\forall
x\in{\mathbb{R}}^{n}\setminus\mathcal{C},$

so that the locally bounded feedback
$K:{\mathbb{R}}^{n}\setminus\mathcal{C}\to U$ given by

(47)
$K(x):=\frac{\hat{w}(x)}{|\hat{w}(x)|}\,\nu^{-1}\left({\frac{|\hat{w}(x)|}{\hat{w}_{0}(x)}}\right)\qquad\forall
x\in{\mathbb{R}}^{n}\setminus\mathcal{C}$

is sample stabilizing for the original system (28) to $\mathcal{C}$.

###### Proof.

From Theorem 4.3 and Proposition 4.2 it follows that, if the extended control
system $y^{\prime}=F(y,w_{0},w)$ is sample stabilizable to $\mathcal{C}$, then
it admits a locally semiconcave CLF $W$. Hence, Proposition 4.3 with reference
to $W$ implies the existence of a continuous function $N:(0,+\infty)\to(0,1]$
such that, fixed a selection $p(x)\in\partial_{L}W(x)$ for any
$x\in{\mathbb{R}}^{n}\setminus\mathcal{C}$, any map
$\hat{K}:{\mathbb{R}}^{n}\setminus\mathcal{C}\to\mathbb{U}$ such that

$\hat{K}(x)=(\hat{w}_{0}(x),\hat{w}(x))\in\underset{(w_{0},w)\in\mathbb{U}_{N(W(x))}}{\arg\min}\Big{\\{}\langle
p(x),F(x,w_{0},w)\rangle\Big{\\}},$

satisfies (46) and the inequality

$\langle p(x),F(x,w_{0},w)\rangle<-\gamma(W(x))\qquad\forall
x\in{\mathbb{R}}^{n}\setminus\mathcal{C}.$

As shown in [LM20, Sect. 3], this implies that $\hat{K}$ is a sample
stabilizing feedback for the extended system (34). Consider now the feedback
$K:{\mathbb{R}}^{n}\setminus\mathcal{C}\to U$ associated to such $\hat{K}$,
defined as in (47). It is locally bounded, since

$|K(x)|\leq\nu^{-1}\left({\frac{1}{N(W(x))}}\right)\qquad\forall
x\in{\mathbb{R}}^{n}\setminus\mathcal{C}.$

Furthermore, $K$ is sample stabilizing for the rescaled control system
$y^{\prime}=\bar{f}(y,u)$, in view of the identity
$\bar{f}(x,K(x))=F(x,\hat{K}(x))$ for all
$x\in{\mathbb{R}}^{n}\setminus\mathcal{C}$, from which it follows that
sampling trajectories associated to $\hat{K}$ for the extended system coincide
with the sampling trajectories associated to $K$ for the rescaled system.
Since by [LM20, Theorem 2.5] the rescaled and the original system share the
same stabilizing feedbacks, $K$ is sample stabilizing also for
$\dot{x}=f(x,u)$ and the proof is concluded. ∎

######  Remark 4.4.

Observe that the converse relation, namely the fact that, given a stabilizing
feedback $K$ for the original system (28), it is possible to derive a feedback
$\hat{K}$ for the impulsive extension, is quite obvious. Indeed, it is easy to
see that the map

$\hat{K}:x\mapsto\left(\frac{1}{1+\nu(|K(x)|)}\,,\,\frac{K(x)\,\nu(|K(x)|)}{|K(x)|(1+\nu(|K(x)|))}\right)\quad\forall
x\in{\mathbb{R}}^{n}\setminus\mathcal{C}$

is a stabilizing feedback for the extended system (34), again by the identity
$F(x,\hat{K}(x))=\bar{f}(x,K(x))$ together with [LM20, Theorem 2.5].

##  5\. An example

In this section, we introduce a simple control system which is neither
globally asymptotically controllable nor sample stabilizable to the origin by
means of bounded controls, whereas it is by means of unbounded strategies.
Furthermore, we show how to construct a stabilizing feedback for the original
system, given a control Lyapunov function (and an associated stabilizing
feedback) of the extend system.

Consider the target $\mathcal{C}:=\\{0\\}$ and the one-dimensional control
system

(48) $\dot{x}(t)=x(t)-x^{3}(t)u(t),\quad u(t)\in
U:=[0,+\infty)\quad\text{a.e}.$

Using only controls taking values in a bounded subset $[0,M]$ of $U$ for some
$M>0$, the best strategy to approach the origin is clearly to implement the
constant control $u\equiv M$ and solve the differential equation

$\dot{x}(t)=x(t)-Mx^{3}(t).$

But this way, for every initial point $z\neq 0$ we get the trajectory

(49) $x(t)=\frac{ze^{t}}{\sqrt{z^{2}M(e^{2t}-1)+1}},$

so we have
$\displaystyle\lim_{t\to+\infty}x(t)=\frac{\text{sign}(z)}{\sqrt{M}}\neq 0$.
Therefore, the control system (48) with controls in any given bounded subset
of $U$ is not GAC to $\\{0\\}$.

In view of Theorem 4.3, the global asymptotic controllability and the sample
stabilizability of system (48) to $\\{0\\}$ when admissible pairs $(x,u)$ with
controls $u\in L^{\infty}_{loc}([0,T_{x}),[0,+\infty))$ are allowed, is
equivalent to the global asymptotic controllability to $\\{0\\}$ of the
impulsive extension

(50)
$y^{\prime}(s)=F(y(s),w_{0}(s),w(s))=y(s)\,w_{0}(s)-y^{3}(s)w(s),\quad(w_{0},w)(s)\in\overline{\mathbb{U}}\quad\text{a.e.},$

where $\mathbb{U}:=\\{(w_{0},w)\in(0,+\infty)\times[0,+\infty),\ \
w_{0}+w=1\\}$. Here, by choosing the constant control
$(w_{0},w)(s)\equiv(0,1)$ for every $s\geq 0$, for each starting point $z\neq
0$ we get an extended trajectory (describing in the original time variable an
instantaneous jump from $z$ to the target) that satisfies

$\mathbf{d}(y(s))=|y(s)|=\frac{1}{\sqrt{2s+\frac{1}{z^{2}}}}=:\beta(|z|,s)\qquad\forall
s\geq 0,$

where $\beta\in{\mathcal{KL}}$. So, the original system is GAC and sample
stabilizable to the origin, because the extended system is GAC to $\\{0\\}$.

Again Theorem 4.3 together with Propositions 4.2, 4.4 guarantees that there is
a locally semiconcave control Lyapunov function for the extended system, which
makes it possible to build both a sample stabilizing feedback $\hat{K}$ for
(50) and a locally bounded sample stabilizing feedback $K$ for (48). In
particular, a locally semiconcave CLF for (50) is given by the function
$W(x):=|x|$ for all $x\in{\mathbb{R}}$. Indeed, for every $x\neq 0$, one has

$H_{F,\mathbb{U}}(x,\partial_{L}W(x))=\inf_{(w_{0},w)\in\mathbb{U}}\left\\{\frac{x}{|x|}\,(xw_{0}-x^{3}w)\right\\}=-|x|^{3}<-\gamma(W(x)),$

if we choose the decrease rate $\gamma(r):=\frac{r^{3}}{2(2+r^{2})}$, $r>0$.
At this point, setting $N(r):=\frac{r^{2}}{2+r^{2}}$, we can define the
feedback $\hat{K}:{\mathbb{R}}\setminus\\{0\\}\to\mathbb{U}$, given by

$\hat{K}(x)=(\hat{w}_{0}(x),\hat{w}(x)):=(N(|x|),1-N(|x|))=\left(\frac{x^{2}}{2+x^{2}},\frac{2}{2+x^{2}}\right),$

which is sample stabilizing for the extend system, since

$\frac{x}{|x|}\,(x\hat{w}_{0}(x)-x^{3}\hat{w}(x))=\frac{x}{|x|}\left(x\frac{x^{2}}{2+x^{2}}-x^{3}\frac{2}{2+x^{2}}\right)=-\frac{|x|^{3}}{2+x^{2}}<-\gamma(W(x)),$

for any $x\neq 0$. At this point, from Proposition 4.4 it follows that the
locally bounded feedback $K:{\mathbb{R}}\setminus\\{0\\}\to U$, defined by

$K(x)=\frac{\hat{w}(x)}{\hat{w}_{0}(x)}=\frac{2}{x^{2}}\qquad\forall x\neq 0,$

is sample stabilizing for the original control system (48). In particular, an
associated descent rate is $\beta(R,t):=Re^{-t/2}$ for all
$(R,t)\in[0,+\infty)^{2}$. Indeed, let $(R,r)$ be a pair with $0<r<R$, and
choose any sampling time $\delta(R,r)>0$, which is continuous, $r$-increasing
and $R$-decreasing, such that $\delta(R,r)\leq\ln(\varphi)$, where
$\varphi:=(1+\sqrt{5})/2$ is the Golden Mean. Then, for any partition
$\pi=(t_{i})_{i}$ of $[0,+\infty)$ with sampling time $\delta(R,r)$, the
$\pi$-sampling trajectory from each $z\not=0$ with $\mathbf{d}(z)\leq R$
associated to $K$, satisfies the recursive relation

$\dot{x}(t)=x(t)-\frac{1}{(x(t_{n}))^{2}}x^{3}(t)\quad
t\in[t_{n},t_{n+1}],\quad x(0)=z.$

In view of the definition of $\delta(R,r)$, one has
$e^{t-t_{n}}\in[1,\varphi]$ for all $t\in[t_{n},t_{n+1}]$ and, in particular,
this implies $e^{3(t-t_{n+1})}-2e^{2(t-t_{n})}+1\leq 0$ for all
$t\in[t_{n},t_{n+1}]$. Using (49) (with $M=2(x(t_{n}))^{-2}$) we then obtain,
after few computations, the estimate

$|x(t)|=|x(t_{n})|\frac{e^{t-t_{n}}}{\sqrt{2e^{2(t-t_{n})}-1}}\leq|x(t_{n})|e^{-(t-t_{n})/2}\quad\forall
t\in[t_{n},t_{n+1}].$

In particular $|x(t_{n+1})|\leq|x(t_{n})|e^{-(t_{n+1}-t_{n})/2}$, for all
$n\geq 0$, therefore

$|x(t)|\leq|z|e^{-t/2}=\beta(|z|,t)\qquad\forall t\geq 0.$

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