Erick De La Barra

Mauricio Allendes

Emilio Vilches

Valentina Giaconi

Evelyn Cueva

Juan Carlos López

Pedro Perez

Antoine Haddon

Ignacio Brevis

Rodrigo Quezada

Alexis Fuentes

Viviana Solano

Beddani Abdallah

Nouredine Mouslim

Abdeldjabar Bourega

Ba Khiet

Cam Tu

Dinh Hoang

Thuy Phuong

The Cang

Gia Bao

Thi Nga

Duy Khanh

Phuoc Tai

Trong Phong

Van Thanh

Van Duc

Van Vu

Ngoc Nguyen

Nang Thieu

Nhu Thao

Quynh Tram

Bao Tran Nguyen

Keywords: Differential inclusion, maximal monotone operators, Lyapunov function, invariant set, prox-regular sets, Cusco mapping, boundary points

W ≡ 0 and a ≡ 0, one recovers the classical concept of Lyapunov functions V : H → R ∪ {+∞}, that satisfy V (x(t; x 0 )) ≤ V (x 0 ) for all t ≥ 0.

Lyapunov functions are fundamental for the study of different stability concepts of dynamical systems, including Lyapunov stability, asymptotic stability, exponential stability and so on. From the mechanical point of view, Lyapunov functions are interpreted as energy-like functions whose decreasingness along trajectories of the systems drives the system to its equilibrium state. They are also important for control theory where they play a crucial role within the theory of Hamilton-Jacobi equations.

General introduction

The main objective of this work is the study in Hilbert spaces setting of nonsmooth Lyapunov functions and pairs associated to dynamical systems represented as firstorder differential inclusions of the following form ẋ(t; x 0 ) ∈ F (x(t; x 0 )), t ∈ [0, T ), x(0; x 0 ) = x 0 , (

for appropriate initial conditions x 0 ∈ H, and different types of multifunctions F : H ⇒ H defined on a real Hilbert space H. Namely, we provide primal and dual explicit criteria for a pair of given lower semi-continuous extended real-valued functions V, W : H → R ∪ {+∞}, and a nonnegative real number a, to be an a-Lypunov pair associated to differential inclusion (1.1); that is, e at V (x(t; x 0 )) + t 0 W (x(τ ; x 0 ))dτ ≤ V (x 0 ), for all t ≥ 0 and all x 0 closed to points of definition of multifunction F. In this way, our analysis allows the initial condition x 0 to be possibly a point where F is not well-defined. The inequality above may hold for at least one solution of (1.1), in which case the pair (V, W ) is referred to as a weak a-Lyapunov pair, or for all solutions of (1.1), and in this case we say that (V, W ) is a strong a-Lyapunov pair. The objective of this thesis fits within the main spirit of Lyapunov's nondirect approach to the stability of differential equations, since that we provide criteria for a-Lypunov pairs, which only depend on the involved data, represented by F, and which do not require an apriori knowledge of the solutions. When

The main novelty of this work resides in the consideration of nonsmooth data functions, namely function V which is allowed to be nondifferentiable, even may having extended real-values. The nonsmoothness is handled by the use of general subdifferentials and deep techniques from nonsmooth and variational analysis. We also allow F to be a very general multifunction, so that to include the cases of maximal monotone operators, of Cusco (convex upper semi-continuous, nonempty compact valued) mappings, or both.

We are also concerned with the investigation of explicit criteria of the so-called invariant sets associated to differential inclusion (1.1); that is, sets S ⊂ H such that x 0 ∈ S =⇒ x(t; x 0 ) ∈ S for all t ≥ 0.

As for Lyapunov pairs, when the relation above is satisfied for all solutions, the set S is said to be a strong invariant set, and a weak invariant or viable when such a relation is satisfied for at least one solution of (1.1). Invariant sets and Lyapunov pairs or functions associated to general differential inclusions/equations of the form of (1.1) have been the subject of extensive research during the last decades; namely, in relation with differential inclusions involving Cusco mappings in their right-hand side (see, e.g., [START_REF] Aubin | Viability Theory, Systems & Control: Foundations & Applications[END_REF]), or (possibly unbounded) maximal monotone operators (see, e.g., [START_REF] Aubin | Viability Theory, Systems & Control: Foundations & Applications[END_REF][START_REF] Barbu | Nonlinear differential equations of monotone types in Banach spaces[END_REF][START_REF] Brézis | Operateurs Maximaux Monotones et Semi-Groupes de Contractions Dans Les Espaces de Hilbert[END_REF][START_REF] Pazy | The Lyapunov Method for Semigroups of Nonlinear Contractions in Banach Space[END_REF]).

The problem of a-Lyapunov pairs will be investigated in different setting, relying on the nature of the multifunction F governing the dynamical system in (1.1). Due to these various situations depending on the right-hand side F, the scope of this work covers different topics of analysis and optimization theory, including the theory of maximal monotone operators, differential inclusions and Chapter 1 : Introduction equations, nonsmooth and variational analysis, and stability theory.

The need of more explicit conditions for a-Lyapunov pairs and invariant sets, depending only on the data F and the Lyapunov candidate functions and invariant candidates sets, is important for many reasons. For example, inclusion (1.1) above is sometimes evoked as a companion tool to analyze other differential inclusions, in which case the operator F may not be known explicitly, and the access to its semi-group (in the case of maximal monotone operators) can be more complicate.

In [START_REF] Adly | A convex approach to differential inclusions with prox-regular sets: Stability analysis and observer design[END_REF] we investigated the existence of solutions to a differential inclusion governed by the normal cone to a prox-regular set [START_REF] Poliquin | Local differentiability of distance functions[END_REF], by rewriting it in the form of (1.10) with A being some intrinsic maximal monotone operator to this prox-regular set. Such an operator A is not known explicitly but it processes enough information in order to check the invariance of the involved prox-regular set with respect to (1.1). This was sufficient to get the desired existence results; see Section 4.4 of Chapter 4.

Invariant sets are also referred to, in the wide literature, as viable sets [START_REF] Aubin | Viability Theory, Systems & Control: Foundations & Applications[END_REF][START_REF] Aubin | Differential Inclusions. Set-Valued Maps and Viability Theory[END_REF][START_REF] Aubin | Set-Valued Analysis[END_REF], and are of crucial use in many domains, as in economic, renewable resources, biology, diseases propagation, control processes of species and so on. Lyapunov pairs and functions are used extensively in dynamic systems and control theory, among many other applications; see, e.g., [START_REF] Acary | Nonsmooth modeling and simulation for switched circuits[END_REF][START_REF] Brogliato | Nonsmooth mechanics. Models, dynamics and control[END_REF].

Characterizations of Lyapunov pairs for the system (1.1) have been studied in the case of maximal monotone operators by Pazy [START_REF] Pazy | The Lyapunov Method for Semigroups of Nonlinear Contractions in Banach Space[END_REF], and next, extended to single-valued Lipschitz perturbations of maximal monotone operators by Carja and Montreanu [START_REF] Carja | Characterization of Lyapunov pairs in the nonlinear case and applications[END_REF], Kocan and Soravia [START_REF] Kocan | Lyapunov Functions for Infinite-Dimensional Systems[END_REF][START_REF] Kocan | Nonlinear, Dissipative, Infinite dimensional systems[END_REF], Adly, Hantoute and Théra [START_REF] Adly | Nonsmooth Lyapunov pairs for infinite-dimensional first order differential inclusions[END_REF][START_REF] Adly | Nonsmooth Lyapunov pairs for differential inclusions governed by operators with nonempty interior domain[END_REF], among many other contributions. Pazy [START_REF] Pazy | The Lyapunov Method for Semigroups of Nonlinear Contractions in Banach Space[END_REF] proved some sufficient criterias for Lyapunov pairs in the homogeneous case (f ≡ 0), by taking into account that the solution has the following explicit form:

x(t; x 0 ) = lim n→∞ (I + λ n A) -kn (x 0 ) (1.2)
whenever λ n k n → t as n → ∞ (see [START_REF] Crandall | Generation of Semi-Groups of Nonlinear Transformations on General Banach Spaces[END_REF]). Observe that since the operator A is maximal monotone, this expression makes sense, actually (I + λ n A) -1 is a welldefined single-valued and Lipschitz mapping.

In the case of single-valued Lipschitz perturbations of maximal monotone operators, Carja and Motreanu [START_REF] Carja | Characterization of Lyapunov pairs in the nonlinear case and applications[END_REF] proved a characterization of Lyapunov pairs for (1.1) in the Banach spaces setting where A is a multi-valued m-accretive operator. The characterizations of [START_REF] Carja | Characterization of Lyapunov pairs in the nonlinear case and applications[END_REF] rely on the flow invariance and the contingent derivative associated to the operator A. Kocan and Soravia [START_REF] Kocan | Lyapunov Functions for Infinite-Dimensional Systems[END_REF][START_REF] Kocan | Nonlinear, Dissipative, Infinite dimensional systems[END_REF] 1.1. General introduction provided another characterization using nonlinear unbounded Hamilton-Jacobi partial equations, whose viscosity solutions turn to be Lyapunov functions. These two approaches use the semigroup generated by the operator A. In Adly, Hantoute and Théra [START_REF] Adly | Nonsmooth Lyapunov pairs for infinite-dimensional first order differential inclusions[END_REF][START_REF] Adly | Nonsmooth Lyapunov pairs for differential inclusions governed by operators with nonempty interior domain[END_REF], the authors provided a characterization which does not involve the semi-group generated by A. The case of Cusco mappings was treated for example in [START_REF] Clarke | Nonsmooth Analysis and Control Theory[END_REF][START_REF] Clarke | Qualitative properties of trajectories of control systems: a survey[END_REF][START_REF] Clarke | Approximate invariance and differential inclusions in Hilbert spaces[END_REF], where the authors use Euler approximations to provide criteria for strong and weak invariance in terms of the associated Hamiltonian. All these results will be reviewed at the end of this chapter.

An overview of Lyapunov methods

This work goes in the spirit of the nondirect Lyapouv's method to approach stability problems of complex dynamical systems, whose solutions are not easily accessible or that the associated calculation are expensive way. Roughly speaking, in front of lack of explicit information on the solutions of (1.1), the original Lyapunov's idea to check wether a given dynamical system is stable, consists of looking for an associated nonnegative real-valued function V, hopefully regular, which is "strongly" continuous in the sense that x → θ iff V (x) → 0, and such for each trajectory x(•; x 0 ) of (1.1) t → V (x(t; x 0 )) is non-increasing.

(1.

3)

The existence of such a function easily ensures the stability of the system at its equilibrium point θ; that is, for every ε > 0, there exists δ > 0 such that for every y ∈ B(θ, δ) and every solution x(•; y) of (1.1), we have that x(t; y) ∈ B(θ, ε) for all t ≥ 0. Since we do not dispose of explicit calculus of the solutions of (1.1), a natural question is then to find accurate criteria depending only on V, and which guarantee the validity of relation (1.3). Suppose for instance that our setting is finite-dimensional, H = R n , and that (1.1) is a classical differential equation; that is, F = f with f being continuous. If V is smooth, it can be easily verified that the following condition written by means of the derivative of V, V , or equivalently, the gradient of V, V (x), f (x) = ∇V (x), f (x) ≤ 0, for all x ∈ R n , (1.4) leads to relation (1.3). Indeed, in this case, function x(•; x 0 ) is C 1 and by the classical chain rule we obtain that d dt V (x(t; x 0 )) = ∇V (x(t; x 0 )), f (x(t; x 0 )) ≤ 0, which means that V (x(•; x 0 )) is non-increasing. By arguing similarly as above, the use of a-Lyapunov's pairs instead of functions allows analyzing other concepts of stability as asymptotic and exponential stability. Now, to extend this analysis to differential inclusion (1.1) one has to handle the following difficulties:

The solution x(•; x 0 ) of (1.1) may not be sufficiently smooth; for instance, it is generally only continuous when F = -A, with A a maximal monotone 1.1. General introduction operator, and x 0 in the closure of the domain of A. In this case, the solutions of (1.1) are understood in the weak sense. Even under existence of strong solutions that are absolutely continuous, all what can be expected is that they are differentiable almost everywhere.

The domain of F does not need to be closed, nor the values of F are necessarily bounded or even nonempty. This makes the scope of the equation above going beyond the differential inclusions treated in [START_REF] Adly | Nonsmooth Lyapunov pairs for differential inclusions governed by operators with nonempty interior domain[END_REF][START_REF] Adly | Nonsmooth Lyapunov pairs for infinite-dimensional first order differential inclusions[END_REF][START_REF] Aubin | Set-Valued Analysis[END_REF][START_REF] Barbu | Nonlinear differential equations of monotone types in Banach spaces[END_REF][START_REF] Bauschke | Convex Analysis and Monotone Operator Theory in Hilbert Spaces[END_REF], where the right-hand side is generally represented by a Cusco set-valued mapping (in particular, with nonempty and weak*-compact multi-valued operator). The monotonicity assumption of F will compensate the lack of compacity in our differential inclusion (1.1), while the maximality of this operator guarantees, among other properties, the existence and the regularity of solutions. These two facts are also essential when checking the invariance of closed sets.

The Lyapunov candidate functions are generally only lower semi-continuous, while the solution x(•; x 0 ) is only continuous, or at most absolutely continuous. Thus, one need to use tools of nonsmooth analysis, like general subdifferentials, calculus subdiferential rules, mean value theorems and so on, in order to provide criteria written by means of first-order approximations of the Lyapunov functions and invariant sets.

In this thesis, we follow the last ideas above and try to find characterizations in the line of (1.4) when V is only an extended-real-valued lower semi-continuous function. The main tools that allow us to overcome the difficulties listed above come from general subdifferentials theory and techniques of variational analysis.

Our contributions

Our contributions are listed below:

A-Lyapunov stability of differential inclusions with Lipschitz perturbations of maximal monotone operators

We study differential inclusion (1.1) when the right-hand side F is given by

F = f -A,
with A : H ⇒ H being a maximal monotone operator defined on the real Hilbert space H, and f : H → H a Lipschitz perturbation. Hence, the right-hand side may be empty, non-compact or even unbounded, and possibly non uppersemi-continuous. Also, the initial condition x 0 may be not in the domain of definition of operator A, giving rise to more general concept of solutions called weak solutions. This model includes and covers many typical partial differential equations, as well as control problems dealing with differential equations of the form ẋ(t; x 0 ) = f (x(t; x 0 )). A typical example of maximal monotone operators is the Fenchel subdifferential mapping of lower semi-continuous convex proper functions, namely, the normal cone to closed convex sets. However, it is known that there are maximal monotone operators which are not necessarily of subdifferential's type. The problem of the existence of solutions was completely solved since the sixties (see, e.g. [START_REF] Aubin | Viability Theory, Systems & Control: Foundations & Applications[END_REF][START_REF] Barbu | Nonlinear differential equations of monotone types in Banach spaces[END_REF][START_REF] Brézis | Operateurs Maximaux Monotones et Semi-Groupes de Contractions Dans Les Espaces de Hilbert[END_REF][START_REF] Pazy | The Lyapunov Method for Semigroups of Nonlinear Contractions in Banach Space[END_REF]) and many important results have been done, regarding the regularity of solutions. We investigate in this thesis different primal and dual criteria for closed invariant sets and lower semi-continuous (extended realvalued) a-Lyapunov pairs with respect to differential inclusion (5.7), now written as ẋ(t; x 0 ) ∈ f (x(t; x 0 )) -A(x(t; x 0 )), t ∈ [0, T ), x(0; x 0 ) = x 0 ∈ domA.

(1.5)

We provide the following sharp criteria for the associated invariant closed sets S ⊂ domA ∩ S, using proximal normal cones: for every x ∈ S ∩ domA there exists a large enough m > 0 such that for all y ∈ S closed to x it holds ξ, f (y) -y * ≤ 0, (1.6) where S m is some appropriate subset of S, which reflects in some sense the farness of the values of A from the origin θ. This criterion leads to more explicit invariance criteria in many natural situations. For instance, we have proved that when the minimal norm mapping A • is locally bounded, each one of the following conditions (i)-(ii)-(iii), provides a characterization for a closed set S ⊂ domA to be invariant for (1.5):

(i) For every x ∈ S ∩ domA

f (x) -Π A(x) (f (x)) ∈ T B S (x),
where T B S (x) is the Bouligand tangent cone and Π A(x) is the orthogonal projection onto A(x); in particular, when A is a normal cone mapping, A = N C for some closed convex set C ⊂ H, this last relation is also equivalent to Π T C (x) (f (x)) ∈ T B S (x), ξ, f (x) -Π A(x) (f (x)) ≤ 0, where N S stands for either the proximal normal cone N P S or the Fréchet normal cone N F S . It is worth observing that when S is a closed convex set, this last condition reads f (x) -Π A(x) (f (x)) ∈ (N S (x)) • = T S (x), and one goes back to condition (i) above. However, this argument cannot be extended outside convex sets since that the relation (N S (x)) • = T S (x) is not true in general for closed sets which are not convex. This is to say that the last condition is meaningful and may have an interpretation which differs from the one in (i); however, both are equivalent.

(iii) For every x ∈ S ∩ domA sup

ξ∈N S (x) inf x * ∈f (x)-A(x) ξ, x * ≤ 0.
This condition is very practical since it only appeals to the values of the data, which are the mapping f and the operator A, and thus no projection is needed.

The main feature of criteria (i)-(ii)-(iii) above is that they only involve the position of the set S regarding the values of A and f, as the figure above shows.

The generality of our setting, dealing with general lower semi-continuous extended-real-valued functions, allows us to make an exact correspondence between Chapter 1 : Introduction a-Lyapunov pairs and invariant sets. For instance, it is not difficult to verify that a lower semi-continuous function V : H → R ∪ {+∞} is Lyapunov for (1.5) if and only if the epigraph of V is invariant with respect to the following augmented differential inclusion given in H × R, ( ẋ(t; x 0 ), α(t; α 0 )) ∈ (f (x(t; x 0 )) -A(x(t; x 0 )), 0), t ∈ [0, T ).

Hence, then invariance criterion above is naturally rewritten into a criterion for a-Lyapunov pairs in the following form: a pair (V, W ) of two proper lower semicontinuous functions forms an a-Lyapunov pair for differential inclusion (1.5) whenever for every x ∈ domV ∩ domA there exists a large enough m > 0 such that for all y closed to x we have that sup ξ∈∂ P (V +I Am )(y) inf y * ∈A(y)∩B(θ,m) ξ, f (y) -y * + aV (x) + W (x) ≤ 0, where the set A m plays a similar role as the set S m above. Similarly, when A is locally bounded, we prove that (V, W ) forms an a-Lyapunov pair with respect to system (1.5) if and only if one of the following conditions is satisfied: V (x; f (x) -v) + aV (x) + W (x) ≤ 0.

Let us observe that, according to theses equivalences, when looking for complete characterizations of Lyapunov pairs it doesn't matter to consider either the proximal or the Fréchet subdifferentials. However, when verifying the validity of conditions (i ) or (ii ), it is more natural in practice to check the inequalities in (i ) and (ii ) only for the proximal subdifferential, which is in general smaller than 1.1. General introduction the Fréchet subdifferential. For instance, we know that for differentiable functions in finite dimensions we always have that ∂ F V = ∇V , while ∂ P V may be empty at some differentiability points of V even for C 1 functions. We also observe that for differentiable function V , each one of the relations above is equivalent to inf x * ∈A(x) ∇V (x), f (x) -x * + aV (x) + W (x) ≤ 0.

However, as we have just commented above, it is enough to verify this last inequality only for points x where ∂ P V (x) is not empty. It is then clear that the main advantage in using the proximal mapping comes from being the smallest one among well-known subdifferentials.

Condition (ii ) is a kind of Hamilton-Jacobi inequality. Let us denote h(x, p) := inf x * ∈A(x) p, f (x) -x * , so that condition (ii ) is written as h(x, ∂V (x)) := sup p∈∂V (x) h(x, p) ≤ -aV (x) -W (x), and V is seen as a lower solution of the Hamilton-Jacobi inequality ( [START_REF] Clarke | Nonsmooth Analysis and Control Theory[END_REF]) h(x, ∂V (x)) ≤ -aV (x) -W (x).

As with the invariance criteria above, only the data of system (1.5) are used. When the minimal norm mapping A is not necessarily locally bounded, one needs to consider in the criteria above the singular subddifferential of V, ∂ ∞ V (see [START_REF] Adly | Nonsmooth Lyapunov pairs for infinite-dimensional first order differential inclusions[END_REF][START_REF] Adly | Nonsmooth Lyapunov pairs for differential inclusions governed by operators with nonempty interior domain[END_REF]).

The results above extend the ones in [START_REF] Adly | Nonsmooth Lyapunov pairs for infinite-dimensional first order differential inclusions[END_REF][START_REF] Adly | Nonsmooth Lyapunov pairs for differential inclusions governed by operators with nonempty interior domain[END_REF] by removing the assumption of the weak closedness of the candidate invariant sets. Only the data of the system represented by A and f are appealed to within the presented criteria and, so, no need to solve explicitly the differential inclusion (1.1). The invariant results of this work are then rewritten as criteria for lower semi-continuous a-Lyapunov pairs which are non-necessarily weakly lower semi-continuous functions. Because the sets we consider are not necessarily convex or smooth, and the candidate Lyapunov functions are not necessarily sufficiently regular, we use techniques from nonsmooth analysis (e.g. [START_REF] Clarke | Nonsmooth Analysis and Control Theory[END_REF][START_REF] Mordukhovich | Variational Analysis and Generalized Differentiation I: Basic Theory, Grundlehren der Mathematischen Wissenschaften[END_REF][START_REF] Rockafellar | Variational Analysis, Grundlehren der Mathematischen Wissenschaften[END_REF]), including general subdifferentials. The main invariance criterion is given by means of the normal cone to the nominal set. Other invariance results are given by means of primal and dual conditions. These results are next Chapter 1 : Introduction applied to obtain criteria for a-Lyapunov pairs associated to differential inclusion (1.5), including primal conditions using the directional derivatives of the Lyapunov candidate functions, and dual ones using general subdifferentials of such functions like the Proximal, the Fréchet, the singular, and the limiting subdifferentials. The result of this part are presented in Chapter 3.

B. Lyapunov stability of differential inclusions with prox-regular sets

We study the case when F = f -N C , so that differential inclusion (1.1) takes the form ẋ(t; x 0 ) ∈ f (x(t; x 0 )) -N C (x(t; x 0 )) a.e. t ≥ 0, x(0

; x 0 ) = x 0 ∈ C, (1.7)
where C is a uniformly prox-regular set and f : H → H is a Lipschitz mapping.

Here, N C refers to the proximal normal cone to the set C. In general, the set C may depend on the time parameter, in which case the underlying system is refered to as a sweepping process, a name coined in the sixties by Moreau who studied the convex case (C(t) convex) and applied it in mathematical models of elastoplacity mechanical problems. Differential inclusion (1.7) appears in the modeling of many concrete problems in economics, unilateral mechanics, electrical engineering as well as optimal control (see, eg., [START_REF] Acary | Nonsmooth modeling and simulation for switched circuits[END_REF][START_REF] Colombo | The minimum time function for the controlled Moreau's sweeping process[END_REF][START_REF] Mazade | Differential variational inequalities with locally prox-regular sets[END_REF][START_REF] Stewart | Dynamics with inequalities: Impacts and hard constraints[END_REF] and references therein). The model above is also used as a companion system for differential equations of the form ẋ = f (x), for which C is not necessarily invariant. In this case, system (1.7) above is a reasonable approximation of this differential equation, since the corresponding solution naturally remains in the set C ( [START_REF] Cornet | Existence of slow solutions for a class of differential inclusions[END_REF]). The family of uniformly proxregular sets contains and is larger than the family of convex set; for example, the union of two disjoint convex sets is uniformly prox-regular, but obviously is not necessarily convex. Also, the graph of C 2 -functions are prox-regular sets ( [START_REF] Bernard | Uniform prox-regularity of functions and epigraphs in Hilbert spaces[END_REF]).

In the current thesis, we restrict ourselves to time-independent constraint sets, in order to provide a new and natural approach to prove existence of solutions. We also establish new criteria for the associated a-Lyapunov pairs. This model inherits the main difficulties of differential inclusion (1.7), namely, the right-hand side is naturally unbounded, and may even be empty (at points outside the set C). As well, the multimapping -N C is not upper semi-continuous in general.

Existence of solutions of (1.7) are known to occur for general uniformly prox-regular time-depending sets; indeed, in finite-dimension, (1.7) has solutions without any regularity assumption on C ( [START_REF] Benabdellah | Existence of solutions to the nonconvex sweeping process[END_REF][START_REF] Benabdellah | Nonconvex sweeping process[END_REF]). However, the methods used in the literature for this prox-regular setting are very similar to the convex one, 1.1. General introduction originated by Moreau for convex sweeping processes. In our case, we shall follow a different and direct approach, which could use the results already known in the convex case. For this aim, we transform (1.7) into a differential inclusion of the form (1.5), so that we can use and apply invariance and a-Lyapunov criteria established for (1.5) to investigate the existence and properties of solutions, and to give explicit criteria for the invariance and a-Lyapunov pairs associated to (1.7). Then we prove in this case that a closed subset S of C is invariant with respect to system (1.7) if and only if one of the following conditions is satisfied:

(i) For every x ∈ S (f (x) -N C (x)) • ∈ T B S (x).
(ii) For every x ∈ S

[f (x) -N C (x)] ∩ T B S (x) ∩ B(θ, f (x) ) = ∅.
(iii) For every x ∈ S inf

x * ∈[f (x)-N C (x)]∩B(θ, f (x) )
ξ, x * ≤ 0.

Similarly, we also have the following primal and dual characterizations for a-Lyapunov pairs of proper lower semi-continuous functions real-extended-valued functions V, W, associated to differential inclusion (1.7):

(i) For every x ∈ domV and ξ ∈ ∂V (x)

ξ, (f (x) -N C (x)) • + aV (x) + W (x) ≤ 0.
(ii) For every x ∈ domV and ξ ∈ ∂V (x) min

x * ∈N C (x)∩B(θ, f (x) ) ξ, f (x) -x * + aV (x) + W (x) ≤ 0. (iii) For every x ∈ domV inf x * ∈N C (x)∩B(θ, f (x) ) V (x; f (x) -x * ) + aV (x) + W (x) ≤ 0.
These results are next applied to study the stability and observers design of Lur'e systems involving non-monotone set-valued nonlinearities.

C. Lyapunov stability of differential inclusions with Lipschitz Cusco perturbations of maximal monotone operators

Chapter 1 : Introduction

In this part, we study differential inclusion (1.1) when

F = f -A,
with A : R n ⇒ R n being a maximal monotone operator defined on R n , and f : R n ⇒ R n is a Lipschitz multifunction. The study of the Lyapunov stability of this problem by means of criteria in the form given in paragraphs above A and B, has not been adressed before in the current generality. This is why we restrict ourselves in this work to the finite-dimensional setting. Compared to paragraphs A and B above, we face here the problem of having multiple solutions, and this leads us to consider weak and strong invariant sets, as well as weak and strong a-Lyapunov pairs for our differential inclusion (1.1), which takes the form

ẋ(t; x 0 ) ∈ f (x(t; x 0 )) -A(x(t; x 0 )) a.e. t ≥ 0, x(0; x 0 ) = x 0 ∈ domA. (1.8)
In this case, we show that a set S ⊂ domA is strong invariant for this differential inclusion if and only if one of the following conditions holds:

(i) For every x ∈ S ∩ domA v -Π A(x) (v) ∈ T B S (x) ∀v ∈ F (x). (ii) For every x ∈ S ∩ domA [v -A(x)] ∩ T B S (x) = ∅ ∀v ∈ F (x). (iii) For every x ∈ S ∩ domA sup ξ∈N S (x) sup v∈F (x) ξ, v -Π A(x) (v) ≤ 0. (iv) For every x ∈ S ∩ domA sup ξ∈N S (x) sup v∈F (x) inf x * ∈A(x) ξ, v -x * ≤ 0.
It is worth obersving that we do not assume here that the minimal section A • is locally bounded. On the other hand, a closed S ⊂ domA such that A • is locally bounded on S, is weak invariant for differential inclusion (1.8) if and only if one of the following conditions holds:

(i) For every x ∈ S, there exist v ∈ F (x), x * ∈ A(x) ∩ B(θ, F (x) + m(x)) 1.1. General introduction such that v -x * ∈ T B S (x). (ii) For every x ∈ S sup ξ∈N S (x) inf v∈F (x) inf x * ∈A(x)∩B(θ, F (x) +m(x)) ξ, v -x * ≤ 0.
where m(x) := lim sup y→x,y∈S

A • (y) .
Concerning Lyapunov pairs, we obtain that a pair (V, W ) of two proper lower semi-continuous extended-real-valued functions such that domV ⊂ domA, forms a strong a-Lyapunov pair for differential inclusion (1.8) if and only if for every x ∈ domV sup

ξ∈∂ P V (x) sup v∈F (x) inf x * ∈ A(x) ξ, v -x * + aV (x) + W (x) ≤ 0 and sup ξ∈∂ P,∞ V (x) sup v∈F (x) inf x * ∈A(x) ξ, v -x * ≤ 0.
In this case, we need to consider the singular proximal subdifferential of function V, ∂ P,∞ V. However, if A • is locally bounded on domV , then (V, W ) forms a weak a-Lyapunov pair for differential inclusion (1.8) if and only if one of following hold

(i) For any x ∈ domV sup ξ∈∂V (x) inf v∈F (x) inf x * ∈A(x)∩B(θ,m(x)) ξ, v -x * + aV (x) + W (x) ≤ 0,
where ∂ stands for either ∂ P , ∂ F , or ∂ L .

(ii) For any x ∈ domV inf

v∈F (x) inf x * ∈A(x)∩B(θ, F (x) +m(x)) V (x; v -x * ) + aV (x) + W (x) ≤ 0.
This result is also applied to systems involving uniformly prox-regular sets, given in the form

ẋ(t; x 0 ) ∈ f (x(t; x 0 )) -N C (x(t; x 0 )) a.e. t ≥ 0, x(0; x 0 ) = x 0 ∈ C, (1.9)
with f being a Lipschitz Cusco multifunction. We show that for V, W : H → R as above, such that domV ⊂ C, form a strong a-Lyapunov pair (a ≥ 0) for differential inclusion (1.9) if and only if one of the following conditions holds:

Chapter 1 : Introduction (i) for every x ∈ domV sup ξ∈∂V (x) sup v∈F (x) ξ, v -Π N C (x) (v) + aV (x) + W (x) ≤ 0.
(ii) for every x ∈domV

sup ξ∈∂V (x) sup v∈F (x) inf x * ∈N C (x)∩B(θ, F (x) ) ξ, v -x * + aV (x) + W (x) ≤ 0;
(iii) for every x ∈domV

sup v∈F (x) V (x; v -Π N C (x) (v)) + aV (x) + W (x) ≤ 0;
(iv) for every x ∈domV

sup v∈F (x) inf x * ∈N C (x)∩B(θ, F (x) ) V (x; v -x * ) + aV (x) + W (x) ≤ 0.

D. Application to the geometry of maximal monotone operators:

In this part, we characterize the boundary of the values of maximal monotone operators defined in Hilbert spaces, by means only of the values at nearby points, which are closed enough to the reference point but distinct of it. This allows to write the values of such operators using finite convex (2-)combinations of the values at such nearby points. We also provide similar characterizations for the normal cone to prox-regular sets.

For instance, given a maximal monotone operator A : H ⇒ H, defined on a Hilbert space H, for every x ∈ H we have bd

(A(x)) = Limsup y→ = x bd(A(y)) = Limsup y→ = x A(y),
and, consequently, for every x ∈ domA such that bd(A(x)) = ∅ we obtain

A(x) = N cl(domA) (x) + co 2 Limsup y→ = x A(y) .

Previous results from the literature

First results dealing with Lyapunov pairs and functions associated to (1.5) have been established by Pazy in [START_REF] Pazy | On the Lyapunov method for evolution equations governed by accretive operators[END_REF][START_REF] Pazy | The Lyapunov Method for Semigroups of Nonlinear Contractions in Banach Space[END_REF] in the case of homogeneous systems governed

V (J λ (x)) -V (x) λ + W (x) ≤ 0,
where J λ := (I + λA) -1 is the resolvent operator A.

Pazy's results have been extended by Kocan and Soravia [START_REF] Kocan | Lyapunov Functions for Infinite-Dimensional Systems[END_REF] (see, also, [START_REF] Carja | Characterization of Lyapunov pairs in the nonlinear case and applications[END_REF]) to the non-homogeneous case; that is, when (1.1) is written as

ẋ(t; x 0 ) ∈ f (x(t; x 0 )) -A(x(t; x 0 )) t ∈ [0, T ), x(0; x 0 ) = x 0 ∈ H, (1.10)
with f being a Lipschitz mapping. The approach of [START_REF] Carja | Characterization of Lyapunov pairs in the nonlinear case and applications[END_REF][START_REF] Kocan | Lyapunov Functions for Infinite-Dimensional Systems[END_REF] uses implicit criteria depending heavily on the semi-group generated by the maximal monotone operator A. For instance, Kocan and Soravia show that (V, W ) is a Lyapunov pair for differential inclusion (1.10) if and only if (V, W ) is a solution in the viscosity sense of the following differential inequality

A(x), DV (x) -f (x), DV (x) ≥ W (x).
Recently, always dealing with Lipschitz perturbations of maximal monotone operators, different criteria for weak lower semi-continuous a-Lyapunov pairs have been investigated in [START_REF] Adly | Nonsmooth Lyapunov pairs for infinite-dimensional first order differential inclusions[END_REF][START_REF] Adly | Nonsmooth Lyapunov pairs for differential inclusions governed by operators with nonempty interior domain[END_REF], using the condition that for any x in the domain of V (which is a subset of the closure of the domain of A, domA) and δ > 0 it holds sup

ξ∈∂ P V (x) lim inf y domA -→ x inf y * ∈A(y) ξ + δ(y -x), f (y) -y * + aV (x) + W (x) ≤ 0, sup ξ∈∂ P,∞ V (x) lim inf y domA -→ x inf y * ∈A(y) ξ + y -x, f (y) -y * ≤ 0,
where ∂ P and ∂ P,∞ are the proximal and the proximal singular subdifferential operators, respectively.

More early at the beginning of the twentieth century, in his the pioneering work [START_REF] Haddad | Nonlinear Dynamical Systems and Control: A Lyapunov-Based Approach[END_REF][START_REF] Lasalle | Stability of Dynamical Systems[END_REF], Lyapunov studied stability properties of linear systems that he extended to nonlinear differential equations. These results are known as the first and the second methods of Lyapunov. Since then, this approach continues to be fundamental in the study of dynamical systems from both the theoritecal and applicable points of view. There are other famous results like the ones of Zubov and Krasovskii on the stability of differential equations; i.e., F = f, and which use Lyapunov functions. Lyapunov himself also investigated instability of differential equations (see [START_REF] Haddad | Nonlinear Dynamical Systems and Control: A Lyapunov-Based Approach[END_REF][START_REF] Kong | A short course in ordinary differential equations[END_REF][START_REF] Lasalle | Stability of Dynamical Systems[END_REF][START_REF] Liao | Stability of Dynamical Systems[END_REF]).

We may distinguish in this study two main important types of differential inclusion (1.1), the one involving maximal monotone operators, and the other one governed by upper semi-continuous multifunctions F with nonempty, bounded (or compact), and closed values, which are refereed to as Cusco mappings. We shall also consider in some cases, specially in finite-dimensions, coupled systems covering both situations.

In the case of Cusco multifunctions F defined on R n , considered as the natural extension of classical differential equations, under some standard linear growth hypothesis, differential inclusion (5.7) has solutions (may be not unique) [START_REF] Aubin | Viability Theory, Systems & Control: Foundations & Applications[END_REF][START_REF] Filippov | Differential equations with multi-valued discontinuous righthand side,(Russian)[END_REF]. In this case, (strong and weak) invariant sets and Lyapunov pairs associated to (5.7) have been studied in Clarke et all [START_REF] Clarke | Nonsmooth Analysis and Control Theory[END_REF] (and references therein), using Eulerlike approximations that in the finite-dimensional setting lead to the required solution. It is worth observing that strong invariant sets and strong Lyapunov pairs require in [START_REF] Clarke | Nonsmooth Analysis and Control Theory[END_REF] Lipschitz assumptions on multifunction F . Donchev, Ríos, Wolenski [START_REF] Donchev | Strong invariant and one-sided Lipschitz multifunctions[END_REF] extended the strong invariant results to the class of one-side Lipschitz time-dependent multifunctions, a family which is less restrictive than the class of Lipschitz multifunctions. Colombo, Palladino [START_REF] Colombo | The minimum time function for the controlled Moreau's sweeping process[END_REF] provided similar results to [START_REF] Clarke | Nonsmooth Analysis and Control Theory[END_REF] for classes of time-depend multifunctions of the form F (t, x) = G(t, x) -N C(t) (x), where G is Lipschitz with respect to the second variable x, and C(t) is a uniformly prox-regular set with Lipschitz dependence on t. This class of problems is called Sweeping process. This problem has been introduced and studied by Moreau [START_REF] Moreau | Evolution problem associated with a moving convex set in a Hilbert space[END_REF]. More precisely, Moreau studied the existence of solution of the differential inclusion which has the form

ẋ(t) ∈ -N C(t) (x(t)) t ≥ 0, x(0) = x 0 ∈ C(0), (1.11)
where C(t) is closed convex set in a Hilbert space. This problem has been developed by Castaing and his collaborators to study the following differential inclusion

ẋ(t) ∈ -N C(t) (x(t)) + F (t, x(t)) t ≥ 0, x(0) = x 0 ∈ C(0), (1.12)
where all the sets C(t) are either convex or complements of open convex sets (in finite-dimensional settings). Differential inclusion (1.12) was studied by Mazade, Thibault [START_REF] Mazade | Differential variational inequalities with locally prox-regular sets[END_REF][START_REF] Mazade | Regularization of differential variational inequalities with locally prox-regular sets[END_REF] for the case when C(t) = C is a uniformly prox-regular subset of a Hilbert space, while in Adly, Haddad and Thibault [START_REF] Adly | Convex sweeping process in the framework of measure differential inclusions and evolution variational inequalities[END_REF], the authors consider the case when C(t) are closed convex sets. These last papers only studied the existence of solutions, but recently, Mazade and Hantoute [START_REF] Mazade | Lyapunov functions for evolution variational inequalities with uniformly prox-regular sets[END_REF] have given characterizations for Lyapunov pairs (V, W ) (with V being weak lower semi-continuous) with respect to (1.12), for the cases F = -N C + f , where C is uniformly prox-regular subset of a Hilbert space.

Chapter 2

Notation and main concepts

Throughout the thesis, we frequently work in a real Hilbert space H which is endowed with the inner product •, • and associated norm • , and identified to its dual space H .

We use the notations →, to the denote strong and weak convergence, respectively, and write

S

→,

S if the convergence is restricted to a set S ⊂ H. We denote by B(x, r) the closed ball in H with center x and radius r, in particular, we denote B = B(θ, 1) and by B r the ball B(θ, r). Given a subset S of H, we denote by S ( or cl(S)), co(S), co(S), cone(S), int(S), bd(S), the closure, the convex hull, the closed convex hull, the conic hull, the interior, and the boundary of the set S, respectively. S • denotes the set of points of minimal norm in S, i.e., S • := {x ∈ S | x ≤ s , for all s ∈ S}.

Suppose that K is a convex, closed subset of H. We denote

(S -K) • := {(s -K) • | s ∈ S}.
The dual cone set of S is the set

S * := {x * ∈ H | x * , x ≤ 0 for all x ∈ S}.
The indicator function and the distance function are respectively given by I S (x) := 0 if x ∈ S; +∞ otherwise, and d S (x) := inf{ x -y | y ∈ S}.

The support function of a non-empty set S is given by σ S (x) := sup{ x, s | s ∈ S}.

For δ ≥ 0, we denote Π δ S the (orthogonal ) δ-projection mapping onto S defined as

Π δ S (x) := {y ∈ S : x -y 2 ≤ d 2 S (x) + δ 2 }.
When δ = 0, we simply write Π 0 S (x) =: Π S (x). It is known that Π S is nonemptyvalued on a dense subset of H \ S (see [START_REF] Clarke | Nonsmooth Analysis and Control Theory[END_REF]). We have the following theorem: Theorem 2.1. [START_REF] Clarke | Approximate invariance and differential inclusions in Hilbert spaces[END_REF][START_REF] Radulescu | Geometric Approximation of Proximal Normals[END_REF] Suppose that S is closed. Then, for any x ∈ H and any s ∈ Π δ S (x), with δ > 0, there exist s δ ∈ S and y ∈ H such that

     y -s δ ∈ N P S (s δ ), y -s δ -(x -s) ≤ 2δ, s -s δ ≤ δ, x -y ≤ δ. (2.1)
In addition, if x ∈ B(x 0 , σ) for some x 0 ∈ S and σ > 0, then s δ satisfies

s δ -x 0 2 ≤ 6σ 2 + 8δ 2 . (2.2)
Proof. We can find the proof of the first part of the theorem in [START_REF] Clarke | Approximate invariance and differential inclusions in Hilbert spaces[END_REF][START_REF] Radulescu | Geometric Approximation of Proximal Normals[END_REF]. We now suppose that x ∈ B(x 0 , σ). One has

s δ -x 0 2 ≤ 2 s δ -x 2 +2 x -x 0 2 ≤ 4 s δ -s 2 +4 x -s 2 +2σ 2 ≤ 4δ 2 + 4(d 2 S (x) + δ 2 ) + 2σ 2 ≤ 8δ 2 + δ 2 + 4 x -x 0 2 +2σ 2 ≤ 8δ 2 + 6σ 2 ,
which completes the proof of the theorem.

Finally, let a function ϕ :

H → R := R ∪ {+∞} be given. The domain of ϕ is domϕ := {x ∈ H | ϕ(x) < +∞},
and the epigraph of ϕ is

epiϕ := {(x, α) ∈ H × R | ϕ(x) ≤ α}.

Basic definitions and some properties of nonsmooth analysis

In this section, we remind some basic concepts of nonsmooth analysis theory and the theory of solutions of differential inclusions which play an important role in our work.

First, let us remind some concepts and properties of convex analysis. (ii) A function ϕ : H → R is called convex if for any x, y ∈ domϕ and any α ∈ [0, 1], one has

ϕ(αx + (1 -α)y) ≤ αϕ(x) + (1 -α)ϕ(y) (with 0.∞ = ∞).
Example 2.5. (i) The closed balls B(x 0 , r), r ≥ 0 are convex.

(ii) The function V (x) := x p is convex whenever p ≥ 1.

(iii) Suppose that ϕ ∈ F(H). Then the conjugate function ϕ * of ϕ which is defined by

ϕ * (x) = sup y∈H { x, y -ϕ(y)} 2.
1. Basic definitions and some properties of nonsmooth analysis is convex.

We have

I * S = σ S , σ * S = I co(S) , ( 1 2 x 2 ) * = 1 2 x 2 .
(iv) A subset S is convex if and only if the function I S (•) is convex.

Next, we remind some basic concepts in nonsmooth analysis Definition 2.6. Let ϕ ∈ F(H) and let x ∈ domϕ. The contingent directional

derivative of ϕ at x ∈ domϕ in the direction v ∈ H is ϕ (x; v) := lim inf t→0 + ,w→v ϕ(x + tw) -ϕ(x) t . (2.3) Definition 2.7. Let ϕ ∈ F(H) and let x ∈ domϕ. (i) A vector ξ ∈ H is called a proximal subgradient of ϕ at x, written ξ ∈ ∂ P ϕ(x), if there exist ρ > 0 and σ ≥ 0 such that ϕ(y) ≥ ϕ(x) + ξ, y -x -σ x -y 2 ∀y ∈ B(x, ρ). (ii) A vector ξ ∈ H is called a Fréchet subgradient of ϕ at x, written ξ ∈ ∂ F ϕ(x), if the following inequality holds ϕ(y) ≥ ϕ(x) + ξ, y -x + o( y -x ) ∀y ∈ H. (iii) A vector ξ ∈ H is called a basic/limiting subgradient of ϕ at x, written ξ ∈ ∂ L ϕ(x), if there are sequences x i ϕ → x, (i.e., x i → x, ϕ(x i ) → ϕ(x)) and ξ i ξ such that ξ i ∈ ∂ P ϕ(x i ) for all i ∈ N. (iv) A vector ξ ∈ H is called a singular subdifferential of ϕ at x, written ξ ∈ ∂ ∞ ϕ(x), if there exist sequences (α i ) i ⊂ R + and (x i ) i , (ξ i ) i ⊂ H such that α i ↓ 0, x i ϕ → x, ξ k ∈ ∂ P ϕ(x k ), α i ξ k ξ. (v) A vector ξ ∈ H is called a Clarke subdifferential of ϕ at x, written ξ ∈ ∂ C ϕ(x), if ξ belongs to the set ∂ C ϕ(x) := co(∂ L ϕ(x) + ∂ ∞ ϕ(x)).
In the case x / ∈ domϕ, we put

∂ P ϕ(x) = ∂ F ϕ(x) = ∂ L ϕ(x) = ∂ C ϕ(x) = ∅.
From the definitions above, it is clear that

∂ P ϕ(x) ⊂ ∂ F ϕ(x) ⊂ ∂ L ϕ(x) ⊂ ∂ C ϕ(x) ∀x ∈ H, (2.4)
and for any x ∈ domV , one has

σ ∂ P ϕ(x) (•) ≤ σ ∂ F ϕ(x) (•) ≤ ϕ (x; •).
When ϕ ∈ F(H) and is convex, then for every x, one has

∂ P ϕ(x) = ∂ F ϕ(x) = ∂ L ϕ(x) = ∂ C ϕ(x) = ∂ϕ(x),
where ∂ϕ(x) is Fenchel subdifferential of ϕ at x which is defined by

∂ϕ(x) := {ξ ∈ H | ϕ(y) ≥ ϕ(x) + ξ, y -x ∀y ∈ H}.
In general, the subdifferential of ϕ at x ∈ domϕ may be empty. However, the set dom(∂ P ϕ) of points where ∂ P ϕ is nonempty dense in domϕ.

Theorem 2.8. [START_REF] Clarke | Nonsmooth Analysis and Control Theory[END_REF] Let ϕ ∈ F(H), x 0 ∈ domϕ and let ε > 0 be given. Then there exists a point y ∈ B(x 0 , ε) such that

∂ P ϕ(y) = ∅ and ϕ(x 0 ) -ε ≤ ϕ(y) ≤ ϕ(x 0 ).
In particular, dom ∂ P ϕ is dense in dom ϕ.

From this theorem and the inclusions in (2.4), the domains of all the subdifferentials defined above are dense in domϕ. Given a closed set S and x ∈ S, we define the proximal normal cone, Fréchet normal cone, limiting normal cone, Clarke normal cone, respectively, by

N P S (x) := ∂ P I S (x), N F S (x) := ∂ F I S (x), N L S (x) := ∂ L I S (x), N C S (x) := ∂ C I S (x).
From inclusions in (2.4), we derive that (refer [START_REF] Mordukhovich | Variational Analysis and Generalized Differentiation I: Basic Theory, Grundlehren der Mathematischen Wissenschaften[END_REF])

N P S (x) ⊂ N F S (x) ⊂ N L S (x) ⊂ co(N L S (x)) = N C S (x) ∀x ∈ S. (2.5)
We have the relationships between subdifferentials and geometric characterizations 2.1. Basic definitions and some properties of nonsmooth analysis of normal cones:

ξ ∈ ∂ ∞ ϕ(x) ⇔ (ξ, 0) ∈ N L epiϕ (x, ϕ); ξ ∈ ∂ † ϕ(x) ⇔ (ξ, -1) ∈ N † epiϕ (x, ϕ(x)), (2.6)
where " † " stands for P, F, L, C respectively.

In the thesis, we also define the singular proximal subdifferential of ϕ at x as follows:

∂ P,∞ ϕ(x) := {ξ | (ξ, 0) ∈ N P epiϕ (x; ϕ(x))}.
According to [START_REF] Mordukhovich | Variational Analysis and Generalized Differentiation I: Basic Theory, Grundlehren der Mathematischen Wissenschaften[END_REF], if ξ ∈ ∂ P,∞ ϕ(x), then there exist sequences (x n ) ⊂ domϕ, (ξ n ), (α n ) such that

x n ϕ → x, ξ n ∈ ∂ P ϕ(x n ), α n ↓ 0, α n ξ n → ξ as n → ∞.
Proposition 2.9. [START_REF] Clarke | Nonsmooth Analysis and Control Theory[END_REF] Let ϕ ∈ F(H) and let (x, α) ∈ epiϕ. We have that

N P epiϕ (x, α) ⊂ N P epiϕ (x, ϕ(x)). (2.7)
Moreover, if α > ϕ(x) and (ξ, -κ) ∈ N P epiϕ (x, α), κ ≥ 0 then κ = 0.

Proof. If α = ϕ(x) then (2.7) holds. We now suppose that α > ϕ(x) and (ξ, -κ) ∈ N P epiϕ (x, α). There exist η > 0, δ ≥ 0 such that

(ξ, -κ), (y, β) -(x, α) ≤ δ( y -x 2 +(β -α) 2 ) ∀(y, β) ∈ B((x, α), η) ∩ epiϕ.
(2.8) Since α > ϕ(x), then exists η ∈ (0, η) such that (x, β) ∈ epiϕ whenever |β-α|≤ η . Hence, from the last inequality, we obtain that

-κ(β -α) ≤ (β -α) 2 ∀β ∈ (α -η , α + η ), which implies that κ = 0. We now suppose that (y, β) ∈ B((x, ϕ(x)), η) ∩ epiϕ. Then it is clear that (y, β + α -ϕ(x)) ∈ B((x, α), η) ∩ epiϕ.
Hence, from inequality (2.8), one has

(ξ, 0), (y, β) -(x, ϕ(x)) = (ξ, 0), (y, β + α -ϕ(x)) -(x, α) ≤ δ( y -x 2 +(β -ϕ(x)) 2 )
which implies that (ξ, -κ) ∈ N P epiϕ (x, ϕ), and we complete the proof of the proposition.

We will continue with the concept tangent cones Definition 2.10. Let S ⊂ H and let x ∈ S be given.

(i) The Bouligand tangent cone to S at x is defined by

T B S (x) := {v ∈ H | ∃ x i S → x, ∃ t i ↓ 0 such that x i -x t i → v as i → ∞}.
(ii) The Clarke tangent cone to S at x is defined by

T C S (x) := {v ∈ H | ∀ x i S → x, ∀ t i ↓ 0, ∃ v i → v such that x i +t i v i ∈ S for all i ∈ N}.
From this definition, it is clear that T C S (x) is always closed, convex and

T C S (x) ⊂ T B S (x) ∀x ∈ S. (2.9)
Furthermore, we have

N C S (x) = (T C S (x)) * , T C S (x) = (N C S (x)) * .
When S is convex, we have that

N P S (x) = N C S (x) = {x * | x * , y -x ≤ 0 ∀y ∈ S}, T B S (x) = T C S (x) = R + (S -x).
(2.10)

Next, we remind the definition of prox-regular sets and some of their properties.

Definition 2.11. [START_REF] Mazade | Regularization of differential variational inequalities with locally prox-regular sets[END_REF][START_REF] Poliquin | Local differentiability of distance functions[END_REF] For positive numbers r and α, a closed set S is said to be (r, α)-prox-regular at x ∈ S provided that one has x = Π S (x + v), for all x ∈ S ∩ B(x, α) and all v ∈ N P S (x) such that v < r. The set S is r-prox-regular (resp., prox-regular) at x when it is (r, α)-prox-regular at, x for some real α > 0 (resp., for some numbers r, α > 0). The set S is said to be r-uniformly prox-regular when α = +∞. From the definition above, it is clear that if S is r-uniformly prox-regular then S is also r -uniformly prox-regular for every r ≤ r.

Example 2.12. (i) Any closed convex set of H is r-uniformly prox-regular for any r ≥ 0.

2.1. Basic definitions and some properties of nonsmooth analysis (ii) The unit sphere S of H is not convex, but it is r-uniformly prox-regular for any r ≤ 1.

When S is r-uniformly prox-regular, then all of normal cones that mention above coincide, i.e.,

N P S (x) = N F S (x) = N L S (x) = N C S (x) ∀x ∈ S and T B S (x) = T C S (x) ∀x ∈ S;
indeed, we have that

T B S (x) ⊂ (N P S (x)) * = (N C S (x)) * = T C S (x) ⊂ T B S (x).
Hence, we denote T S (x) for these tangent cones. According to Poliquin, Rockafellar and Thibault [START_REF] Poliquin | Local differentiability of distance functions[END_REF], we have the following property of r-uniformly proxregular sets:

Proposition 2.13. Let S be a closed subset of H. If S is r-uniformly prox-regular, then the set-valued mapping defined by x → N P S (x) ∩ B is 1 r -hypomonotone, i.e., for any x, y ∈ C, x * ∈ N P S (x) ∩ B, y * ∈ N P S (y) ∩ B, one has

x * -y * + 1 r (x -y), x -y ≥ 0.
Lemma 2.14. Let C be a r-uniformly-prox-regular set of R n and let κ > 0.

Suppose that A C is any maximal monotone extension of the mapping x → N C (x) ∩ B(θ, κ) + κ r x, then for every x ∈ C, one has

N C (x) ∩ B(θ, κ) + κ r x ⊂ A C (x) ⊂ N C (x) + κ r x (2.11)
and for any v such that v ≤ κ,

(v -N C (x)) • = (v + κ r x -A C (x)) • .
(2.12)

Proof. We refer [START_REF] Adly | A convex approach to differential inclusions with prox-regular sets: Stability analysis and observer design[END_REF] for the first part of the lemma. We now justify the second part of the lemma. By the first part of the lemma, we have that

v -N C (x) ∩ B(θ, κ) ⊂ v + κ r x -A C (x) ⊂ v -N C (x). Since v ≤ κ, it is clear that (v -N C (x) ∩ B(θ, κ)) • = (v -N C (x)) • . Hence, one
Chapter 1 : Notation and main concepts

has (v -N C (x) ∩ B(θ, κ)) • = (v + κ r x -A C (x)) • = (v -N C (x)) • .

Maximal monotone operators

In this section, we remind the concept of maximal monotone operators.

Let A : H ⇒ H be a multivalued operator. We define the domain and the graph of operator A, respectively, as

domA := {x ∈ H | A(x) = ∅}, Gr(A) := {(x, x * ) ∈ H × H | x * ∈ A(x)}.
To simplify, we identify A to its graph. The inverse operator A -1 : H ⇒ H of A defined by Let A be a maximal monotone and let λ > 0. We define the resolvent and the Yoshida approximation of A, respectively, by

A -1 (y) := {x ∈ H | y ∈ A(x)}.
J λ := (I + λA) -1 , A λ := I -J λ λ . (2.14)
According to Bauschke and Combettes [START_REF] Bauschke | Convex Analysis and Monotone Operator Theory in Hilbert Spaces[END_REF], Brézis [START_REF] Brézis | Operateurs Maximaux Monotones et Semi-Groupes de Contractions Dans Les Espaces de Hilbert[END_REF], Phelps [START_REF] Phelps | Convex Functions, Monotone Operators and Differentiability[END_REF][START_REF] Phelps | Lectures on maximal monotone operators[END_REF], etc, for any λ > 0, J λ and A λ are singular and maximal monotone with domJ λ = domA λ = H. Moreover, J λ is 1-Lipschitz, A λ is 1 λ -Lipschitz and

A λ (x) ∈ A(J λ (x)) ∀x ∈ H.
Theorem 2.21. [START_REF] Bauschke | Convex Analysis and Monotone Operator Theory in Hilbert Spaces[END_REF][START_REF] Brézis | Operateurs Maximaux Monotones et Semi-Groupes de Contractions Dans Les Espaces de Hilbert[END_REF][START_REF] Phelps | Convex Functions, Monotone Operators and Differentiability[END_REF][START_REF] Phelps | Lectures on maximal monotone operators[END_REF] Let A : H ⇒ H be a maximal monotone operator.

The following assertions hold:

(i) For every x ∈ H, one has

J λ (x) → Π domA (x) as λ ↓ 0.
In particular, for any x ∈ domA, then J λ (x) → x as λ ↓ 0.

(ii) For any λ, µ > 0, one has (A λ ) µ = A λ+µ .

(iii) For every x ∈ domA, then A λ (x) → A • (x) and A λ (x) → A • (x) as λ ↓ 0.

More generally, we have that

A λ (x) -A • (x) 2 ≤ A • (x) 2 -A λ (x) 2 ,
where A • (x) := Π A(x) (θ).

(iv) If x / ∈ domA then A λ (x) ↑ +∞ as λ ↓ 0.

Differential inclusions with maximal monotone perturbations

In this section, we provide some basic knowledges of the following differential inclusion

   ẋ(t) ∈ f (x(t)) -A(x(t)) t ≥ 0, x(0) = x 0 ∈ domA, (2.15)
where f : H → H is an l-Lipschitz mapping and A : H ⇒ H is a maximal monotone operator. We refer to [START_REF] Aubin | Viability Theory, Systems & Control: Foundations & Applications[END_REF][START_REF] Aubin | Differential Inclusions. Set-Valued Maps and Viability Theory[END_REF][START_REF] Brézis | Operateurs Maximaux Monotones et Semi-Groupes de Contractions Dans Les Espaces de Hilbert[END_REF] for all of results of this section. it can be expressed in the form

x(t) = x(a) + t 0 v(τ )dτ ∀t ∈ [a, b],
for some integrable function v. In this case, we have

ẋ(t) = v(t) a.e. t ∈ [a, b].
Theorem 2.23. [START_REF] Barbu | Nonlinear differential equations of monotone types in Banach spaces[END_REF][START_REF] Brézis | Operateurs Maximaux Monotones et Semi-Groupes de Contractions Dans Les Espaces de Hilbert[END_REF] We consider the differential inclusion (2.15). For any exists and

x 0 ∈ domA, there exists a unique function x : [0, ∞) → H such that (i) x(0) = x 0 , x(t) ∈ domA ∀t ≥ 0, (
d + x(t) dt = f (x(t)) -Π A(x(t)) (f (x(t)) ∀t ≥ 0.
Furthermore, the function t → d + x(t) dt is right continuous and satisfies

d + x(t) dt ≤ e lt (f (x 0 ) -A(x 0 )) • ∀t ≥ 0.
In particular, we have

d + x(0) dt = (f (x 0 ) -A(x 0 )) • .
(iv) Suppose that y 0 ∈ domA and y(•) also satisfies (i), (ii), (iii) with respect to the initial value y 0 . Then we have that (ii) For any x 0 ∈ domA, system (2.15) always has a unique weak solution x(•; x 0 ) which satisfies x(t; x(s; x 0 )) = x(t + s; x 0 ) ∀t ≥ 0, ∀s ≥ 0.

x(t) -y(t) ≤ e lt x 0 -y 0 ∀t ≥ 0. ( 2
(iii) Suppose that x(•; x 0 ) and x(•; y 0 ) are two weak solutions of (2.15) with respect to the two initial values x 0 , y 0 ∈ domA, we have inequality x(t; x 0 ) -x(t; y 0 ) ≤ e lt x 0 -y 0 ∀t ≥ 0.

For each λ > 0, A λ is a Lipschitz mapping. Hence, the following differential inclusion

   ẋ(t) = f (x(t)) -A λ (x(t)) a.e. t ≥ 0, x(0) = x 0 ∈ H
always has a unique strong solution. The proposition below proves that the solution of (2.15) is the limit of the solutions of the differential equations above.

Proposition 2.26. For any x 0 ∈ domA and T > 0, differential inclusion (2.15) has a unique continuous solution, which is the uniform limit on [0, T ] of x λ (•; x 0 ) (as λ ↓ 0), where x λ (•; x 0 ) is the solution of the following differential equation

ẋ(t) = f (x(t)) -A λ (x(t)) a.e. t ∈ [0, T ], x(0) = x 0 .
Proof. We can refer to Brézis [START_REF] Brézis | Operateurs Maximaux Monotones et Semi-Groupes de Contractions Dans Les Espaces de Hilbert[END_REF] and Barbu [14] for the case x 0 ∈ domA. Now we consider the case x 0 / ∈ domA. Let us fix ε > 0 and any sequence λ ↓ 0 and any z ∈ domA such that z -x 0 ≤ e -lT ε. According to Theorem 2.23 then

max{ x λ (t; x 0 ) -x λ (t; z) , x(t; x 0 ) -x(t; z) } ≤ e lt x 0 -z ≤ ε ∀ t ∈ [0, T ].
Since the proposition holds whenever the initial-value belongs to domA, there exists λ 0 > 0 such that for every λ < λ 0 , one has

x(t; z) -x λ (t; z) ≤ ε ∀t ∈ [0, T ].
Combining the above results, we obtain that for any λ ≤ λ 0 , one has

x(t; x 0 ) -x λ (t; x 0 ) ≤ 3ε ∀t ∈ [0, T ],
we complete the proof of the proposition.

Example 2.27. (i) If int(domA) = ∅, then every weak solution of differential inclusion (2.15) is a strong solution. In particular, the conclusion still hold when dimH < +∞.

(ii) The weak solution of the following differential inclusion

   ẋ(t) ∈ -∂ϕ(x(t)) a.e. t ≥ 0, x(0) = x 0 ∈ domϕ,
where ϕ ∈ F(H) and is convex, is a strong solution (see [START_REF] Aubin | Viability Theory, Systems & Control: Foundations & Applications[END_REF][START_REF] Brézis | Operateurs Maximaux Monotones et Semi-Groupes de Contractions Dans Les Espaces de Hilbert[END_REF]).

Differential inclusions with maximal monotone perturbations

Let A : R n ⇒ R n be a maximal monotone operator, g ∈ L 1 (0, T ; R n ) (space of integrable functions from [0, T ] to R n ), and consider the following differential inclusion

   ẋ(t) ∈ g(t) -A(x(t)) a.e. t ∈ [0, T ],
x(0) = x 0 ∈ domA.

(2.17)

Because of the finite-dimension, we can suppose that int(domA) = ∅ (see [START_REF] Brézis | Operateurs Maximaux Monotones et Semi-Groupes de Contractions Dans Les Espaces de Hilbert[END_REF]). Hence, the differential inclusion above always has a unique continuous solution satisfying (2.17). The following theorem provides the exact valued of the rightderivative of the solution ( Brézis [START_REF] Brézis | Operateurs Maximaux Monotones et Semi-Groupes de Contractions Dans Les Espaces de Hilbert[END_REF] and Barbu [14]).

Theorem Chapter 3

Invariant sets and Lyapunov pairs for differential inclusions with maximal monotone operators

We give different conditions for the invariance of closed sets with respect to differential inclusions governed by a maximal monotone operator defined on Hilbert spaces, which is subject to a Lipschitz continuous perturbation depending on the state. These sets are not necessarily weakly closed as in [START_REF] Adly | Nonsmooth Lyapunov pairs for infinite-dimensional first order differential inclusions[END_REF][START_REF] Adly | Nonsmooth Lyapunov pairs for differential inclusions governed by operators with nonempty interior domain[END_REF], while the invariance criteria are still written by using only the data of the system. So, no need to the explicit knowledge of neither the solution of this differential inclusion, nor the semi-group generated by the maximal monotone operator. These invariant/viability results are next applied to derive explicit criteria for a-Lyapunov pairs of lower semi-continuous (not necessarily weakly lower semi-continuous) functions associated to these differential inclusions. The lack of differentiability of the candidate Lyapunov functions and the consideration of general invariant sets (possibly not convex or smooth) are carried out by using techniques from nonsmooth analysis.

Introduction

We provide sufficient and, in many different interesting situations, necessary criteria for the invariance property of closed subsets with respect to the following differential inclusion, given in a Hilbert space H,

ẋ(t) ∈ f (x(t)) -A(x(t)), x(0) = x 0 ∈ domA, a.e. t ≥ 0 (3.1)
where A is a maximal monotone operator which is subject to a Lipschitzian perturbation f . Equivalently, we establish many primal and dual explicit criteria for a-Lyapunov pairs and functions associated to the differential inclusion above. The current work extends and improves some of the results given in [START_REF] Adly | Nonsmooth Lyapunov pairs for infinite-dimensional first order differential inclusions[END_REF][START_REF] Adly | Nonsmooth Lyapunov pairs for differential inclusions governed by operators with nonempty interior domain[END_REF] on weakly closed invariant sets and weakly lower semi-continuous a-Lyapunov pairs.

The domain of A does not need to be closed, nor the values of A are supposed to be bounded or even nonempty. Thus, the scope of the equation above goes beyond the differential inclusions treated in [START_REF] Aubin | Viability Theory, Systems & Control: Foundations & Applications[END_REF][START_REF] Aubin | Differential Inclusions. Set-Valued Maps and Viability Theory[END_REF][START_REF] Clarke | Nonsmooth Analysis and Control Theory[END_REF][START_REF] Clarke | Qualitative properties of trajectories of control systems: a survey[END_REF][START_REF] Frankowska | A measurable upper semicontinuous viability theorem for tubes[END_REF], where the righthand side is generally represented by a cusco set-valued mapping (in particular, with nonempty and weak*-compact multi-valued operator). It is the monotonicity of A which compensates the lack of compacity in our differential inclusion, while the maximality of this operator guaranties, among other properties, the existence and the regularity of solutions. These two facts are also essential when checking the invariance of closed sets.

In front of the lack to a direct access to the explicit calculus of either the solution of the inclusion above or to the semi-group generated by A, the current work aims at finding weaker conditions for the invariance of closed sets, which only appeal to the fresh input data, namely the maximal monotone operator and the Lipschitz mapping. These conditions are applicable to a large variety of closed sets which do not need to be convex or smooth. Our approach fits the general scope and the main ideas behind Lyapunov's stability, which consists of looking for an adjacent function to the system described by the inclusion above; namely, an energy-like function which decreases along the trajectories and, so, under some extra usual conditions, forces the system to converge towards its equilibrium state and to remain there. Since our analysis allows to deal with extended-real valued functions, the invariance of a set occurs as long as the associated indicator function is a Lyapunov's function. However, our approach is more geometric since we first establish criteria for the invariance property and next deduce the adequate conditions for Lyapunov pairs and functions.

Invariant sets associated to general differential inclusions/equations have been the subject of extensive research during the last decades; namely, in relation with differential inclusions involving cusco mappings in their right-hand side (see, e.g., [START_REF] Aubin | Viability Theory, Systems & Control: Foundations & Applications[END_REF]). First results dealing with Lyapunov pairs and functions associated to the differential inclusions above have been first established in [START_REF] Pazy | Semigroup of linear operators and applications to partial differential equations[END_REF][START_REF] Pazy | The Lyapunov Method for Semigroups of Nonlinear Contractions in Banach Space[END_REF] in the case of homogeneous systems; that is, f ≡ 0. Pazy's criteria for a-Lyapunov pairs are given by means of directional-like derivative using the Moreau-Yoshida approximation Invariant sets and Lyapunov pairs of the operator A. This result has been extended to the general inclusion above in [START_REF] Carja | Characterization of Lyapunov pairs in the nonlinear case and applications[END_REF][START_REF] Kocan | Lyapunov Functions for Infinite-Dimensional Systems[END_REF], with the use of implicit criteria depending heavily on the semi-group generated by the maximal monotone operator A. Recently, different criteria for weakly lower semi-continuous a-Lyapunov pairs have been investigated in [START_REF] Adly | Nonsmooth Lyapunov pairs for infinite-dimensional first order differential inclusions[END_REF][START_REF] Adly | Nonsmooth Lyapunov pairs for differential inclusions governed by operators with nonempty interior domain[END_REF].

The need of more explicit conditions, not depending on the semi-group generated by A, is of utmost importance for many reasons, one of which is that the inclusion above is sometimes evoked as a companion tool to analyze other differential inclusions. In that case, the operator A may not be known explicitly, and this facts makes the access to its semi-group more complicated. For instance, in our work [START_REF] Adly | A convex approach to differential inclusions with prox-regular sets: Stability analysis and observer design[END_REF] we have investigated the existence of solutions to a differential inclusion governed by the normal cone to a prox-regular set ( [START_REF] Poliquin | Local differentiability of distance functions[END_REF]), by rewriting it in the form of (3.1) with A being some intrinsic maximal monotone operator to this prox-regular set. Such an operator A is not known explicitly but it processes enough information in order to check the invariance of the involved prox-regular set with respect to (3.1). This was sufficient to get the desired existence results; for more details, we refer the reader to [START_REF] Adly | A convex approach to differential inclusions with prox-regular sets: Stability analysis and observer design[END_REF].

Invariant sets are also referred to in the wide literature as viable sets [START_REF] Aubin | Viability Theory, Systems & Control: Foundations & Applications[END_REF][START_REF] Aubin | Differential Inclusions. Set-Valued Maps and Viability Theory[END_REF][START_REF] Aubin | Set-Valued Analysis[END_REF], and are of crucial use in many domains, as in economic, renewable resources, biology, diseases propagation, control processes of species and so on. It is manifest, in recent papers [START_REF] Yorke | A continuous differential equation in Hilbert space without existence[END_REF], that the investigation of certain algebraic varieties is sufficient to characterize invariant sets forced by symmetries. Lyapunov pairs and functions are used extensively in dynamic systems and control theory, among many other applications; see, e.g., [START_REF] Acary | Nonsmooth modeling and simulation for switched circuits[END_REF][START_REF] Brogliato | Nonsmooth mechanics. Models, dynamics and control[END_REF].

In this work, we provide different criteria to characterize those sets which are invariant with respect to the differential inclusion (3.1). Only the data, A and f, will be appealed to and no need to solve explicitly the equation. These invariant results are then rewritten as criteria for a-Lyapunov pairs, which are crucial for Lyapunov stability of (3.1). Because the sets we consider are not necessary convex or smooth, and the candidate Lyapunov functions are not necessarily sufficiently regular, we use techniques of nonsmooth analysis (e.g.. [START_REF] Clarke | Nonsmooth Analysis and Control Theory[END_REF][START_REF] Mordukhovich | Variational Analysis and Generalized Differentiation I: Basic Theory, Grundlehren der Mathematischen Wissenschaften[END_REF][START_REF] Rockafellar | Variational Analysis, Grundlehren der Mathematischen Wissenschaften[END_REF]).

The organization of the paper is as follows. After an introductory section to present the main notations and tools which are used through this work, we give in Section 3.3 the main invariance criterion in Theorem 3.6, using the normal cone to the nominal set. Other corollaries follow in order to simplify this invariance criterion and provide equivalent primal and dual conditions. In Section 3.4, we 3.2. Notation and preliminary results apply the previous invariance result to investigate a-Lyapunov pairs associated to differential inclusion (3.1).

Notation and preliminary results

Let (H, •, • , • ) be a Hilbert space, with origin θ. Given a set S ⊂ H, by S and S * we denote the closure of S and the polar of S, respectively, where

S * := {x * ∈ H | x * , x ≤ 0, for all x ∈ S}.
The indicator and the distance functions are respectively given by I S (x) := 0 if x ∈ S; +∞ otherwise, and d S (x) := inf{ x -y : y ∈ S} (in the sequel we shall adopt the convention inf ∅ = +∞). For δ ≥ 0, we denote Π δ S the (orthogonal) δ-projection mapping onto S defined as

Π δ S (x) := {y ∈ S : x -y 2 ≤ d 2 S (x) + δ 2 };
for δ = 0, we simply write Π S (x) := Π 0 S (x). It is known that Π S is nonemptyvalued on a dense subset of H \ S ( [START_REF] Clarke | Nonsmooth Analysis and Control Theory[END_REF]). For an extended-real valued function ϕ : H → R := (-∞, +∞] , we denote domϕ := {x ∈ H | ϕ(x) < +∞} and epiϕ :

= {(x, α) ∈ H × R | ϕ(x) ≤ α}. Function ϕ is lower semi-continuous if epiϕ is closed. The contingent directional derivative of ϕ at x ∈ domϕ in the direction v ∈ H is ϕ (x; v) := lim inf t→0 + ,w→v ϕ(x + tw) -ϕ(x) t .
A vector ξ ∈ H is called a proximal subgradient of ϕ at x ∈ H, written ξ ∈ ∂ P ϕ(x), if there are ρ > 0 and σ ≥ 0 such that

ϕ(y) ≥ ϕ(x) + ξ, y -x -σ y -x 2 , ∀y ∈ B ρ (x),
where B ρ (x) ( =: B(x, ρ)) is the closed ball centered at x ∈ H of radius ρ > 0. The vector ξ is called a Fréchet subgradient of ϕ at x,

written ξ ∈ ∂ F ϕ(x), if ϕ(y) ≥ ϕ(x) + ξ, y -x + o( y -x ) ∀y ∈ H;
Invariant sets and Lyapunov pairs and a basic (or Limiting) subgradient of ϕ at x, written ξ ∈ ∂ L ϕ(x), if there exist sequences (x k ) k and (ξ k ) k such that

x k ϕ → x, ξ k ∈ ∂ P ϕ(x k ) ξ k ξ,
where refers to the weak convergence in H, and x k ϕ → x means that x k → x together with ϕ(x k ) → ϕ(x).

If x /

∈ domϕ, we write

∂ P ϕ(x) = ∂ F ϕ(x) = ∂ L ϕ(x) = ∅. If S is
T B S (x) := v ∈ H | ∃ x k ∈ S, ∃ t k → 0, st. t -1 k (x k -x) → v as k → +∞ .
We also define the Clarke subgradients of ϕ at x as the vectors ξ ∈ H such that (ξ, -1) ∈ N C epiϕ (x, ϕ(x)), and denote ∂ C ϕ(x) the Clarke subdifferential of ϕ at x. The singular subdifferential of ϕ at x, written ∂ ∞ ϕ(x), is the set of vectors ξ ∈ H for which there are sequences [START_REF] Mordukhovich | Variational Analysis and Generalized Differentiation I: Basic Theory, Grundlehren der Mathematischen Wissenschaften[END_REF]Theorem 2.38]). It is known that every ξ ∈ H such that (ξ, 0) ∈ N P epiϕ (x, ϕ(x)) belongs to ∂ ∞ ϕ(x) and, moreover, there exist sequences as in the definition before but with λ k ξ k → ξ instead of λ k ξ k ξ (see [START_REF] Mordukhovich | Variational Analysis and Generalized Differentiation I: Basic Theory, Grundlehren der Mathematischen Wissenschaften[END_REF]Lemma 2.37]). Observe that

x k ϕ → x, ξ k ∈ ∂ P ϕ(x k ) and λ k → 0 + such that λ k ξ k ξ; equivalently, ξ ∈ ∂ ∞ ϕ(x) iff (ξ, 0) ∈ N L epiϕ (x, ϕ(x)) (see
∂ P ϕ(x) ⊂ ∂ F ϕ(x) ⊂ ∂ L ϕ(x) ⊂ ∂ C ϕ(x).
For all these concepts and properties we refer to [START_REF] Mordukhovich | Variational Analysis and Generalized Differentiation I: Basic Theory, Grundlehren der Mathematischen Wissenschaften[END_REF][START_REF] Rockafellar | Variational Analysis, Grundlehren der Mathematischen Wissenschaften[END_REF].

We shall use the following version of Gronwall's Lemma:

Lemma 3.1. (Gronwall's Lemma [2]) Let T > 0 and a, b ∈ L 1 (t 0 , t 0 + T ; R) such that b(t) ≥ 0 a.e. t ∈ [t 0 , t 0 + T ]. If an absolutely continuous function w : [t 0 , t 0 + T ] → R + satisfies, for 0 ≤ α < 1, (1 -α)w (t) ≤ a(t)w(t) + b(t)w α (t) a.e. t ∈ [t 0 , t 0 + T ], then w 1-α (t) ≤ w 1-α (t 0 )e t t 0 a(τ )dτ + t t 0 e t s a(τ )dτ b(s)ds, ∀t ∈ [t 0 , t 0 + T ].

Notation and preliminary results

Next, we review some facts about monotone and maximal monotone operators. Given a set-valued operator A : H ⇒ H, which we identify with its graph, we denote its domain by domA :

= {x ∈ H | A(x) = ∅}. Operator A is monotone if x 1 -x 2 , y 1 -y 2 ≥ 0 for all (x 1 , y 1 ), (x 2 , y 2 ) ∈ A.
We say that A is maximal monotone if A is monotone and coincides with every monotone operator containing its graph. In such a case, it is known that A(x) is convex and closed for every x ∈ H; moreover, for every λ > 0 there exists a unique vector J λ (x) ∈ (id + λA) -1 (x), which is the resolvent of the (maximal monotone) operator A, while

A λ (x) := x-J λ x λ is the Moreau-Yoshida approximation of A. If S ⊂ H is a closed convex set, we denote S • := {y ∈ S | y = min z∈S z }; in particular, we write A • (x) := (A(x)) • , x ∈ domA.
Associated with a maximal monotone operator A : H ⇒ H we consider the differential inclusion given in (3.1) :

ẋ(t) ∈ f (x(t)) -A(x(t)), a.e. t ≥ 0, x(0) = x 0 ∈ domA,
where f : H → H is a given (l-)Lipschitz continuous mapping. Every solution of differential inclusion (3.1) will be denoted by x(•; x 0 ). We introduce the concept of invariant sets (see, e.g., [START_REF] Aubin | Viability Theory, Systems & Control: Foundations & Applications[END_REF][START_REF] Clarke | Nonsmooth Analysis and Control Theory[END_REF][START_REF] Clarke | Approximate invariance and differential inclusions in Hilbert spaces[END_REF]): Definition 3.2. A set S ⊂ domA is said to be invariant for (3.1) provided that x(t; x 0 ) ∈ S for every x 0 ∈ S and every t ≥ 0.

We also recall the following result on the existence of solutions of (3.1); for more details, we refer to [START_REF] Brézis | Operateurs Maximaux Monotones et Semi-Groupes de Contractions Dans Les Espaces de Hilbert[END_REF]. Proposition 3.3. For any x 0 ∈ domA and T > 0, system (3.1) has a unique continuous solution, which is the uniform limit on [0, T ] of x λ (•; x 0 ) (as λ ↓ 0), where x λ (•; x 0 ) is the solution of the differential equation

ẋλ (t) = f (x λ (t)) -A λ (x λ (t)), x λ (0) = x 0 .
Moreover, the following holds:

(i) For all s, t ≥ 0 and all y 0 ∈ domA we have that x(s; x(t; x 0 )) = x(t + s; x 0 ), x(t; x 0 ) -x(t; y 0 ) ≤ e lt x 0 -y 0 .

Invariant sets and Lyapunov pairs (ii) If x(t 0 , x 0 ) ∈ domA for some t 0 ≥ 0, then

d + x(t 0 ; x 0 ) dt = (f (x(t 0 ; x 0 )) -A(x(t 0 ; x 0 ))) • .
(iii) The function t → d + x(t;x 0 ) dt is right-continuous at every t ≥ t 0 , where t 0 ≥ 0 is such that x(t 0 ; x 0 ) ∈ domA, and we have

d + x(t; x 0 )
dt ≤ e l(t-t 0 ) d + x(t 0 ; x 0 ) dt .

Invariant sets

In this section, we achieve our first goal to characterize those closed sets in the Hilbert space H, which are invariant with respect to differential inclusion (3.1):

ẋ(t) ∈ f (x(t)) -A(x(t)), t ∈ [0, ∞), x(0) = x 0 ∈ domA;
the unique solution of this inclusion is written x(•; x 0 ).

It is worth observing that whenever differential inclusion (3.1) possesses a strong solution starting from S (x 0 ∈ S), which is an absolutely continuous function such that x(t; x 0 ) ∈ domA for all t > 0, each invariant closed set S ⊂ domA satisfies the condition

S = domA ∩ S. (3.2)
However, this condition may not be true when only weak solutions exist. This is why we shall assume in what follows that our invariance candidate sets satisfy this "almost necessary" condition.

Remark 3.4. Theorem 3.6 below gives the main invariance criterion, given in (3.3), for closed sets with respect to differential inclusion (3.1), using only the data in (3.1) which are the operator A and the mapping f. Hence, explicit calculus of either the solution or the semigroup generated by A are not required. Criterion (3.3) extends and adapts some of the results given in [START_REF] Adly | Nonsmooth Lyapunov pairs for infinite-dimensional first order differential inclusions[END_REF][START_REF] Adly | Nonsmooth Lyapunov pairs for differential inclusions governed by operators with nonempty interior domain[END_REF] on weakly closed invariant sets. Its geometric meaning is very similar to the classical ones established in [START_REF] Clarke | Nonsmooth Analysis and Control Theory[END_REF][START_REF] Clarke | Qualitative properties of trajectories of control systems: a survey[END_REF] for differential inclusions of the form ẋ(t) ∈ F (x(t)), with a w * -compact, nonempty and convex multifunction F . In our case, condition (3.3) takes into account that the right-hand side in (3.1) , which is governed by a general maximal monotone operator, may have empty or unbounded values. As well, another crucial difference between (3.1) and the last inclusion above is that our analysis also allows the initial condition in (3.1) to start from the larger set domA. Thus, the scope of our analysis goes beyond the differential inclusions treated in [START_REF] Aubin | Viability Theory, Systems & Control: Foundations & Applications[END_REF][START_REF] Aubin | Differential Inclusions. Set-Valued Maps and Viability Theory[END_REF][START_REF] Clarke | Nonsmooth Analysis and Control Theory[END_REF][START_REF] Clarke | Qualitative properties of trajectories of control systems: a survey[END_REF][START_REF] Frankowska | A measurable upper semicontinuous viability theorem for tubes[END_REF]. First invariance criteria for differential inclusions involving maximal monotone operators have been given in [START_REF] Pazy | The Lyapunov Method for Semigroups of Nonlinear Contractions in Banach Space[END_REF] (see, also, [START_REF] Brézis | Operateurs Maximaux Monotones et Semi-Groupes de Contractions Dans Les Espaces de Hilbert[END_REF]) without considering the Lipschitzian perturbation. Such results have been extended in [START_REF] Carja | Characterization of Lyapunov pairs in the nonlinear case and applications[END_REF][START_REF] Kocan | Lyapunov Functions for Infinite-Dimensional Systems[END_REF] to maximal monotone operators which are subject to Lipschitz perturbations, using criteria which depend on the semi-group of contractions generated by -A. Compared to [START_REF] Carja | Characterization of Lyapunov pairs in the nonlinear case and applications[END_REF][START_REF] Kocan | Lyapunov Functions for Infinite-Dimensional Systems[END_REF] (see, also, references therein), condition (3.3) relies exclusively on the geometry of C as in [START_REF] Clarke | Nonsmooth Analysis and Control Theory[END_REF][START_REF] Clarke | Qualitative properties of trajectories of control systems: a survey[END_REF].

Before we state the main theorem of this section, Theorem 3.6 below, we give the following lemma. Lemma 3.5. Given a closed set S ⊂ H and an m ≥ 0, we denote

S m := {x ∈ S ∩ domA | (f (x) -A(x)) • ≤ m} .
Then the set S m is closed.

Proof. Take a sequence (x k ) k ⊂ S m such that x k → x (∈ S). Without loss of generality, and taking into account the norm-weak upper semi-continuity of the maximal monotone operator A, we conclude that the sequence (Π

A(x k ) (f (x k ))) k weakly converges to some z ∈ A(x). Then (f (x) -A(x)) • ≤ f (x) -z ≤ lim inf k→∞ f (x k ) -Π A(x k ) (f (x k )) = lim inf k→∞ (f (x k ) -A(x k )) • ≤ m, so that x ∈ S m .
Theorem 3.6. Given a closed set S ⊂ domA ∩ S, we assume that for every x ∈ S ∩ domA there exist m, r > 0 such that Π A(x) (f (x)) ≤ m and

sup ξ∈N P Sm (y) min y * ∈A(y)∩B(θ,m) ξ, f (y) -y * ≤ 0 for all y ∈ B(x, r). (3.3)
Then S is invariant for (3.1).

Proof. We fix x 0 ∈ S ∩ domA and ε > 0. Let m, r > 0 be as in the current assumption (with x = x 0 ), and choose an M > 0 such that

f (y) -A(y) ∩ B(θ, m) ⊂ B(θ, M ) for all y ∈ K := S m ∩ B(x 0 , r). (3.4)
We also choose sufficiently small numbers t, δ > 0 and a sufficiently large integer

N such that max{6M 2 t2 , 8δ 2 } < r 2 2 , δ < t N , (3.5)
max (M 2 + 4M + 1) t2 N , M 2 t2 N 2 + 2δ 2 < ε 2 4 . (3.6)
We denote by π := {t 0 , t 1 , ..., t N } the uniform partition of the interval [0, t]. We put d(π

) := max 0≤i≤N -1 (t i+1 -t i ) = t N and, by (3.4), we choose an element s * 0 ∈ f (x 0 ) -A(x 0 ) such that s * 0 ≤ M . We consider the the function z 0 (t), t ∈ [t 0 , t 1 ] such that ż0 (t) = s * 0 , t ∈ [t 0 , t 1 ], z 0 (0) = x 0 , and denote z 1 := x 0 + s * 0 t 1 . We pick ŝ1 ∈ Π δ K (z 1 )
. Then there exists a pair (y 1 , s 1 ) such that s 1 ∈ K, y 1 -s 1 ∈ N P K (s 1 ) and (see, e.g., [START_REF] Clarke | Approximate invariance and differential inclusions in Hilbert spaces[END_REF][START_REF] Radulescu | Geometric Approximation of Proximal Normals[END_REF])

max{ y 1 -z 1 , s 1 -ŝ1 } ≤ δ, (y 1 -s 1 ) -(z 1 -ŝ1 ) ≤ 2δ,
as well as (see [START_REF] Adly | A convex approach to differential inclusions with prox-regular sets: Stability analysis and observer design[END_REF]Lemma 4], also Theorem 2.1)

s 1 -x 0 2 ≤ 6 z 1 -x 0 2 + 8δ 2 = 6t 2 1 s * 0 2 + 8δ 2 < 6 t2 M 2 + 8δ 2 < r 2 ;
hence, s 1 ∈ int(B(x 0 , r)) and, so, N P K (s 1 ) = N P Sm (s 1 ). Consequently, by the current assumption of the theorem, we find s

* 1 ∈ (f (s 1 ) -A(s 1 )) ∩ B(θ, M ) such that y 1 -s 1 , s * 1 ≤ 0. With this vector s * 1 in hand, we consider the function z 1 (t), t ∈ [t 1 , t 2 ], such that ż1 (t) = s * 1 , t ∈ [t 1 , t 2 ] z 1 (t 1 ) = z 1 .
By repeating the arguments used above, for each i ∈ 2, N -1, we consider the

3.3. Invariant sets function z i (t), t ∈ [t i , t i+1 ], such that żi (t) = s * i , t ∈ [t i , t i+1 ] z i (t i ) = z i-1 (t i ) =: z i ,
and the corresponding elements

(ŝ i , y i , s i , s * i ) such that ŝi ∈ Π δ K (z i ), y i -s i ∈ N P K (s i ) = N P Sm (s i ), s * i ∈ [f (s i ) -A(s i )] ∩ B(θ, M ), y i -s i , s * i ≤ 0, max{ y i -z i , s i -ŝi } ≤ δ, (y i -s i ) -(z i -ŝi ) ≤ 2δ.
Now, we are going to prove that the absolute continuous trajectory z(•), defined on [0, t] as z(t

) := z i (t) = z i + (t -t i )s * i for t ∈ [t i , t i+1 ], satisfies d S (z(t)) ≤ ε, ∀t ∈ [0, t], (3.7)
s i -z(t) ≤ 2ε, ∀t ∈ [t i , t i+1 ]. (3.8)
Indeed, for any 1 ≤ i ≤ N -1, one has

d 2 K (z i+1 ) ≤ z i+1 -ŝi 2 = z i+1 -z i 2 + z i -ŝi 2 + 2 z i+1 -z i , z i -ŝi = (t i+1 -t i )s * i 2 + d 2 K (z i ) + δ 2 + 2d(π) s * i , z i -ŝi ≤ M 2 d 2 (π) + d 2 K (z i ) + δ 2 + 2d(π) s * i , y i -s i + 2d(π) s * i , (z i -ŝi ) -(y i -s i ) ≤ d 2 K (z i ) + (M 2 + 4M + 1)d(π)(t i+1 -t i ),
which gives us

d 2 K (z i+1 ) ≤ d 2 K (z 1 ) + (M 2 + 4M + 1)d(π)(t i+1 -t 1 ) ≤ z 1 -x 0 2 + (M 2 + 4M + 1)d(π)(t i+1 -t 1 ) ≤ (M 2 + 4M + 1)d(π) t ≤ (M 2 + 4M + 1) t2 N < ε 2 4 . (3.9)
This shows that, for every t ∈ [t i , t i+1 ],

d 2 S (z(t)) ≤ d 2 K (z(t)) = d 2 K (z i (t))) = d 2 K (z i (t i ) + (t -t i )s * i ) ≤ 2d 2 K (z i ) + 2(t -t i ) 2 M 2 ≤ ε 2 2 + 2d 2 (π)M 2 ≤ ε 2
Invariant sets and Lyapunov pairs and (3.7) follows. Inequality (3.8) also follows since that for every t ∈ [t i , t i+1 ]

s i -z(t) 2 ≤ 2 z(t) -z i 2 + 2 s i -z i 2 ≤ 2(t -t i ) 2 M 2 + 4 s i -ŝi 2 + 4 z i -ŝi 2 ≤ 2(t -t i ) 2 M 2 + 4d 2 K (z i ) + 8δ 2 ≤ 2d 2 (π)M 2 + ε 2 + 8δ 2 ≤ 2ε 2 ,
where in the last inequality we used (3.9). Now, let x(t) be the (strong) solution of (3.1) starting at x 0 , and denote

l i (t) := s i -z(t), t ∈ [t i , t i+1 ], so that ż(t) = s * i ∈ f (s i ) -A(s i ) = f (z(t) + l i (t)) -A(z(t) + l i (t))
. Hence, by using the monotonicity of A we get

f (z(t) + l i (t)) -ż(t) -f (x(t)) + ẋ(t), z(t) + l i (t) -x(t) ≥ 0,
which leads us, using (3.7) and (3.8) together with the l-Lipschitzianity of f , to

ż(t) -ẋ(t), z(t) -x(t) ≤ 2ε f (z(t) + l i (t)) -ż(t) -f (x(t)) + ẋ(t) + z(t) -x(t) f (z(t) + l i (t)) -f (x(t)) ≤ 2ε ż(t) -ẋ(t) + 2εl z(t) + l i (t) -x(t) + l z(t) -x(t) z(t) + l i (t) -x(t) . So, if C is any constant such that ż(t) -ẋ(t) ≤ C for all t ∈ [0, t] (as ż(t) ≤ M, and x(•) is Lipschitz on [0, t]), we get ż(t) -ẋ(t), z(t) -x(t) ≤ 2εC + 4εl z(t) -x(t) + l z(t) -x(t) 2 + 4ε 2 l.
Next, by applying Lemma 3.1 to the function z(

•) -x(•) 2 + 2εC+4ε 2 l l we get, for all t ∈ [0, t] z(t) -x(t) ≤ 4ε 2 l + 2εC l 2 e lt + 4ε(e lt -1),
implying that, in view of (3.7) and (3.8),

d S (x(t)) ≤ d S (z(t)) + z(t) -x(t) ≤ 4ε 2 l + 2εC l 2 e l t + 4εe l t.
Consequently, by the arbitrariness of ε we conclude that x(t) ∈ S for every t ∈ [0, t]. Moreover, as x( t; x 0 ) ∈ S ∩ domA, by the same argument as above we find t > 0 3.3. Invariant sets such that for every t ∈ [0, t] (recall Proposition 3.3)

x(t + t; x 0 ) = x(t; x( t; x 0 )) ∈ S ∩ domA;
that is, x(t) ∈ S for every t ∈ [0, t + t]. This proves that x(t) ∈ S for every t ≥ 0. Finally, if x 0 ∈ S ∩ domA, we take a sequence (x k ) ⊂ S ∩ domA such that x k → x 0 . As we have just shown, for every k ≥ 1 we have that x(t; x k ) ∈ S for every t ≥ 0. Thus, since S is closed, as k → +∞ we deduce that x(t; x 0 ) ∈ S for every t ≥ 0.

The proof of Theorem 3.6 shows actually the following:

Corollary 3.7. Given a closed set S ⊂ domA ∩ S and x 0 ∈ S ∩ domA, we assume that for some m, r > 0 such that

Π A(x) (f (x 0 )) ≤ m it holds sup ξ∈N P Sm (y) min y * ∈A(y)∩B(θ,m) ξ, f (y) -y * ≤ 0 for all y ∈ B(x 0 , r).
Then there exists t > 0 such that x(t; x 0 ) ∈ S for all t ∈ [0, t].

As we show in the corollary below the criterion of Theorem 3.6 becomes necessary if the maximal monotone operator A has a minimal norm section, which is locally bounded relative to its domain. As typical examples of such operators there are normal cones to closed convex sets, and the subdifferential mapping of convex, lower semi-continuous functions, which are Lipschitz relative to their domains. To fix this concept we say that the operator A is locally minimally bounded on S, if for every x ∈ S ∩ domA there exist m, r > 0 such that

A • (y) ≤ m for all y ∈ S ∩ domA ∩ B(x, r). (3.10)
This condition is less restrictive compared with the local boundedness of A relative to S, which means that for every x ∈ S ∩ domA there exist m, r > 0 such that

y * ≤ m, ∀y * ∈ Ay, y ∈ S ∩ domA ∩ B(x, r). (3.11)
Obviously every locally bounded operator is locally minimally bounded.

Then the following result gives necessary and sufficient simpler criteria for the invariance of closed sets with respect to differential inclusion (3.1), using the normal cone mapping to S, N S , which stands for either the proximal normal cone N P S or the Fréchet normal cone N F S .

Invariant sets and Lyapunov pairs Corollary 3.8. Let S ⊂ H be a closed set satisfying (3.2). Then the following statements are equivalent, provided that A is locally minimally bounded on S, (i) S is an invariant set for (3.1);

(ii) for every x ∈ S ∩ domA

f (x) -Π A(x) (f (x)) ∈ T B S (x); (iii) for every x ∈ S ∩ domA sup ξ∈N S (x) ξ, f (x) -Π A(x) (f (x)) ≤ 0; (iv) for every x ∈ S ∩ domA and every m ≥ f (x) -Π A(x) (f (x)) sup ξ∈N S (x) inf x * ∈(f (x)-A(x))∩B(θ,m) ξ, x * ≤ 0;
and the following assertion, when A is locally bounded relative to S,

(v) for every x ∈ S ∩ domA sup ξ∈N S (x) inf x * ∈f (x)-A(x)
ξ, x * ≤ 0.

Proof. We fix x ∈ S ∩ domA. The implication (iii) =⇒ (iv) is immediate, while the implication (ii) =⇒ (iii) follows because T B S (x) ⊂ (N S (x)) * . In the same line, implication (i) ⇒ (ii) follows easily by observing that

(f (x) -A(x)) • = d + x(•; x) dt (0) = lim t↓0 x(t; x) -x t ∈ T B S (x).
Thus, we only need to prove that (iv) ⇒ (i). If (iv) holds, by the current local boundedness assumption of ξ, x * ≤ 0, and (i) follows, according to Theorem 3.6.

A • on S ∩ domA we pick m, r > 0 such that (f (y) -A(y)) • ≤ m for all y ∈ B(x, 2r) ∩ S ∩ domA. Hence, B(x, 2r) ∩ S ∩ domA = S m ∩ B(x,
Suppose now that A is locally bounded on S ∩ domA, and consider the intermediate assertion (iv) for every x ∈ S ∩ domA and every large enough m ≥ (f (x) -A(x)) • we have that sup

ξ∈N S (x) inf x * ∈(f (x)-A(x))∩B(θ,m) ξ, x * ≤ 0.
As we see from the proof above (namely, the implication (iv) ⇒ (i)), we have that (iv) ⇒ (i), so that (v) ⇒ (iv) ⇒ (i). The proof of the corollary is finished because the implication (iv) =⇒ (v) is immediate.

In the following corollary we deduce another sufficient condition for the invariance of closed sets, using the Moreau-Yoshida approximations of A. Observe that we do not require here that set S satisfies condition (3.2). Corollary 3.9. Given a closed set S ⊂ H, we suppose that for every bounded subsets B of S lim inf

λ↓0 sup y∈B sup ξ∈N P S (y) ξ, f (y) -A λ (y) ≤ 0.
Then S is invariant set for (3.1).

Proof. Fix an x ∈ S and let x(•; x) be the corresponding solution of (3.1). Given an r > 0 we let λ k , k ≥ 1, be such that λ k ↓ 0 and sup

ξ∈N P S (y) ξ, f (y) -A λ k (y) ≤ 0 for all k ≥ 1 and y ∈ B(x, r) ∩ S. (3.12) If ε < r 4 and t > 0 are such that x(t; x) ∈ B(x, r 4 ) for all t ∈ [0, t], then for large enough k ≥ 1 the solution x λ k (•; x) of the differential equation ẋ(t) = f (x(t)) - A λ k (x(t)), x(0) = x, satisfies (see Proposition 3.3) x(t; x) -x λ k (t; x) ≤ ε < r 4 ; (3.13) hence, x λ k (t; x) ∈ B(x, r 2 ) for all t ∈ [0, t]. On the other hand, since A λ k is Lipschitz continuous, for large enough m > 0 we have B(x, r) ∩ S = {z ∈ B(x, r) ∩ S | A λ k (z) ≤ m}.
So, according to Corollary 3.7, (3.12) ensures that for some t > 0, say t ∈ (0, t), it holds

x λ k (t; x) ∈ S for all t ∈ [0, t]. Since x λ k (t; x) ∈ B(x, r
2 ) for all t ∈ [0, t], we infer that x λ k (t; x) ∈ B(x, r

2 ) ∩ S for all t ∈ [0, t]. Consequently, by (3.13) we get d S (x(t; x)) ≤ ε for all t ∈ [0, t]. Then, as ε → 0, we deduce that x(t; x) ∈ S for all t ∈ [0, t]. Finally, the invariance of S follows by using the semi-group property of the solution x(•; x) (see again Proposition 3.3).

We consider now the special case where f ≡ θ, so that our differential inclusion (3.1) takes the simpler form

ẋ(t) ∈ -A(x(t)), x(0) = x 0 ∈ domA. (3.14)
In this case, the criterion of Theorem 3.6 becomes also necessary as the following corollary shows. Here too N Sm stands for either N P Sm or N F Sm .

Corollary 3.10. Let S ⊂ H be a closed set satisfying (3.2). Then the following statements are equivalent:

(i) S is an invariant set of (3.14); (ii) for every x ∈ S ∩ domA -A • (x) ∈ T B Sm (x) for all m ≥ A • (x) ;
(iii) for every x ∈ S ∩ domA and for every m ≥ A

• (x) sup ξ∈N Sm (x) ξ, -A • (x) ≤ 0;
(iv) for any x ∈ S ∩ domA and every m ≥ A

• (x) sup ξ∈N Sm (x) inf x * ∈(-A(x))∩B(θ,m) ξ, x * ≤ 0.
Proof. As in the proof of Corollary 3.8, the implications (ii) =⇒ (iii) and (iii) =⇒ (iv with N Sm = N F Sm ) =⇒ (iv with N Sm = N P Sm ) are immediate. For the implication (i) =⇒ (ii), we assume that S is an invariant set of (3.14). If x ∈ S ∩ domA, then for a given m ≥ A • (x) we have

A • (x(t; x)) = d + x(t; x) dt ≤ d + x(0; x) dt = A • (x) ≤ m, for all t ≥ 0.
Hence, x(t; x) ∈ S m for all t ≥ 0 and we deduce that

-A • (x) = d + x(0;x) dt ∈ T B Sm (y),
yielding (ii). Finally, the implication (iv with N Sm = N P Sm ) =⇒ (i) is direct from Theorem 3.6.

To show how can our Theorem 3.6 be applied we consider the following example, which is treated in details in [START_REF] Adly | A convex approach to differential inclusions with prox-regular sets: Stability analysis and observer design[END_REF] in order to study the existence and the stability of solutions of differential inclusions involving the normal cone to a prox-regular set.

Recall that a closed set C ⊂ H is said to be uniformly r-prox-regular (r > 0) if for every x ∈ C and ξ ∈ N P C (x) ∩ B(θ, 1) we have ( [START_REF] Poliquin | Local differentiability of distance functions[END_REF])

ξ, y -x ≤ 1 2r y -x 2 for all y ∈ C.
Example 3.11. Let C ⊂ H be a uniformly r-prox-regular set and consider the associated differential inclusion

ẋ(t) ∈ g(x(t)) -N C (x(t)) a.e. t ∈ [0, T ], x(0) = x 0 ∈ C, (3.15)
where g is a Lipschitz mapping on H. According to [5, Lemma 6(c)], let L : H ⇒ H be a maximal monotone operator such that for some m ≥ 0 it holds, for all y ∈ C,

N C (y) ∩ B(0, m) + m r y ⊂ L(y) ⊂ N C (y) + m r y,
and consider the associated differential inclusion

   ẋ(t) ∈ g(x(t)) + m r x(t) -L(x(t)) a.e. t ∈ [0, T ], x(0) = x 0 ∈ C (⊂ domL). (3.16)
This inclusion perfectly fits the form of differential inclusion (3.1). Then we make appeal to Theorem 3.6 to prove that the set C is invariant for (3.1), so that

ẋ(t) ∈ g(x(t)) + m r x(t) -L(x(t)) ⊂ g(x(t)) -N C (x(t)),
providing us with a solution for (3.15). We refer to [START_REF] Adly | A convex approach to differential inclusions with prox-regular sets: Stability analysis and observer design[END_REF] for more details.

Lyapunov pairs and functions

In this section, we apply the results of the previous section to derive different criteria for a-Lyapunov pairs with respect to differential inclusion (3.1) :

ẋ(t) ∈ f (x(t)) -A(x(t)), t ∈ [0, ∞), x(0) = x 0 ∈ domA,
whose unique solution is written x(•; x 0 ). Similar criteria to ours have been established recently in [START_REF] Adly | Nonsmooth Lyapunov pairs for infinite-dimensional first order differential inclusions[END_REF][START_REF] Adly | Nonsmooth Lyapunov pairs for differential inclusions governed by operators with nonempty interior domain[END_REF] in the case of weakly lower semi-continuous Lyapunov pairs. Definition 3.12. We say that a pair (V, W ) of proper lower semi-continuous functions V, W : H → R with W ≥ 0, is (or forms) an a-Lyapunov pair (a ≥ 0) with respect to system (3.1) if, for every x 0 ∈ domA,

e at V (x(t; x 0 )) + t s W (x(τ ; x 0 ))dτ ≤ e as V (x(s; x 0 )), for all t ≥ s ≥ 0.
Observe that (V, W ) is an a-Lyapunov pair with respect to system (3.1) iff for every x 0 ∈ domA there exists a t > 0 such that (see, e.g., [START_REF] Adly | Nonsmooth Lyapunov pairs for infinite-dimensional first order differential inclusions[END_REF]Proposition 3.2])

e as V (x(s; x 0 )) + s 0 W (x(τ ; x 0 ))dτ ≤ V (x 0 ), for all s ∈ [0, t] .
We may assume without loss of generality that W is Lipschitz continuous on every bounded set (see, e.g., [START_REF] Adly | Nonsmooth Lyapunov pairs for infinite-dimensional first order differential inclusions[END_REF]Lemma 3.1] or [30, Theorem 1.5.1]). While, concerning function V, one need to suppose the following condition

V (x) = lim inf y domA -→ x V (y) for every x ∈ domV, (3.17)
which is in fact necessary for V to be a Lypunov function in many important cases (for instance, when differential inclusion (3.1) possesses a strong solution).

Theorem 3.13. Given two proper lower semi-continuous functions V : H → R satisfying (3.17), W : H → R + , and a real number a ≥ 0, we assume that for every x ∈ domV ∩ domA there are m, r > 0 such that Π A(x) (f (x)) ≤ m and, for all y ∈ B(x, r),

sup ξ∈∂ P (V +I Am )(y) inf y * ∈A(y)∩B(θ,m) ξ, f (y) -y * + aV (x) + W (x) ≤ 0.
Then (V, W ) forms an a-Lyapunov pair with respect to system (3.1).

Proof. We fix T > 0 and x 0 ∈ domV ∩domA. Following the discussion made before the current theorem we may suppose without loss of generality that W is Lipschitz continuous on every bounded set containing the trajectory {x(t; x 0 ), t ∈ [0, T ]}.

Let us define the maximal monotone operator  : H × R 4 ⇒ H × R 4 and the

Lyapunov pairs and functions

Lipschitz function f :

H × R 4 → H × R 4 as Â(x, µ) := (A(x), θ R 4 ), f (x, µ) := (f (x), 1, 0, 1, 0),
and, given a fixed µ 0 ∈ R 4 , consider the associated differential inclusion given in

H × R 4 by ẏ(t) ∈ f (y(t)) -Â(y(t)) a.e. t ∈ [0, T ], y(0) = (x 0 , µ 0 ), (3.18)
whose unique solution is y(t

) := (x(t), t, 0, t, 0) + (θ, µ 0 ), t ∈ [0, T ] (with x(t) := x(t; x 0 )).
For each n ≥ 1, we consider the lower semi-continuous function V n : H×R 3 → R defined as

V n (x, α, β, γ) := e aγ V (x) + (α -β)g n (α) + l 2 (α -β) 2 , (3.19)
where g n is an l -Lipschitz extension of the function

W (x(•; x 0 )) -1 n from [0, T ] to [-1, T + 1]; hence, ∂ C g n (α) ⊂ B(0, l ) for all α ∈ [0, T + 1]. (3.20)
We denote S := epiV n , so that S = S ∩ dom Â, by (3.17), and

epi(V n + I Am×R 3 ) = S ∩ Âm =: S m . (3.21)
We also denote

y 0 := (x 0 , θ R 3 , V (x 0 )) ∈ S ∩ dom Â.
Let m, r > 0 be as in the current assumption, corresponding to x 0 , and choose r < r small enough such that for all (x, α, β, γ) ∈ B((x 0 , θ R 3 ), r) (ξ, -κ) ∈ N P Sm (y) = N P epi(Vn+I Am×R 3 ) (y) ⊂ N P epi(Vn+I Am×R 3 ) (y 1 , V n (y 1 )) ;

g n (α) -e aγ W (x) + 2l |α -β| ≤ -1 2n . ( 3
Invariant sets and Lyapunov pairs hence, κ ≥ 0. If κ > 0, say κ = 1 for simplicity, then ξ ∈ ∂ P (V n + I Am×R 3 )(y 1 ) and, thanks to (3.19), we find

ξ 1 ∈ ∂ P (V + I Am )(x 1 ) and ς ∈ ∂ P g n (α 1 ) ⊂ ∂ C g n (α 1 ) such that ξ ∈ (e aγ 1 ξ 1 , g n (α 1 ) + (α 1 -β 1 )(ς + l ), -g n (α 1 ) + l (β 1 -α 1 ), ae aγ 1 V (x 1 )) .
Since y ∈ B(y 0 , r) ∩ S m we have that x 1 ∈ B(x 0 , r) ∩ A m ∩ domV and, so, by the current assumption, there exists an

x * 1 ∈ A(x 1 ) ∩ B(θ, m) (this last set being weak*-compact) such that ξ 1 , f (x 1 ) -x * 1 + aV (x 1 ) + W (x 1 ) ≤ 0.
Then we obtain (recall (3.20) and (3.22))

(ξ, -1), (f (x 1 ) -x * 1 , 1, 0, 1, 0) = e aγ 1 ξ 1 , f (x 1 ) -x * 1 + g n (α 1 ) + (α 1 -β 1 )(ς + l ) + ae aγ 1 V (x 1 ) = e aγ 1 ( ξ 1 , f (x 1 ) -x * 1 + aV (x 1 ) + W (x 1 )) + g n (α 1 ) -e aγ 1 W (x 1 ) + (α 1 -β 1 )(ς + l ) ≤ g n (α 1 ) -e aγ 1 W (x 1 ) + 2l |α 1 -β 1 | ≤ - 1 2n .
(3.23)

If κ = 0, then thanks to (3.19) we find ξ 2 ∈ H such that ξ = (ξ 2 , θ R 3 ), with the property that there are sequences

λ k ↓ 0, z k V +I Am -→ x 1 , ζ k ∈ ∂ P (V + I Am )(z k ) such that λ k ζ k → ξ 2 as k → ∞. By the current assumption, for each large enough k so that z k ∈ B(x 0 , r) there exists z * k ∈ A(z k ) ∩ B(θ, m) such that ζ k , f (z k ) -z * k + aV (z k ) + W (z k ) ≤ 0. Because A is maximal monotone and (z * k ) k is bounded, we can find an x * 2 ∈ A(x 1 )∩ B(θ, m) such that ξ 2 , f (x 1 ) -x *
2 ≤ 0; hence, by multiplying the last inequality above by λ k and taking the limit as k → ∞,

(ξ, 0), (f (x 1 ) -x * 2 , 1, 0, 1, 0) = ξ, f (x 1 ) -x * 2 ≤ 0. (3.24)
According to Corollary 3.7, (3.23) and (3.24) imply the existence of some t := t(n) ∈ (0, T ] such that for every t ∈ [0, t],

(x(t), t, 0, t, V (x 0 )) ∈ S;

in other words, e at V (x(t)) + tg n (t) + l 2 t 2 ≤ V (x 0 ) and, so, for every t ∈ [0, t]

e at V (x(t)) + t 0 W (x(τ ))dτ ≤ e at V (x(t)) + t 0 (g(t) + l (t -τ ))dτ + t n ≤ V (x 0 ) + t n . (3.25)
Now, we claim that for all t ∈ [0, T ]

e at V (x(t)) + t 0 W (x(τ ))dτ ≤ V (x 0 ) + e (1+a)t n . (3.26)
To prove this claim we define

t * := sup{t ∈ [0, T ] | inequality (3.26) holds on [0, t]}.
Indeed, from (3.25) and the lower semi-continuous of V, it follows that (3.26) holds at t * . If t * < T, we denote y * := (x(t * ), θ R 3 , V (x(t * ))) and we easily check that y * ∈ S ∩ dom Â. Then, arguing as with y 0 above, we arrive at a relation which is similar to (3.25); that is, there is some t > 0 such that for all t ∈ [0, t]

e at V (x(t; x(t * ))) + t 0 W (x(τ ; x(t * )))dτ ≤ V (x(t * )) + t n . (3.27)
Hence,

e a(t+t * ) V (x(t + t * )) + t+t * 0 W (x(τ ))dτ ≤ e a(t+t * ) V (x(t + t * )) + t+t * 0 W (x(τ ))dτ + (e at * -1) t 0 W (x(τ + t * ))dτ = e at * (e at V (x(t + t * )) + t 0 W (x(τ + t * ))dτ -t n ) + t * 0 W (x(τ ))dτ + e at * t n ≤ e at * V (x(t * )) + t * 0 W (x(τ ))dτ + e at * t n ≤ V (x 0 ) + e (1+a)t * n + e at * t n .
Consequently, due to the inequality e γ ≥ 1 + γ, we obtain that for all t ∈ [0, t]

e a(t+t * ) V (x(t + t * )) + t+t * 0 W (x(τ ))dτ ≤ V (x 0 ) + e (1+a)(t+t * ) n ,
leading us to a contradiction with the definition of t * . Now, the claim being true, we take the limit in (3.26) as n goes to +∞ to obtain that

e at V (x(t)) + t 0 W (x(τ ))dτ ≤ V (x 0 ) for all t ∈ [0, T ].
Finally, if x 0 ∈ domV, then by the current assumption (3.17), there exists a sequence (x k ) k≥1 ⊂ domV ∩ domA such that x k V → x 0 . Thus, from the last inequality above we conclude that

e at V (x(t; x k )) + t 0 W (x(τ ; x k ))dτ ≤ V (x k ) for all t ∈ [0, T ] and all k ≥ 1.
Hence, as k goes to +∞, the lower semi-continuous of V and Proposition 3.3 ensure that

e at V (x(t; x 0 )) + t 0 W (x(τ ; x 0 ))dτ ≤ V (x 0 ) for all t ∈ [0, T ],
showing that (V, W ) is an a-Lyapunov pair.

As in the case of the invariance of closed sets, the criterion of Theorem 3.13 takes a more simpler form when the maximal monotone operator A, or its minimal norm section, A • , is locally bounded (see (3.10)). Here, ∂V stands for either ∂ P V or ∂ F V.

Corollary 3.14. Given two proper lower semi-continuous functions V, W : H → R, such that W ≥ 0 and (3.17) holds, and a number a ≥ 0, we assume that A is minimally locally bounded relative to domV . Then the following statements are equivalent.

(i) (V, W ) is an a-Lyapunov pair for (3.1); (ii) for any x ∈ domV ∩ domA sup ξ∈∂V (x) ξ, (f (x) -A(x)) • + aV (x) + W (x) ≤ 0; (iii) for any x ∈ domV ∩ domA V (x; (f (x) -A(x)) • ) + aV (x) + W (x) ≤ 0;
Moreover, if in addition, (3.11) holds, then the above statements are also equivalent to 3.4. Lyapunov pairs and functions (iv) for any x ∈ domV ∩ domA sup

ξ∈∂V (x) inf x * ∈A(x) ξ, f (x) -x * + aV (x) + W (x) ≤ 0; (v) for any x ∈ domV ∩ domA inf v∈A(x) V (x; f (x) -v) + aV (x) + W (x) ≤ 0. Proof. First, the implications (iii)(with ∂ = ∂ F )⇒ (iii)(with ∂ = ∂ P ) ⇒ (ii) follow since that ∂ P ⊂ ∂ F and oe ∂ F V (x) ≤ V (x; •). (i) ⇒ (iii). Fix x 0 ∈ domV ∩ domA. Since (V, W ) is an a-Lyapunov for (3.1), we have that for all t > 0 V (x(t; x 0 )) -V (x 0 ) t + e at -1 t V (x(t; x 0 )) + 1 t t 0 W (x(τ ; x 0 ))dτ ≤ 0, while Proposition 3.5 ensures that lim t↓0 x(t; x 0 ) -x 0 t = d + x(0; x 0 ) dt = (f (x 0 ) -A(x 0 )) • .
Hence, using the lower semi-continuous of V together with the continuity of x(•; x 0 ),

V (x 0 ; (f (x 0 ) -A(x 0 )) • ) ≤ lim inf t↓0 V (x(t; x 0 )) -V (x 0 ) t ≤ -aV (x 0 ) -W (x 0 ), (3.28
) leading us to (ii).

(ii)(with ∂ = ∂ P ) ⇒ (i). We fix x 0 ∈ domV ∩ domA. From the one hand, by the boundedness assumption of A • , for a large m ≥ 0 there exists an r > 0 such that B(x 0 , r)

∩ domV ∩ domA ⊂ A m . (3.29)
On the other hand, we have that

∂ P (V + I Am )(x) ⊂ ∂ P V (x) for all x ∈ B(x 0 , r 2
).

(3.30)

Indeed, if ξ ∈ ∂ P (V + I Am )(x) for x ∈ B(x 0 , r 2
), there exist δ > 0 and ρ ∈ (0, r 2 ) such that

(V + I Am )(z) ≥ V (x) + ξ, z -x -δ z -x 2 ∀z ∈ B(x, ρ).

Invariant sets and Lyapunov pairs

Take z ∈ B(x, ρ 4 ) ∩ domV (⊂ B(x 0 , r)). By (3.17) together with (3.29), there exists a sequence (z

n ) n ⊂ B(x, ρ) ∩ domV ∩ A m such that z n → z and V (z n ) → V (z).
Since each z n satisfies the last inequality above, by taking the limit as n → ∞ we arrive at V (z) ≥ V (x) + ξ, z -x -δ z -x 2 and the inclusion (3.30) follows.

At this stage, from (3.29) and the Lipschitzianity of f there exists some M ≥ m such that, for all x ∈ B(x 0 , r), (3.30), assumption (ii)(with ∂ = ∂ P ) implies that, for every x ∈ B(x 0 , r 2 )

Π A(x) (f (x)) ≤ f (x) + A • (x) ≤ f (x) + m ≤ M, which shows that (f (x) -A(x)) • ∈ f (x) -A(x) ∩ B(θ, M ). Since ∂ P (V + I A M ) ⊂ ∂ P (V + I Am ), in view of
sup ξ∈∂ P (V +I A M )(x) inf x * ∈A(x)∩B(θ,M ) ξ, f (x) -x * + aV (x) + W (x) ≤ sup ξ∈∂ P V (x) ξ, (f (x) -A(x)) • + aV (x) + W (x) ≤ 0.
Thus, (i) follows from Theorem 3.13. Finally, if A is locally bounded on domV, then from the first part of the proof one only needs to verify the implication (iv) =⇒ (i), the proof of which is similar to the one of "(ii) ⇒ (i)" that we did above.

In the following corollary we provide criteria for a-Lyapunov pairs, which use the Moreau-Yoshida approximation of A.

Corollary 3.15. Let V, W and a be as in Corollary 3.14, and let ∂ be such that

∂ P ⊂ ∂ ⊂ ∂ C . If there exists λ 0 > 0 such that for all λ ∈ (0, λ 0 ] sup ξ∈∂V (x) ξ, f (x) -A λ (x) + aV (x) + W (x) ≤ 0 ∀x ∈ domV , then (V, W ) is an a-Lyapunov pair for (3.1). Proof. Fix x 0 ∈ domV and t ≥ 0. If x λ (•; x 0 ) is the solution of the differential equation ẋλ (t) = f (x λ (t)) -A λ (x λ (t)), x λ (0) = x 0 (λ ∈ (0, λ 0 ]), (3.31)
then, according to Corollary 3.14(ii), the pair (V, W ) is an a-Lyapunov pair of (3.31); that is,

e at V (x λ (t)) + t 0 W (x λ (τ ))dτ ≤ V (x 0 ) for all t ≥ 0.
Hence, the conclusion follows as λ ↓ 0.

We consider now the case when f ≡ 0 so that differential inclusion (3.1) reads

ẋ(t) ∈ -A(x(t)), x(0) = x 0 ∈ domA. (3.32)
In the following theorem ∂ stands for either ∂ P or ∂ F .

Corollary 3.16. Let V, W : H → R be two proper lower semi-continuous functions, such that W ≥ 0 and (3.17) holds, and let a ≥ 0. Then the following statements are equivalent:

(i) (V, W
) is an a-Lyapunov pair for (3.32);

(ii) for every x ∈ domV ∩ domA and every m ≥ A

• (x) sup ξ∈∂(V +I Am )(x) ξ, -A • (x) + aV (x) + W (x) ≤ 0;
(iii) for every x and m as in (ii)

sup ξ∈∂(V +I Am )(x) inf x * ∈-A(x)∩B(θ,m) ξ, x * + aV (x) + W (x) ≤ 0;
(iv) for every x and m as in (ii)

(V + I Am ) (x; -A • (x)) + aV (x) + W (x) ≤ 0;
(v) for every x and m as in

(ii) inf v∈-A(x)∩B(θ,m) (V + I Am ) (x; v) + aV (x) + W (x) ≤ 0.
Proof. The implications (ii) ⇒ (iii), (iv) ⇒ (v), (iv) ⇒ (ii), and (v) ⇒ (iii) are immediate. To prove that (i) ⇒ (iv), we fix x 0 ∈ domV ∩ domA and m ≥ A • (x 0 ) . According to Proposition 3.3, for any t ≥ 0 we have that

-A • (x(t; x 0 )) = d + x(t; x 0 ) dt ≤ d + x(0; x 0 ) dt = -A • (x 0 ) ≤ m;
that is, x(t, x 0 ) ∈ A m for all t ≥ 0. Hence, since x(t,x 0 )-x 0 t → -A • (x 0 ) as t ↓ 0, provided that (V, W ) is an a-Lyapunov pair for (3.32) we obtain, by arguing as in the proof of (3.28),

(V + I Am ) (x 0 ; -A • (x 0 )) ≤ lim inf t↓0 (V + I Am )(x(t; x 0 )) -(V + I Am )(x 0 ) t = lim inf t↓0 V (x(t; x 0 )) -V (x 0 ) t ≤ -aV (x) -W (x),
giving rise to (iv).

Finally, the conclusion of the corollary follows because the implication (iii) ⇒ (i) holds according to Theorem 3.13.

We obtain the following corollary, which can be find in [START_REF] Kocan | Lyapunov Functions for Infinite-Dimensional Systems[END_REF]; the original version of this result was established in [START_REF] Pazy | The Lyapunov Method for Semigroups of Nonlinear Contractions in Banach Space[END_REF] Corollary 3.17. Let V, W : H → R be two proper lower semi-continuous functions, such that W ≥ 0, and let a ≥ 0. If condition (3.17) and, for every

x ∈ domV , lim inf λ↓0 V (J λ (x)) -V (x) λ + aV (x) + W (x) ≤ 0, then (V, W ) is an a-Lyapunov pair for (3.32).
Proof. We fix x ∈ domV ∩ A m for some large m ≥ 1. Since A λ (x) ∈ A(J λ x) and A λ (x) ≤ A • (x) ≤ m, we infer that J λ (x) ∈ A m and, so, using the current assumption,

(V + I Am ) (x 0 ; -A • (x 0 )) ≤ lim inf t↓0 V (J λ (x)) -V (x) t ≤ -aV (x) -W (x).
The conclusion follows then from Corollary 3.16(iv).

Corollary 3.14 obviously covers the case when A is the null operator, where (3.1) becomes a usual differential equation stated in the Hilbert space H as

ẋ(t) = f (x(t)) a.e. t ≥ 0, x(0) = x 0 ∈ H. (3.33)
The following characterization is known when ∂ is the viscosity subdifferential as defined in [54, Definition 2.7], while the case of weakly lower semi-continuous a-Lyapunov pairs can be found in [START_REF] Adly | Nonsmooth Lyapunov pairs for infinite-dimensional first order differential inclusions[END_REF].

Lyapunov pairs and functions

Corollary 3.18. Let V, W and a be as in Corollary 3.14, and let ∂ be such that

∂ P ⊂ ∂ ⊂ ∂ C .
Then the following statements are equivalent:

(i) (V, W ) is an a-Lyapunov pair for differential equation (3.33), (ii) for every x ∈ domV sup ξ∈∂V (x) ξ, f (x) + aV (x) + W (x) ≤ 0, (3.34)
(iii) for every x ∈ domV

V (x; f (x)) + aV (x) + W (x) ≤ 0.
Proof. In view of Corollary 3.14, we only need to check that (i) =⇒ (ii) (with ∂ = ∂ C ), and this easily follows from the relation

∂ C V = co(∂ L V + ∂ ∞ V ). Indeed, assume that (i) holds and take ξ ∈ ∂ L V (x) and ζ ∈ ∂ ∞ V (x)
. By the definition of

∂ L V (x) we choose sequences ξ k ∈ ∂ P V (x k ) such that x k V → x and ξ k ξ. Then, by (i), ξ k , f (x k ) + aV (x k ) + W (x k ) ≤ 0 for all k ≥ 1,
and, so, as k → ∞, we deduce that ξ, f (x) + aV (x) + W (x) ≤ 0. Similarly, we choose sequences

x k V → x and λ k ↓ 0 such that ζ k ∈ ∂ P V (x k ) and λ k ζ k ζ.
Then, by arguing as above we deduce that ζ, f (x) ≤ 0, which in turn yields

ξ + ζ, f (x) + aV (x) + W (x) ≤ 0,
and this gives us (ii) (with ∂ = ∂ C ) by convexification.

We close this section by analyzing a typical example of Lyapunov pairs. Example 3.19. Assume that a function V : H → R is a proper, convex and lower semi-continuous, and consider the differential inclusion ẋ(t) ∈ -∂V (x(t)).

Then the pair (V, (∂V ) • 2 ) is a Lyapunov pair, so that for every x 0 ∈ domV

V (x(t; x 0 )) + t 0 ẋ(τ ; x 0 ) 2 dτ ≤ V (x 0 ) for all t > 0.
To see this fact we fix x ∈ domA ∩ dom∂V . Since A λ (x) ∈ A(J λ (x))for every λ > 0 Invariant sets and Lyapunov pairs (A = ∂V ), condition (3.17) holds and one has that

V (J λ (x)) -V (x) ≤ -A λ (x), x -J λ (x) = - 1 λ x -J λ (x) 2 .
Hence, lim inf

λ↓0 V (J λ (x))-V (x) λ + A • (x) 2 ≤ lim inf λ↓0 V (J λ (x))-V (x) λ + 1 λ 2 x -J λ (x) 2 ≤ 0 ,
and Corollary 3.17 (together with Proposition 3.3) applies.

Conclusion and further research

We gave different conditions for the invariance of closed sets, which only involve the input data, represented by the maximal monotone operator and the Lipschitz mapping. These conditions are applicable to a large variety of closed sets which do not need to be convex or smooth. The current work extends and improves some of the results given in [START_REF] Adly | Nonsmooth Lyapunov pairs for infinite-dimensional first order differential inclusions[END_REF][START_REF] Adly | Nonsmooth Lyapunov pairs for differential inclusions governed by operators with nonempty interior domain[END_REF] and dealing with weakly closed invariant sets and weakly lower semi-continuous a-Lypunov pairs. It will be our aim in a forthcoming work to apply the current results to specific differential equations/inclusions where the underlying maximal monotone operator is not known explicitly. This will make the access to the corresponding semi-group more easier, namely regarding the behavior at infinity of trajectories.

Chapter 4

A convex approach to differential inclusions with prox-regular sets:

Stability analysis and observer design

We study the existence and stability of solutions for differential inclusions governed by the normal cone to a prox-regular set and subject to a Lipschitz perturbation.

We prove that such, apparently, more general systems can be indeed remodeled into the classical theory of differential inclusions involving maximal monotone operators. This result is new in the literature and permits us to make use of the rich and abundant achievements in this class of monotone operators to derive the desired existence result and stability analysis, as well as the continuity and differentiability properties of the solutions. This going back and forth between these two models of differential inclusions is made possible thanks to a viability result for maximal monotone operators. As an application, we study a Luenbergerlike observer, which is shown to converge exponentially to the actual state when the initial value of the state's estimation remains in a neighborhood of the initial value of the original system.

Introduction

We consider in this paper the existence and stability of solutions problem of the following differential inclusion, given in a Hilbert space H,

   ẋ(t) ∈ f (x(t)) -N C (x(t)) for almost every t ≥ 0, x(0; x 0 ) = x 0 ∈ C, (4.1)
where N C is the normal cone to an r-uniformly prox-regular closed subset C of H.

The dynamical system driven by the set C is subject to a l-Lipschitz continuous perturbation mapping f defined on H. By a solution of (4.1) we mean an absolutely continuous function x(•; x 0 ) : [0, +∞) → H, with x(0; x 0 ) = x 0 , which satisfies (4.1) for almost every (a.e.) t ≥ 0; hence, in particular, x(t) ∈ C for all t ≥ 0. Indeed, such a solution is necessarily Lipschitz continuous on each interval of the form [0, T ] for T ≥ 0 (see Theorem 4.14). Differential inclusion (4.1) appears in the modeling of many concrete problems in economics, unilateral mechanics, electrical engineering as well as optimal control (see eg. [START_REF] Acary | Nonsmooth modeling and simulation for switched circuits[END_REF][START_REF] Colombo | The minimum time function for the controlled Moreau's sweeping process[END_REF][START_REF] Mazade | Differential variational inequalities with locally prox-regular sets[END_REF][START_REF] Stewart | Dynamics with inequalities: Impacts and hard constraints[END_REF] and references therein.)

It was recently shown in [START_REF] Mazade | Regularization of differential variational inequalities with locally prox-regular sets[END_REF] and [START_REF] Mazade | Differential variational inequalities with locally prox-regular sets[END_REF] that (4.1) has one and only one (absolutely continuous) solution, which satisfies the imposed initial condition. These authors employed a regularization approach based on the Moreau-Yosida approximation, and use the nice properties of uniform prox-regularity to show that the approximate scheme converges to the required solution. In this way, such an approach repeats those arguments of approximation ideas which, previously, were extensively used in the setting of differential inclusions with maximal monotone operators.

Problems dealing with the stability of solutions of (4.1), namely the characterization of weakly lower semi-continuous Lyapunov pairs and functions, have been developed in [START_REF] Mazade | Lyapunov functions for evolution variational inequalities with uniformly prox-regular sets[END_REF] following the same strategy, also based on Moreau-Yosida approximations. Most of works on these problems use indeed this natural approximation approach; see, e.g. [START_REF] Mazade | Lyapunov functions for evolution variational inequalities with uniformly prox-regular sets[END_REF][START_REF] Mazade | Differential variational inequalities with locally prox-regular sets[END_REF][START_REF] Mazade | Regularization of differential variational inequalities with locally prox-regular sets[END_REF].

In this paper, at a first glance we provide a different, but quite direct, approach to tackle this problem. We prove that problem (4.1) can be equivalently written as a differential inclusion given in the current Hilbert setting under the form

   ẋ(t) ∈ g(x(t)) -A(x(t)) a.e. t ∈ [0, T ], x(0; x 0 ) = x 0 ∈ domA, (4.2)
where A : H ⇒ H is an appropriate maximal monotone operator defined on H, and g : H → H is a Lipschitz continuous mapping. Then, it will be sufficient to apply the classical theory of maximal monotone operators ( [START_REF] Brézis | Operateurs Maximaux Monotones et Semi-Groupes de Contractions Dans Les Espaces de Hilbert[END_REF]; see, also, [START_REF] Adly | Nonsmooth Lyapunov pairs for infinite-dimensional first order differential inclusions[END_REF][START_REF] Adly | Nonsmooth Lyapunov pairs for differential inclusions governed by operators with nonempty interior domain[END_REF]) to analyze the existence and the stability of solutions for differential inclusion (4.1). The concept of invariant sets will be the key tool to go back and forth between inclusions (4.1) and (4.2). Invariant sets with respect to differential inclusions governed by maximal monotone operators have been studied and characterized in [START_REF] Adly | Nonsmooth Lyapunov pairs for infinite-dimensional first order differential inclusions[END_REF][START_REF] Adly | Nonsmooth Lyapunov pairs for differential inclusions governed by operators with nonempty interior domain[END_REF]. Other references for invariant sets, also referred to as viable sets, and the related theory of Lyapunov stability are [START_REF] Aubin | Differential Inclusions. Set-Valued Maps and Viability Theory[END_REF][START_REF] Carja | Characterization of Lyapunov pairs in the nonlinear case and applications[END_REF][START_REF] Kocan | Lyapunov Functions for Infinite-Dimensional Systems[END_REF][START_REF] Pazy | The Lyapunov Method for Semigroups of Nonlinear Contractions in Banach Space[END_REF] among others. We also refer to [START_REF] Donchev | Strong invariant and one-sided Lipschitz multifunctions[END_REF] for an interesting criterion for weakly invariant sets, which is established in the finite-dimensional setting for differential inclusions governed by one side-Lipschitz multivalued mappings with nonempty convex and compact values. This result has been used in [START_REF] Colombo | The minimum time function for the controlled Moreau's sweeping process[END_REF], always in finite dimensions, to provide weakly and strongly invariance criteria for closed sets with respect to more general differential inclusions where the set C in (4.1) is time dependent and f is a Lipschitzian multivalued mapping.

We shall also provide different criteria for the so-called a-Lyapunov pairs of lower semi-continuous functions to extend some of the results given in [START_REF] Adly | Nonsmooth Lyapunov pairs for infinite-dimensional first order differential inclusions[END_REF][START_REF] Adly | Nonsmooth Lyapunov pairs for differential inclusions governed by operators with nonempty interior domain[END_REF][START_REF] Mazade | Lyapunov functions for evolution variational inequalities with uniformly prox-regular sets[END_REF] to the current setting. It is worth to observe that the assumption of uniformly prox-regularity is required to obtain global solutions of (4.1), which are defined on the whole interval [0, T ]. However, our analysis also works in the same way when the set C is prox-regular at x 0 rather than being a uniformly prox-regular set; but, in this case, we only obtain a local solution defined around x 0 . This paper is organized as follows. After giving the necessary notations and preliminary results in Section 2, we review and study in Section 3 different aspects of the theory of differential inclusions governed by maximal monotone operators, including the existence of solutions, and we provide a stability results dealing with the invariance of closed sets with respect to such differential inclusions. In Sections 4, we provide the new proof of the existence of solutions for differential inclusions involving normal cones to r-uniformly prox-regular sets. Section 5 is devoted to the characterization of lower semi-continuous a-Lyapunov pairs and functions. Inspired from the recent paper [START_REF] Tanwani | Stability and observer design for Lur'e systems with multivalued, non-monotone, time-varying nonlinearities and state jumps[END_REF], we give in section 6 an application of our result to a Luenberger-like observer.

Preliminaries and examples

Preliminaries and examples

Preliminary results

In this paper, H is a Hilbert space endowed with an inner product •, • and an associated norm ||•||. The strong and weak convergences in H are denoted by → and , resp. We denote by B(x, ρ) the closed ball centered at x ∈ H of radius ρ > 0, and particularly we use B for the closed unit ball. The null vector in H is written θ. Given a set S ⊂ H, by coS, coneS and S we respectively denote the convex hull, the conic hull and the closure of S. The dual cone of S is the set

S * := {x * ∈ H | x * , x ≤ 0 for all x ∈ S}.
The indicator and the distance functions are respectively given by I S (x) := 0 if x ∈ S; +∞ otherwise, d S (x) := inf{||x -y||: y ∈ S} (in the sequel we shall adopt the convention inf ∅ = +∞). We shall write S for the convergence when restricted to the set S. For δ ≥ 0, we denote Π δ S the (orthogonal) δ-projection mapping onto S defined as

Π δ S (x) := {y ∈ S : ||x -y|| 2 ≤ d 2 S (x) + δ 2 }.
For δ = 0, we simply write Π S (x) := Π 0 S (x). It is known that Π S is nonemptyvalued on a dense subset of H \ S ( [START_REF] Clarke | Optimization and Nonsmooth Analysis[END_REF]).

For an extended real-valued function ϕ :

H → R, we denote domϕ := {x ∈ H | ϕ(x) < +∞} and epiϕ := {(x, α) ∈ H × R | ϕ(x) ≤ α}. Function ϕ is lower semi-continuous if epiϕ is closed. The contingent directional derivative of ϕ at x ∈ domϕ in the direction v ∈ H is ϕ (x, v) := lim inf t→0 + ,w→v ϕ(x + tw) -ϕ(x) t .
A vector ξ ∈ H is called a proximal subgradient of ϕ at x ∈ H, written ξ ∈ ∂ P ϕ(x), if there are ρ > 0 and σ ≥ 0 such that

ϕ(y) ≥ ϕ(x) + ξ, y -x -σ||y -x|| 2 ∀ y ∈ B ρ (x); Differential inclusions with prox-regular sets a Fréchet subgradient of ϕ at x, written ξ ∈ ∂ F ϕ(x), if ϕ(y) ≥ ϕ(x) + ξ, y -x + o( y -x ) ∀ y ∈ H;
and a basic (or Limiting) subdifferential of ϕ at x, written ξ ∈ ∂ L ϕ(x), if there exist sequences (x k ) k and (ξ k ) k such that

x k ϕ → x, (i.e., x k → x and ϕ(x k ) → ϕ(x)), ξ k ∈ ∂ P ϕ(x k ), ξ k ξ. If x / ∈ domϕ, we write ∂ P ϕ(x) = ∂ F ϕ(x) = ∂ L ϕ(x) = ∅.
In particular, if S is a closed set and s ∈ S, we define the proximal normal cone to S at s as N P S (s) = ∂ P I S (s), the Fréchet normal to S at s as N F S (s) = ∂ F I S (s), the limiting normal cone to S at s as N F S (s) = ∂ F I S (s), and the Clarke normal cone to S at s as N C S (s) = co(N L S (s)). Equivalently, we have that

N P S (s) = cone(Π -1 S (s) -s), where Π -1 S (s) := {x ∈ H | s ∈ Π S (x)}.
The Bouligand and weak Bouligand tangent cones to S at x are defined as

T B S (x) := v ∈ H | ∃ x k ∈ S, ∃ t k → 0, st t -1 k (x k -x) → v as k → +∞ T w S (x) := v ∈ H | ∃ x k ∈ S, ∃ t k → 0, st t -1 k (x k -x) v as k → +∞ , resp.
We also define the Clarke subgradient of ϕ at x, written ∂ C ϕ(x), as the vectors ξ ∈ H such that (ξ, -1) ∈ N C epiϕ (x, ϕ(x)), and the singular subgradient of ϕ at x, written ∂ ∞ ϕ(x), as the vectors ξ ∈ H such that (ξ, 0) ∈ N P epiϕ (x, ϕ(x)); in particular, if ξ ∈ ∂ ∞ ϕ(x), then there are sequences x k ϕ → x, ξ k ∈ ∂ P ϕ(x k ), and

λ k → 0 + such that λ k ξ k → ξ. Observe that ∂ P ϕ(x) ⊂ ∂ F ϕ(x) ⊂ ∂ L ϕ(x) ⊂ ∂ C ϕ(x).
For all these concepts and their properties we refer to the book [START_REF] Mordukhovich | Variational Analysis and Generalized Differentiation I: Basic Theory, Grundlehren der Mathematischen Wissenschaften[END_REF].

We shall frequently use the following version of Gronwall's Lemma:

Lemma 4.1. (Gronwall's Lemma; see, e.g., [2]) Let T > 0 and a, b ∈ L 1 (t 0 , t 0 + T ; R) such that b(t) ≥ 0 a.e. t ∈ [t 0 , t 0 + T ]. If, for some 0 ≤ α < 1, an absolutely continuous function w : [t 0 , t 0 + T ] → R + satisfies (1 -α)w (t) ≤ a(t)w(t) + b(t)w α (t) a.e. t ∈ [t 0 , t 0 + T ], then w 1-α (t) ≤ w 1-α (t 0 )e t t 0 a(τ )dτ + t t 0 e t s a(τ )dτ b(s)ds, ∀t ∈ [t 0 , t 0 + T ].

Some examples

     ∂u ∂t -∆u = f, (t, x) ∈ [0, T ] × Ω, u(0, x) = u 0 (x), x ∈ Ω (initial condition) u ≥ 0, ∂u ∂n ≥ 0 and u ∂u ∂n = 0 for (t, x) ∈ [0, T ] × ∂Ω.
It is well-known that the weak formulation of problem (P ) is given by the following parabolic variational inequalities

(V I)            Find u ∈ C such that Ω u (t)(v(t) -u(t))dx + Ω ∇u(t) • ∇(v(t) -u(t))dx ≥ Ω f (t)(v(t) -u(t))dx, ∀v ∈ C, a.e. t ∈ [0, T ].
Here,

C = {v ∈ L 2 (0, T ; H 1 (Ω)) : v(t) ∈ C for a.e. t ∈ [0, T ]}, where C = {v ∈ H 1 (Ω) : v ≥ 0
C = {v ∈ H 1 (Ω) : g(v(x)) ≥ 0 for x ∈ ∂Ω}.
The set C is no more convex and some sufficient conditions on the function g are necessary to ensure the prox-regularity of the sets C and C (see [START_REF] Adly | Preservation of prox-regularity of sets and application to constrained optimization[END_REF] for more details).

ẋ(t) = f (x(t)) + λ(t), t ∈ [0, T ] λ(t), g(x(t)) ≥ 0, λ(t), g(x(t)) = 0, where f : R n → R n , g : R n → R m are of class C 1 and λ : [0, T ] → R m is a Lagrange
Differential inclusions with prox-regular sets multiplier (unknown function). We have that

λ(t), g(x(t)) ≥ 0, λ(t), g(x(t)) = 0 ⇐⇒ -λ(t) ∈ N R m + (g(x(t))).
Hence, (NDCS) is written as

ẋ(t) ∈ f (x(t)) -N R m + (g(x(t))), with N R m + (g(x(t))) = ∂I R m + (g(x(t))
, where ∂ denotes the subdifferential in the sense of convex analysis. If we suppose a qualification condition such as, e.g., ∇g is surjective, then, using classical chain rules for Clarke generalized subdifferential (see e.g. [START_REF] Rockafellar | Variational Analysis, Grundlehren der Mathematischen Wissenschaften[END_REF]), we get

∂(I R m + • g)(x) = ∇g(x) T N R m + (g(x)).
By setting

C = {x ∈ R n : g(x) ≥ 0}, it is easy to see that problem (NCDS) is equivalent to the following differential inclusion ẋ(t) ∈ f (x(t)) -N C (x(t)),
which is of the form of (4.1). Under some sufficient conditions on the vectorial function g (see [4, Theorem 3.5]), we show that the set C is r-prox-regular.

Many problems in power converters electronics and unilateral mechanics can be modeled by nonlinear differential complementarity problems of the form (NDCS) (see e.g. [START_REF] Acary | Nonsmooth modeling and simulation for switched circuits[END_REF] and [START_REF] Stewart | Dynamics with inequalities: Impacts and hard constraints[END_REF])).

Differential inclusions involving maximal monotone operators

We review in this section some aspects of the theory of differential inclusions involving maximal monotone operators. Namely, we provide an invariance result for associated closed sets that we use in the sequel.

Given a set-valued operator A : H ⇒ H, which we identify with its graph, we denote its domain by domA :

= {x ∈ H | A(x) = ∅}. Operator A is monotone if x 1 -x 2 , y 1 -y 2 ≥ 0 for all (x 1 , y 1 ), (x 2 , y 2 ) ∈ A,
and α-hypomonotone for α ≥ 0 if the operator A + α id is monotone, where id is the identity mapping. We say that A is maximal monotone if A is monotone and coincides with every monotone operator containing its graph. In such a case, it is known that A(x) is convex and closed for every x ∈ H. We shall denote by (A(x)) • , x ∈ domA, the set of minimal norm vectors in A(x); i.e., (A(x)) • := {y ∈ A(x) | y = min z∈A(x) z }; hence, for any vector x ∈ domA and y ∈ H, the set Π A(x) (y) is a singleton and we have that (y -A(x)) • = y -Π A(x) (y).

We consider the following differential inclusion

ẋ(t) ∈ f (x(t)) -A(x(t)) t ∈ [0, ∞), x(0; x 0 ) = x 0 ∈ domA, (4.3)
governed by a maximal monotone operator A : H ⇒ H, which is subject to a perturbation by a (l-)Lipschitz continuous mapping f : H → H. By a strong solution of (4.3) starting at x 0 ∈ domA we refer to an absolutely continuous function x(•; x 0 ) which satisfies (4.3) for a.e. t ≥ 0, together with the initial condition x(0; x 0 ) = x 0 . It is known that (4.3) processes a unique strong solution whenever x 0 ∈ domA, H is finite-dimensional, int(domA) = ∅, or A is the subdifferential of convex, proper, and lower semi-continuous function. More generally, we call x(•; x 0 ) a weak solution of (4.3) starting at x 0 ∈ domA, the unique continuous function which is the uniform limit of strong solutions x(•; x k ) with (x k ) ⊂ domA converging to x 0 .

The following result provides other properties of the solutions of (4.3); for more details we refer to the book [START_REF] Barbu | Nonlinear differential equations of monotone types in Banach spaces[END_REF][START_REF] Brézis | Operateurs Maximaux Monotones et Semi-Groupes de Contractions Dans Les Espaces de Hilbert[END_REF]. To denote the right-derivative whenever it exists we use the notation

d + x(t; x 0 ) dt := lim h↓0 x(t + h; x 0 ) -x(t) h .
Proposition 4.4. Fix x 0 , y 0 ∈ domA. Then system (4.3) has a unique continuous solution x(t) ≡ x(t; x 0 ), t ≥ 0, such that, for all s, t ≥ 0

x(s; x(t; x 0 )) = x(t + s; x 0 ), x(t; x 0 ) -x(t; y 0 ) ≤ e lt ||x 0 -y 0 ||.

Moreover, if x 0 ∈ domA, then d + x(t; x 0 ) dt = [f (x(t; x 0 )) -A(x(t; x 0 ))] • = f (x(t; x 0 )) -Π A(x(t;x 0 )) (f (x(t; x 0 ))),
Differential inclusions with prox-regular sets and the function t

→ d + x(t)
dt is right-continuous at every t ≥ 0 with

d + x(t) dt ≤ e lt d + x(0) dt . (4.4)
We are going to characterize those closed sets which are invariant with respect to differential inclusion (4.3). Definition 4.5. A closed set S ⊂ H is strongly invariant for (4.3) if every solution of (4.3) starting in S remains in this set for all time t ≥ 0. The set S ⊂ H is weakly invariant for (4.3) if for every x 0 ∈ S, there exists a solution x(•; x 0 ) of ( 4.3) such that x(t; x 0 ) ∈ S for all time t ≥ 0. When differential inclusion (4.3) has a unique solution for every given initial condition, both notions coincide, and we simply say in this case that S is invariant.

Due to the semigroup property in Proposition 4.4, it is immediately seen that S is invariant iff every solution of (4.3) starting in S remains in this set for all sufficiently small time t ≥ 0. The issue with these sets, also referred to as viable sets for (4.3); see, [START_REF] Aubin | Differential Inclusions. Set-Valued Maps and Viability Theory[END_REF], is to find good characterizations via explicit criteria, which do not require an a-priori computation of the solution of (4. 3). An extensive research has been done to solve this problem for different kinds of differential inclusions and equations ( [START_REF] Clarke | Optimization and Nonsmooth Analysis[END_REF][START_REF] Clarke | Approximate invariance and differential inclusions in Hilbert spaces[END_REF]). Complete primal and dual characterizations are given in [START_REF] Adly | Nonsmooth Lyapunov pairs for infinite-dimensional first order differential inclusions[END_REF][START_REF] Adly | Nonsmooth Lyapunov pairs for differential inclusions governed by operators with nonempty interior domain[END_REF]. Such a property is almost necessary for the invariance of set S; indeed, it is necessary whenever (4.3) admits strong solutions, as is the case in the finitedimensional setting.

We start by recalling the following lemma (see, e.g., [START_REF] Clarke | Approximate invariance and differential inclusions in Hilbert spaces[END_REF][START_REF] Radulescu | Geometric Approximation of Proximal Normals[END_REF]), which is a consequence of Ekeland's Variational principle [START_REF] Ekeland | On the variational principle[END_REF].

Lemma 4.6. Suppose that S is closed. Then, for any x ∈ H and any s ∈ Π δ S (x), with δ > 0, there exist s δ ∈ S and y ∈ H such that

     y -s δ ∈ N P S (s δ ), y -s δ -(x -s) ≤ 2δ, s -s δ ≤ δ, x -y ≤ δ. (4.6)

Differential inclusions involving maximal monotone operators

In addition, if x ∈ B(x 0 , σ) for some x 0 ∈ S and σ > 0, then s δ satisfies s δ -x 0 ≤ 2σ + δ.

(4.7)

Theorem 4.7. Let S ⊂ H be as in (4.5) and take x 0 ∈ S ∩ domA. Assume there are m, ρ > 0 such that for all x ∈ S ∩ domA ∩ B(x 0 , ρ)

sup ξ∈N P S (x) min x * ∈A(x)∩B(θ,m) ξ, f (x) -x * ≤ 0. (4.8)
Then there exists T > 0 such that the solution x(•; x 0 ) of (4.3) satisfies

x(t; x 0 ) ∈ S for all t ∈ [0, T ].
Consequently, if for every x ∈ S ∩ domA inequality (4.8) holds for some m(x), ρ(x) > 0, then S is invariant for (4.3).

Proof. Fix ε > 0, and let positive real numbers m, ρ be as in (4.8). We start by showing that the set

K := S ∩ domA ∩ B(x 0 , ρ)
is closed and satisfies N P K (x) = N P S (x) for all x ∈ int(B(x 0 , ρ)) ∩ K. (4.9)

Suppose that x n → x for some {x n } ⊂ K; hence, x ∈ S ∩ B(x 0 , ρ) as a consequence of (4.5). From the inequality of the current assumption one has that A(x n ) ∩ B(0, m) = ∅ for all n. Take x * n ∈ A(x n ) ∩ B(0, m) that we assume (w.l.o.g.) weak converging to some x * ∈ B(θ, m). Since A is maximal monotone, it is norm-weak upper semi-continuous, and so we get x * ∈ A(x); that is, x ∈ K. Relation (4.9) follows since, for all x ∈ K such that x -x 0 < ρ, one has (ecall (4.5))

N P K (x) = N P B(x 0 ,ρ)∩S∩domA (x) = N P S∩domA (x) = N P S∩domA (x) = N P S (x).
Let us also observe that in view of condition (4.8), and using the Lipschitzianity of f, there is a constant M > 0 such that

∅ = f (x) -(A(x) ∩ B(0, m)) ⊂ B(0, M ) ∀x ∈ K. (4.10)
Differential inclusions with prox-regular sets

We choose positive numbers t, δ and positive integer N such that 4M t < ρ, max{4 t(M 2 + 4M + 1)ε -2 , 2M tε -1 } < N, 4N δ < min{ t, N ρ, N ε}, (4.11) and denote by π := {t 0 , t 1 , ..., t N } the uniform partition of the interval [0, t]; hence,

d(π) := max 0≤i≤N -1 (t i+1 -t i ) = t N .
We claim the existence of vectors z i , for 0 ≤ i ≤ N, with z 0 = x 0 , and ŝi ∈ H,

y i ∈ H, s i ∈ K, s * i ∈ H, for 0 ≤ i ≤ N -1, such that, for all 0 ≤ i ≤ N -1 ŝi ∈ Π δ K (z i ), (4.12
)

y i -s i ∈ N P K (s i ), (4.13)
z i -ŝi -(y i -s i ) ≤ 2δ, (4.14)
s * i , y i -s i ≤ 0, (4.15)
s * i ∈ (f (s i ) -A(s i )) ∩ B(θ, M ), (4.16)
together with a function Z i (•), defined on [t i , t i+1 ] and satisfying

Żi (t) = s * i , ∀t ∈ (t i , t i+1 ); Z i (t i ) = z i , Z i (t i+1 ) = z i+1 . (4.17)
We proceed by finite induction on k = 0, 1, • • • , N :

The claim is true for all 0 ≤ i ≤ k when k = 0 : indeed, it suffices to take z 0 = ŝ0 = y 0 = s 0 = x 0 and z 1 = (t 1 -t 0 )s * 0 + z 0 . Then the existence of s * 0 comes from (4.10) and the fact that s 0 = x 0 ∈ K ⊂ domA.

We suppose that the claim is true for all 0 ≤ i ≤ k, and we shall prove it for all 0 ≤ i ≤ k + 1; we may suppose that k < N -1, because, for otherwise, we are done. To proceed we first observe that the vector z k+1 , which is already defined at the induction hypothesis, satisfies

z k+1 -x 0 ≤ k i=0 z i+1 -z i = k i=0 Z i (t i+1 ) -Z i (t i ) = k i=0 (t i+1 -t i )s * i (by (4.17)) ≤ M (t k+1 -t 0 ) ≤ M t.

Differential inclusions involving maximal monotone operators

Then, by choosing any element ŝk+1 ∈ Π δ K (z k+1 ), Lemma 4.6 applied with z k+1 and ŝk+1 yields a pair (s k+1 , y k+1 ) ∈ K × H such that

                       y k+1 -s k+1 ∈ N P K (s k+1 ), (y k+1 -s k+1 ) -(z k+1 -ŝk+1 ) ≤ 2δ, y k+1 -z k+1 ≤ δ, ŝk+1 -s k+1 ≤ δ, s k+1 -x 0 ≤ 2M t + δ ≤ 2(δ + tM ) < ρ (by (4.11)).
Next, from the current hypothesis (4.8), together with (4.9) and (4.10), we find

s * k+1 ∈ (f (s k+1 ) -A(s k+1 )) ∩ B(θ, M ) such that y k+1 -s k+1 , s * k+1 ≤ 0. (4.18)
With this vector s * k+1 in hand, we define the function Z k+1 (•) on [t k+1 , t k+2 ] as the unique solution of the following differential equation

   Żk+1 (t) = s * k+1 , a.e. t ∈ [t k+1 , t k+2 ],
Z k+1 (t k+1 ) = z k+1 .

We also introduce the vector

z k+2 := Z k+1 (t k+2 ).
So, the vectors z i , 0 ≤ i ≤ k + 2, and ŝi ∈ H, y i ∈ H, s i ∈ K, s * i ∈ H, for 0 ≤ i ≤ k + 1, together with the functions Z i (•), for 0 ≤ i ≤ k + 1, accomplish with requirements of the claim.

At this stage, based on the claim above, we introduce the continuous piecewise linear function Z(•) defined on [0, t] as

Z(t) := Z i (t) for t ∈ [t i , t i+1 ], i = 0, ..., N -1.
We are going to verify that

d(Z(t), K) ≤ ε for all t ∈ [0, t], (4.19)
s i -Z(t) ≤ 2ε for all i = 0, ..., N -1 and t ∈ [t i , t i+1 ], (4.20)
where s i is defined in (4.13). Indeed, for every j = 0, ..., N -1 one has

d 2 K (Z j+1 (t j+1 )) = d 2 K (Z j (t j+1 )) ≤ Z j (t j+1 ) -ŝj 2 (since ŝj ∈ Π δ K (z j ) ⊂ K by (4.12)) = Z j (t j+1 ) -Z j (t j ) 2 + Z j (t j ) -ŝj 2 + 2 Z j (t j+1 ) -Z j (t j ), Z j (t j ) -ŝj ≤ (t j+1 -t j ) 2 M 2 + z j -ŝj 2 + 2(t j+1 -t j ) s * j , z j -ŝj (by (4.17)) ≤ (t j+1 -t j ) 2 M 2 + d 2 K (z j ) + δ 2 (
≤ (t j+1 -t j ) 2 M 2 + d 2 K (Z j (t j )) + δ 2 + 4δM (t j+1 -t j ) ≤ (t j+1 -t j ) 2 M 2 + d 2 K (Z j (t j )) + d(π)(1 + 4M (t j+1 -t j )) ≤ d 2 K (Z j (t j )) + (t j+1 -t j )d(π)(M 2 + 4M + 1),
which by summing up over j = 0, • • • , i -1 (when i = 1, ..., N -1), and taking into account that x 0 ∈ K, gives us

d 2 K (Z i (t i )) ≤ (t i -t 0 )d(π)(M 2 + 4M + 1) ≤ d(π)t i (M 2 + 4M + 1) ≤ ε 2 4 . (4.21)
This inequality also holds when i = 0, because d 2 K (Z 0 (t 0 )) = d 2 K (x 0 ) = 0. Then, for every i = 0, ..., N -1 and t ∈ (t i , t i+1 ) we write

d 2 K (Z(t)) = d 2 K (Z i (t)) = d 2 K (s * i (t -t i ) + Z i (t i )) (by (4.17)) ≤ ( s * i (t -t i ) + d K (Z i (t i ))) 2 ≤ 2 s * i (t -t i ) 2 + 2d 2 K (Z i (t i )) ≤ 2(t -t i ) 2 M 2 + 2d 2 K (Z i (t i )) (by (4.16)) ≤ 2d(π) 2 M 2 + 2d 2 K (Z i (t i )) ≤ 2d(π) 2 M 2 + ε 2 2 ≤ ε 2 , ( 4
s i -Z(t) 2 = s i -Z i (t) 2 ≤ 2 s i -Z i (t i ) 2 + 2 Z i (t i ) -Z i (t) 2 ≤ 2(2 s i -ŝi 2 + 2 ŝi -Z i (t i ) 2 ) + 2(t -t i ) 2 M 2 ≤ 2(t -t i ) 2 M 2 + 4δ 2 + 4 ŝi -z i 2 ≤ 2(t -t i ) 2 M 2 + 8δ 2 + 4d 2 K (z i ) = 2(t -t i ) 2 M 2 + 8δ 2 + 4d 2 K (Z i (t i )) ≤ 2d(π) 2 M 2 + ε 2 + 8δ 2 < 2ε 2 (by (4.21)).
Now, we consider the (strong) solution x(t) := x(•; x 0 ) of differential inclusion (4.3), and define the absolutely continuous function η : [0, t] → R as

η(t) := Z(t) -x(t) 2 .
Let us prove that for i = 0, ..., N -1 and t in a full-measure subset of

[t i , t i+1 ] it holds 1 2 η(t) ≤ (l + 1)η(t) + 2(4εl) 2 + 4lε(2ε + c), (4.23)
where c := l -1 sup t∈[0, t] ẋ(t) -s * i (c is finite, due to Proposition 4.4).

Differential inclusions with prox-regular sets Indeed, by using the Lipschitz condition of f and relation (4.20),

1 2 η(t) = Ż(t) -ẋ(t), Z(t) -x(t) = s * i -f (s i ) -( ẋ(t) -f (x(t))), Z(t) -s i + f (s i ) -f (x(t)), Z(t) -x(t) + s i -f (s i ) -( ẋ(t) -f (x(t))), s i -x(t)
≤0 (by (4.16) and the monotonicity of A)

≤ s * i -f (s i ) -( ẋ(t) -f (x(t))), Z(t) -s i + f (s i ) -f (x(t)), Z(t) -x(t) ≤ s i -Z(t) f (s i ) -s * i -f (x(t)) + ẋ(t) + Z(t) -x(t) f (s i ) -f (x(t)) ≤ 2εl s i -x(t) + 2ε s * i -ẋ(t) (by (4.20)) + l Z(t) -x(t) s i -x(t) ≤ 2εl( s i -Z(t) + Z(t) -x(t) ) + 2ε s * i -ẋ(t) + l Z(t) -x(t) ( s i -Z(t) + Z(t) -x(t) ) ≤ l Z(t) -x(t) 2 + 4εl Z(t) -x(t) + 2lε(2ε + c) (by (4.20) ≤ (l + 1) Z(t) -x(t) 2 + (4εl) 2 + 2lε(2ε + c) = (l + 1)η(t) + 2(4εl) 2 + 4lε(2ε + c),
and (4.23) follows. Hence, by Gronwall's lemma (Lemma 4.1), we obtain that for very t ∈ [0, t],

η(t) ≤ η(0)e (l+1)t + (2(4εl) 2 + 4lε(2ε + c)) t 0 e (l+1)(t-s) ds = (2(4εl) 2 + 4lε(2ε + c))
t 0 e (l+1)(t-s) ds, and, consequently,

Z(t) -x(t) = η 1 2 (t) ≤ 2t(4εl) 2 + 4lεt(2ε + c) 1 2 e (l+1)t .
By combining this with inequality (4.19), we infer that for every t ∈ [0, t]

d S (x(t)) ≤ d K (x(t)) ≤ d K (Z(t)) + Z(t) -x(t) ≤ 2 t(4εl) 2 + 4lε t(2ε + c) 1 2 e (l+1) t + ε.
Since this inequality holds for any ε > 0, and t does not depend on ε, we conclude that x(t) ∈ S for any t ∈ [0, t]. The first part of the theorem is proved.

We now suppose that for any x ∈ S ∩ domA, there exist positive numbers m, ρ (depending on x) such that inequality (4.8) holds. Let us fix x 0 ∈ S ∩ domA, so that x(t; x 0 ) ∈ domA for every t ≥ 0 (Proposition 4.4). From the first assertion of the theorem, there exists t > 0 such that x(t; x) ∈ S ∩domA for any t ∈ [0, t); moreover, since S is closed, we also have that x( t; x 0 ) ∈ S ∩ domA. By applying the first assertion of the current theorem, and taking into account the semi-group property (again by Proposition 4.4), we find t > 0 such that for any t ∈ [0, t], one has

x(t + t; x 0 ) = x(t; x( t; x 0 )) ∈ S ∩ domA.
Thus, we prove that x(t; x 0 ) ∈ S for every t ≥ 0. Assume now that x 0 ∈ S ∩ domA \ domA, and by (4.5) let (x k ) ⊂ S ∩ domA be such that x k → x. Then, by arguing as in the last paragraph, for each k ≥ 1 we have that x k (t; x k ) ∈ S for every t ≥ 0. But x(•; x k ) converges uniformly to x(•; x 0 ) on each interval [0, t] (see [START_REF] Adly | Invariant sets and Lyapunov pairs for differential inclusions with maximal monotone operator[END_REF][START_REF] Adly | Nonsmooth Lyapunov pairs for infinite-dimensional first order differential inclusions[END_REF][START_REF] Adly | Nonsmooth Lyapunov pairs for differential inclusions governed by operators with nonempty interior domain[END_REF]), and so x(•; x 0 ) also stays in S. The proof of the theorem is complete.

The invariance criterion of Theorem 4.7 takes a simple form when the domain of operator A has a nonempty interior, and S ⊂ int(domA). In this case, because A is locally bounded on int(domA), the number m is dropped from inequality (4.8).

Corollary 4.8. Let S be a nonempty closed subset of int(domA) such that

sup ξ∈N P S (x) min x * ∈A(x) ξ, f (x) -x * ≤ 0 ∀x ∈ S.
Then S is invariant for system (4.3).

The following corollary will be useful in the proof of Theorem 4.19.

Corollary 4.9. Assume that A is a monotone operator, and let S be a closed subset of domA. Suppose that x(•) is an absolutely continuous function such that

ẋ(t) ∈ f (x(t)) -A(x(t)) a.e. t ∈ [0, T ] , x(0) = x 0 ∈ S.

Differential inclusions with prox-regular sets

If there are some numbers m, ρ > 0 such that

sup ξ∈N P S (x) min x * ∈A(x)∩B(θ,m) ξ, f (x) -x * ≤ 0 ∀x ∈ S ∩ B(x 0 , ρ),
then there is some T * ∈ (0, T ] such that x(t) ∈ S for all t ∈ [0, T * ].

Proof. According to [START_REF] Brézis | Operateurs Maximaux Monotones et Semi-Groupes de Contractions Dans Les Espaces de Hilbert[END_REF], there exists a maximal monotone extension of A that we denote in the same way. By the current hypothesis, for every x ∈ S ∩ B(x 0 , ρ) we have that A(x) ∩ B(θ, m) = ∅. Since x(•) is a unique solution, we apply Theorem 4.7, and the conclusion of the corollary follows.

The existence result

In this section, we use tools from convex and variational analysis to prove the existence of a solution for the differential inclusion (4.1),

   ẋ(t) ∈ f (x(t)) -N C (x(t)) a.e. t ≥ 0, x(0, x 0 ) = x 0 ∈ C,
where N C is the proximal, or, equivalently, the limiting, normal cone to an runiformly prox-regular closed subset C of H, and f is a Lipschitz continuous mapping. We shall denote by x(•; x 0 ) the solution of this inclusion.

Definition 4.10. (see [START_REF] Mazade | Regularization of differential variational inequalities with locally prox-regular sets[END_REF][START_REF] Poliquin | Local differentiability of distance functions[END_REF]) For positive numbers r and α, a closed set S is said to be (r, α)-prox-regular at x ∈ S provided that one has x = Π S (x + v), for all x ∈ S ∩ B(x, α) and all v ∈ N P S (x) such that ||v||< r. The set S is r-prox-regular (resp., prox-regular) at x when it is (r, α)-prox-regular at x for some real α > 0 (resp., for some numbers r, α > 0). The set S is said to be r-uniformly prox-regular when α = +∞.

It is well-known and easy to check that when S is r-uniformly prox-regular, then for every x ∈ S, N P S (x) = N C S (x); thus, for such sets we will simply write N S (x) to refer to each one of these cones, and write T S (x) to refer to the Bouligand tangent cone T B S (x) = (N S (x)) * . We have the following property of r-uniformly prox-regular sets, which can be easily checked. Proposition 4.11. Let S be a closed subset of H. If S is r-uniformly prox-regular, then the set-valued mapping defined by x → N P S (x) ∩ B is 1 r -hypomonotone.

The existence result

Before we state the main theorem of this section we give a useful characterization of prox-regularity.

Lemma 4.12. The following statements are equivalent for every closed set C ⊂ H and every m > 0, (a) C is r -uniformly prox-regular for every r < r, (b) the mapping N P C ∩ B(θ, m) + m r id is monotone, (c) there exists a maximal monotone operator A defined on H such that r id, such that C ⊂ domA ⊂ coC (see, e.g., [START_REF] Brézis | Operateurs Maximaux Monotones et Semi-Groupes de Contractions Dans Les Espaces de Hilbert[END_REF]). Moreover, we have that We also need some properties of the solution of (4.1). As pointed out by one of the reviewers, the assertions of the following lemma are very natural and may have already appeared in the literature. For the convenience of the reader, we give a complete proof. Lemma 4.13. If x(•; x 0 ) is a solution of (4.1), then for a.e. t ∈ [0, T ] we have

N P C (x) ∩ B(θ, m) + m r x ⊂ A(x) ⊂ N P C (x) + m r x for every x ∈ C.
N P C (x) ∩ B(θ, m) + m r x ⊂ A(x) ⊂ N P C (x) + m r x, ∀x ∈ C. ( 4
ẋ(t), f (x(t)) -ẋ(t) = 0, (4.25) f (x(t)) -ẋ(t) ≤ f (x(t)) , (4.26
)

|| ẋ(t)||≤ min{||f (x(t))||, ||f (x 0 )||e lt }, ||x(t) -x 0 ||≤ t||f (x 0 )||e lt . (4.27)
Consequently, x(•; x 0 ) is the unique solution of (4.1) on [0, T ].

Differential inclusions with prox-regular sets

Proof. Let t ∈ (0, T ] be a differentiability point of the solution x(•). Then there is some δ > 0 such that

f (x(t)) -ẋ(t), x(s) -x(t) ≤ δ||x(s) -x(t)|| 2 , for all s ∈ [0, T ],
and, so, by dividing on s -t and taking the limit as s ↓ t we derive that

f (x(t)) -ẋ(t), ẋ(t) ≤ 0.
Similarly, when s ↑ t we get f (x(t)) -ẋ(t), ẋ(t) ≥ 0, which yields (4.25).

Since

f (x(t)) -ẋ(t) ∈ N C (x(t)) and ẋ(t) ∈ T B C (x(t)), statement (4.25) means that f (x(t)) -ẋ(t) = Π N C (x(t)) (f (x(t))
) and this yields (4.26), f (x(t)) -ẋ(t) ≤ f (x(t)) . Moreover, using (4.25), we have (for a.e. t ∈ [0, T ])

|| ẋ(t)|| 2 = ẋ(t), ẋ(t) = ẋ(t), f (x(t)) ≤ ẋ(t) f (x(t)) , (4.28)
which gives us || ẋ(t)||≤ ||f (x(t))||. Then

d dt x(t) -x 0 2 = 2 x(t) -x 0 , ẋ(t) ≤ 2||x(t) -x 0 || ||f (x(t))|| ≤ 2||x(t) -x 0 ||(||f (x 0 )||+l||x(t) -x 0 ||) = 2||f (x 0 )|| ||x(t) -x 0 ||+2l||x(t) -x 0 || 2 ,
which by Lemma 4.1 gives us

||x(t) -x 0 ||≤ ||f (x 0 )|| l (e lt -1) ≤ ||f (x 0 )||te lt , (4.29)
so that, using the inequality of the middle together with (4.28),

|| ẋ(t)|| ≤ ||f (x(t))||≤ ||f (x 0 )||+l||x(t) -x 0 || ≤ ||f (x 0 )||+||f (x 0 )||(e lt -1) = ||f (x 0 )||e lt .
This proves (4.26) and (4.27).

To finish we need to check the uniqueness of the solution. Proceeding by contradiction, we assume that y(•) is another solution on [0, T ] of (4.1). Then for all t ∈ [0, T ] such that ||f (x(t))||+||f (y(t)||> 0 and f (y

(t)) -ẏ(t) ∈ N C (y(t)) we have f (y(t)) -ẏ(t) ||f (x(t))||+||f (y(t)|| ∈ N C (y(t)) ∩ B,
and similarly for x(•). Then, by the r-uniformly prox-regularity hypothesis on C,

ẋ(t) -ẏ(t), x(t) -y(t) ≤ l + 1 r (||f (x(t))||+||f (y(t))||) ||x(t) -y(t)|| 2 ; (4.30)
this inequality also holds when ||f (x(t))||+||f (y(t)||= 0 as a consequence of (4.28). By applying Gronwall's Lemma (Lemma 4.1) with the function 1 2 ||x(t)-y(t)|| 2 , and observing that x(0) = y(0) = x 0 , it follows that x(t) = y(t) for every t ∈ [0, T ].

The main result is given in the following theorem, using a convex analysis approach, while Theorem 4.15 below provides more properties of the solution, which will be used later on. Proof. We fix a sufficiently large m > 0 and choose a T 0 > 0 such that

||f (x 0 )||+l(||f (x 0 )||T 0 e (l+ m r )T 0 + 1) ≤ m. (4.31)
By Lemma 4.12(c) we consider a maximal monotone extension A such that, for all x ∈ C,

N C (x) ∩ B(θ, m) + m r x ⊂ A(x) ⊂ N C (x) + m r x. (4.32)
According to [START_REF] Barbu | Nonlinear differential equations of monotone types in Banach spaces[END_REF][START_REF] Brézis | Operateurs Maximaux Monotones et Semi-Groupes de Contractions Dans Les Espaces de Hilbert[END_REF], the differential inclusion

   ẋ(t) ∈ f (x(t)) + m r x(t) -A(x(t)), a.e. t ∈ [0, T 0 ] x(0) = x 0 ∈ C, (4.33)
has a unique solution x(•) such that x(t) ∈ domA (⊂ co(C)) for all t ∈ [0, T 0 ], as well as (see, e.g., [START_REF] Adly | Nonsmooth Lyapunov pairs for infinite-dimensional first order differential inclusions[END_REF])

d + x(t) dt ≤ e (l+ m r )t d + x(0) dt ≤ e (l+ m r )t ||Π A(x 0 ) (f (x 0 ) + m r x 0 )||.
Moreover, since m r x 0 ∈ A(x 0 ) (due to (4.32)), for all t ∈ [0, T 0 ]

d + x(t) dt ≤ e (l+ m r )t ||f (x 0 )||≤ e (l+ m r )T 0 ||f (x 0 )||=: k,
and, hence,

||x(t) -x 0 ||≤ kT 0 , (4.34)
Differential inclusions with prox-regular sets

||f (x(t))||≤ ||f (x 0 )||+l||x(t) -x 0 ||≤ ||f (x 0 )||+lkT 0 ; in particular, x(•) is k-Lipschitz on [0, T 0 ].
Next, we want to show that x(t) ∈ C for every t ∈ [0, T 0 ]. For this aim we shall apply Theorem 4.7. Given y ∈ C ∩ B(x 0 , kT 0 + 1) and ξ ∈ N C (y), we define z := Π N C (y) (f (y)) ∈ N C (y) (z is well defined since N C (y) is closed (and convex)). It is easy to see that

||z||≤ ||f (y)||≤ ||f (x 0 )||+l||y -x 0 ||≤ ||f (x 0 )||+l(kT 0 + 1) ≤ m.
Hence, according to (4.32), we derive that y

* := z + m r y ∈ N C (y) ∩ B(θ, m) + m r y ⊂ A(y), with y * ≤ m := m(1 + 1 r ( x 0 + kT 0 + 1)). Now, since f (y) -z ∈ T C (y) we obtain that ξ, f (y) -z ≤ 0, which shows that inf v * ∈A(y)∩B(θ,m) ξ, f (y) + m r y -v * ≤ ξ, f (y) + m r y -y * ≤ 0. (4.35)
Consequently, according to Theorem 4.7, there is a positive number T ∈ (0, T 0 ) such that x(t) ∈ C for every t ∈ [0, T ]. Moreover, due to (4.34), for all t ∈ [0, T 0 ] we have that x(t) ∈ B(x 0 , kT 0 + 1) and, so, from the argument above we infer that x(t) ∈ C for all t ∈ [0, T 0 ]. Whence, since x(t) ∈ C for t ∈ [0, T 0 ], (4.32) implies that

ẋ(t) ∈ f (x(t)) + m r x(t) -A(x(t)) ⊂ f (x(t)) + m r x(t) -N C (x(t)) - m r x(t) = f (x(t)) -N C (x(t)); that is, x(•) is a solution of (4.1) on [0, T 0 ].

Now, we set

T := sup {T > 0 such that system (4.1) has a solution x(•; x 0 ) on [0, T ]} ; so, T > 0 from the paragraph above. If T is finite, then we take a sequence (T n ) such that T n ↑ T, and denote x n (•; x 0 ) the corresponding solution of (4.1), which is defined on [0, T n ]. Let function x(•; x 0 ) : [0, T ) → H be defined as

x(t; x 0 ) = x n (t) if t ≤ T n .
According x(T n ). Since x(T ) ∈ C, from the first paragraph we find a T 1 > 0 and a solution of (4.1) on [0, T + T 1 ] which coincides with x(•; x 0 ) on [0, T ], contradicting the finiteness of T -this is to say that T = ∞.

An immediate consequence of (the proof of) Theorem 4.14 is that the solution of differential inclusion (4.1) satisfies the so-called semi-group property, x(t; x(s; x 0 )) = x(t + s; x 0 ) for all t, s ≥ 0 and x 0 ∈ C.

(4.36)

The following theorem gathers further properties of the solution of (4.1), that we shall use in the sequel. Relation (4.37) below on the derivative of the solution reinforces the statement of Lemma 4.13.

Theorem 4.15. Let x(•; x 0 ), x 0 ∈ C, be the solution of (4.1). Then the following statements hold true:

(a) For every t ≥ 0, x(•; x 0 ) is right-derivable at t with (c) If y(•; y 0 ), y 0 ∈ C, is the corresponding solution of (4.1), then for every t ≥ 0

d + x(t) dt = (f (x(t)) -N C (x(t))) • = f (x(t)) -Π N C (x(t)) (f (x(t)) = Π T C (x(t)) (f (x(t))), (4.37)
x(t) -y(t) ≤ x 0 -y 0 e lt+ ||f (x 0 )||+||f (y 0 )|| lr (e lt -1) .

Proof. We fix t ≥ 0 (we may suppose that t = 0). From the argument used in the proof of Theorem 4.14 we know that for some m > ||f (x 0 )||+l (l is the Lipschitz constant of f ) there exists a maximal monotone operator A such that x(•) := x(•; x 0 ) is the solution of the following differential inclusion on some interval

[0, δ] , δ > 0, ẋ(t) ∈ f (x(t)) + m r x(t) -A(x(t)), x(0) = x 0 ,
where r comes from the r-uniform prox-regularity of C. W.l.o.g. we may suppose that ||f (x(t))||+l < m for all t ∈ [0, δ] so that (see Proposition 4.4), for every

t ∈ [0, δ] , d + x(t) dt = (f (x(t)) + m r x(t) -A(x(t))) • . (4.40) Since f (x(t)) ∈ B(θ, m) we have that (f (x(t)) -N C (x(t))) • = f (x(t)) -Π N C (x(t)) (f (x(t))) = f (x(t)) -Π N C (x(t))∩B(θ,m) (f (x(t))) = (f (x(t)) -N C (x(t)) ∩ B(θ, m)) • ,
and, so, due to (4.40), and the inclusions (4.32):

f (x(t)) -N C (x(t)) ∩ B(θ, m) ⊂ f (x(t)) + m r x(t) -A(x(t)) ⊂ f (x(t)) -N C (x(t)),
we get the first equality in (4.37). The other two equalities in (4.37) easily follow from the definition of the orthogonal projection. Moreover, statement (b) is also a consequence of Proposition 4.4. Thus, (4.37) follows from Lemma 4.13. Finally, (4.39) and statement (c) follow easily using relation (4.30) (and Lemma 4.1).

The main idea behind the previous existence theorems, Theorems 4.14 and 4.15, as well as the forthcoming results on Lyapunov stability in the next section, is that differential inclusion (4.1) is in some sense equivalent to a differential inclusion governed by a (Lipschitz continuous perturbation of a) maximal monotone operator. This fact is highlighted in the following corollary. Recall, by Lemma 4.12(c), that for every m > 0 the r-uniformly prox-regularity of the set C yields the existence of a maximal monotone operator A C such that

N C (x) ∩ B(θ, m) + m r x ⊂ A C (x) ⊂ N C (x) + m r x for every x ∈ C. (4.41)
Corollary 4.16. An absolutely continuous function x(t) is a solution of (4.1) on

[0, T ]; that is,    ẋ(t) ∈ f (x(t)) -N C (x(t)) a.e. t ∈ [0, T ] x(0) = x 0 ∈ C,
if and only if it is (the unique) solution of the following differential inclusion, for 4.5. Lyapunov stability analysis some m > 0, (DIM )

   ẋ(t) ∈ f (x(t)) + m r x(t) -A C (x(t)) a.e. t ∈ [0, T ] x(0) = x 0 ∈ C,
where the maximal monotone operator A C : H ⇒ H is defined in (4.41).

Proof. According to Theorems 4. [START_REF] Barbu | Nonlinear differential equations of monotone types in Banach spaces[END_REF]

ẋ(t) ∈ f (x(t)) -N C (x(t)) ∩ B(θ, m),
and, so, by the definition of A C above (see (4.41)) we conclude that x(t) is also the solution of differential inclusion (DIM).

Conversely, if x(t) is a solution of differential inclusion (DIM) for some m > 0, then, as it follows from the proof of Theorem 4.14, we get that x(t) ∈ C for all t ∈ [0, T 0 ] for some T 0 > 0. Hence, once again by (4.41), we conclude that x(t) is also a solution of (101) on [0, T 0 ]. Taking into account Lemma 4.14 we show, also as in the proof of Theorem 4.14, that T 0 can be taken to be T .

Lyapunov stability analysis

In this section, we give explicit characterizations for lower semi-continuous a-Lyapunov pairs, Lyapunov functions, and invariant sets associated to differential inclusion (4.1). Recall that x(•; x 0 ) (or x(•), when any confusion is excluded) refers to the unique solution of (4.1), which satisfies x(0; x 0 ) = x 0 . Definition 4.17. Let functions V, W : H → R be lower semi-continuous, with W ≥ 0, and let an a ≥ 0. We say that (V, W ) is (or forms) an a-Lyapunov pair for differential inclusion (4.1) if, for all x 0 ∈ C, e at V (x(t; x 0 )) + t 0 W (x(τ ; x 0 ))dτ ≤ V (x 0 ) for all t ≥ 0.

(4.42)

In particular, if a = 0, we say that (V, W ) is a Lyapunov pair. If, in addition, W = 0, then V is said to be a Lyapunov function.

A closed set S ⊂ C is said to be invariant for (4.1) if the function δ S is a Lyapunov function.

Differential inclusions with prox-regular sets Equivalently, using (4.36), it is not difficult to show that a-Lyapunov pairs are those pairs of functions V, W : H → R such that the mapping t → e at V (x(t; x 0 )) + t 0 W (x(τ, x 0 ))dτ is nonincreasing. In other words (see, e.g. [7, Proposition 3.2]), for any x 0 ∈ C, there exists t > 0 such that e as V (x(s;

x 0 )) + s 0 W (x(τ ; x 0 ))dτ ≤ V (x 0 ) for all s ∈ [0, t]. (4.43)
The failure of regularity in our Lyapunov candidate-like pairs is mainly carried out by the function V , since the function W can be always regularized to a Lipschitz continuous function on every bounded subset of H as the following lemma shows (see, e.g., [START_REF] Clarke | Optimization and Nonsmooth Analysis[END_REF]). Lemma 4.18. Let V , W and a be as in Definition 4.17. Then there exists a sequence of lower semi-continuous functions W k : H → R, k ≥ 1, converging pointwisely to W (for instance, W k W ) such that W k is Lipschitz continuous on every bounded subset of H. Consequently, (V, W ) forms an a-Lyapunov pair for (4.1) if and only if each (V, W k ) does. Now, we give the main theorem of this section, which characterizes lower semicontinuous a-Lyapunov pairs associated to differential inclusion (4.1). Theorem 4.19. Let functions V, W : H → R be lower semi-continuous, with W ≥ 0 and domV ⊂ C, a ≥ 0, and let x 0 ∈ domV . If there is ρ > 0 such that, for any x ∈ B(x 0 , ρ),

sup ξ∈∂ P V (x) min x * ∈N C (x)∩B(θ,||f (x)||) ξ, f (x) -x * + aV (x) + W (x) ≤ 0, (4.44)
then there is some T * > 0 such that

e at V (x(t; x 0 )) + t 0 W (x(τ ; x 0 ))dτ ≤ V (x 0 ) ∀t ∈ [0, T * ].
Consequently, the following statements are equivalent provided that either ∂ ≡ ∂ P or ∂ ≡ ∂ F :

(i) (V, W ) is an a-Lyapunov pair for (4.1);

(ii) for every x ∈ domV and ξ ∈ ∂V (x); ξ, f (x) -x * + aV (x) + W (x) ≤ 0;

ξ, (f (x) -N C (x)) • + aV (x) + W (x) ≤ 0;
(iv) for every x ∈ domV ;

V (x; (f (x) -N C (x)) • ) + aV (x) + W (x) ≤ 0; (v) for every x ∈ domV ; inf x * ∈N C (x)∩B(θ,||f (x)||) V (x; f (x) -x * ) + aV (x) + W (x) ≤ 0.
Moreover, when H is finite-dimensional, all the statements above except (ii) are equivalent when ∂ = ∂ L .

Proof. Let us start with the first part of the theorem. We choose T > 0 such that

T ||f (x 0 )||e lT ≤ ρ 2 ,
2||f (x(t))|| ≤ 2||f (x 0 )||+2l||x(t) -x 0 || < 2 max{||f (x 0 )||e lT , ||f (x 0 )||+lT e lT ||f (x 0 )||+l + 1} = k; that is, ẋ(t) ∈ f (x(t))-(N C (x(t)) ∩ B(θ, k)) . Hence, if A : H ⇒ H is the monotone operator defined as A(x) :=    N C (x) ∩ B(θ, k) + k r x if x ∈ C, ∅ otherwise,
then it is immediately seen that x(•) is also the unique solution of the following Differential inclusions with prox-regular sets differential inclusion,

   ẋ(t) ∈ f (x(t)) + k r x(t) -A(x(t)) a.e. t ∈ [0, T ], x(0) = x 0 ,
where r comes from the r-uniform prox-regularity of the set C. We introduce now the monotone operator A :

H × R 4 ⇒ H × R 4 and the l-Lipschitz function f : H × R 4 → H × R 4 defined as A(x, µ) := (A(x), 0 R 4 ) and f (x, µ) := (f (x) + k r x, 1, 0, 1, 0),
and consider the associated differential inclusion

   ẏ(t) ∈ f (y(t)) -A(y(t)), a.e. t ∈ [0, T ] y(0) = (x 0 , µ 0 ) ∈ C × R 4 , (4.45)
whose unique solution is given by y(t) := (x(t), t, 0, t, 0) + (0, µ 0 ) ⊂ C × R 4 . We define the lower semi-continuous functions

V n : H × R 3 → R, n ≥ 1, as V n (x, α, β, γ) := e aγ V (x) + (α -β)g n (α) + l 2 (α -β) 2 ,
where g n is an l -Lipschitz extension of the (Lipschitz) function

W (x(•)) -1 n from [0, T ] to [-1, T + 1]; hence, ∂ C g n (α) ⊂ B(θ, l ) for all α ∈ [0, T + 1].
Observe that epiV n ⊂ dom A and, for every (x, α, β, γ) ∈ domV n , we have that ∂ P,∞ V n (x, α, β, γ) ⊂ (e aγ ∂ P,∞ V (x), 0, 0, 0) and

∂ P V n (x, α, β, γ) ⊂ (e aγ ∂ P V (x), g n (α), -g n (α), ae aγ V (x)) + (0, (α -β)∂ C g n (α) + l (α -β), l (β -α), 0). (4.46)
At this step, we pick t ∈ [0, T ) and fix n ≥ 1 such that 1 n ≤ ρ 2 . We denote y 0 := (x(t), t, t, 0, V n (x(t), t, t, 0)), and choose ε > 0 such that (recall that W is Lipschitz continuous on B(x 0 , ρ))

g n (α ) + 2l |α -β |-e aγ W (x ) ≤ -1 2n , (4.47)
for any (y , µ ) := (x , α , β , γ , µ ) ∈ U ε := B (y 0 , ε) ∩ epiV n . Take vectors (y, µ) := (x, α, β, γ, µ) ∈ U ε and ξ ∈ N P epiVn (y, µ). Then x ∈ domV ⊂ C and (recall Lemma 4.13 (relation (4.27)))

||x -x 0 ||≤ ||x -x(t)||+||x(t) -x 0 ||≤ ε + ||f (x 0 )||T e lT < ρ; hence, x ∈ domV ∩ int(B(x 0 , ρ)) and ||f (x)||≤ ||f (x 0 )||+l||x -x 0 ||< ||f (x 0 )||+l(1 + ||f (x 0 )||T e lT ) ≤ k 2 ;
thus,

f (x) -(N C (x) ∩ B(0, ||f (x)||)) ⊂ f (x) -(N C (x) ∩ B(θ, k)) = f (x) + k r x -A(x). ( 4
ξ = ι (e aγ ξ 1 , g n (α) + (α -β)(ς + l ), -g n (α) + l (β -α), ae aγ V (x), -1) , (4.49) for some ξ 1 ∈ ∂ P V (x), ς ∈ ∂ C g n (α) and ι ≥ 0, or ξ = ι(e aγ ξ 2 , 0) (4.50)
with ξ 2 ∈ ∂ P,∞ V (x). If (4.49) holds, by the current assumption, there exists an

x * ∈ N C (x) ∩ B(θ, ||f (x)||) such that ξ 1 , f (x) -x * + aV (x) + W (x) ≤ 0.
Hence, due to (4.48),

y * := (x * + k r x, 0 R 4 ) belongs to A(y, µ) ∩ B(0, m) (since ||x * + k r x||≤ ||x * ||+ k r ||x||≤ k 2 + k r (||x 0 ||+ρ) ≤ m), and satisfies ξ, f (y, µ) -y * = ι ( e aγ ξ 1 , f (x) -x * + g n (α) + (α -β)(ς + l ) + ae aγ V (x)) = ιe aγ ( ξ 1 , f (x) -x * + aV (x) + W (x)) + ι (g n (α) + (α -β)(ς + l ) -e aγ W (x)) ≤ ι (g n (α) + 2l |α -β|-e aγ W (x)) ≤ - ι 2n ≤ 0,
where in the last inequality we used (4.47). We now consider the case when ξ satisfies (4.50). Let sequences [START_REF] Mordukhovich | Variational Analysis and Generalized Differentiation I: Basic Theory, Grundlehren der Mathematischen Wissenschaften[END_REF]Lemma 2.37]). Since x ∈ intB(x 0 , ρ), we may assume that x k ∈ B(x 0 , ρ) ∩ domV for all k = 1, 2, ... By hypothesis, for every k, there exists

x k V → x, ν k ∈ ∂ P V (x k ) and α k → 0 + be such that α k ν k → ξ 2 (see
x * k ∈ N C (x k ) ∩ B(θ, ||f (x k )||) such that ν k , f (x k ) -x * k + aV (x k ) + W (x k ) ≤ 0.
Since f is Lipschitz, the sequence (x * k ) is bounded and we may suppose (w.l.o.g.) that it weakly converges to some x * ∈ N C (x) ∩ B(θ, ||f (x)||), due to the r-uniform prox-regularity of C. Consequently, by multiplying the last inequality above by α k and next passing to the limit on k we obtain ξ 2 , f (x) -x * ≤ 0. Hence, the vector z * := (x * + k r x, 0 R 4 ) belongs to A(y, µ)∩B(θ H×R 4 , m) and satisfies ξ, f (y, µ)-z * = ι (e aγ ξ 2 , 0, 0, 0, 0), (f (x)-x * , 1, 0, 1, 0) = ιe aγ ξ 2 , f (x)-x * ≤ 0.

Consequently, according to Corollary 4.9, there is a T 0 > 0 such that the solution of (4.45) on [0, T 0 ] starting at (x(t), t, t, 0, V (x(t))), which is given by y(s) = (x(s+ t), s + t, t, s, V (x(t))), lies in epiV n ; that is, for every s ∈ [0, T 0 ],

V n (x(s + t), s + t, t, s) = e as V (x(t + s)) where in the last inequality we used the fact that 1 + λ ≤ e λ for all λ > 0. This contradicts the definition of T * , and so (4.52) holds true; that is (evaluating at t = 0), for all s ∈ [0, T ]

+ sg n (s + t) + l 2 s 2 ≤ V (x(
e as V (x(s)) + s 0 W (x(τ ))dτ - s n = e as V (x(s)) + s 0 g n (τ )dτ ≤ V (x 0 ) + 2 n e max{1,a}s ;
hence, as n goes to ∞, we get e as V (x(s))+ s 0 W (x(τ ))dτ ≤ V (x 0 ) for all s ∈ [0, T ], and the first part of the theorem is proved.

We turn now to the second part of the theorem. Implications (iv) ⇒ (v) and (ii) ⇒ (iii) follow from the relation (f (x) -N C (x)) • = f (x) -Π N C (x) (f (x)), x ∈ C, and the fact that Π N C (x) (f (x)) ≤ f (x) .

(i) ⇒ (iv). Assuming that (V, W ) is an a-Lyapunov pair, for any x 0 ∈ domV and t > 0 the solution x(•) = x(•; x 0 ) satisfies

0 ≥ t -1 (V (x(t)) -V (x 0 )) + t -1 (e at -1)V (x(t)) + t 0 t -1 W (x(τ ))dτ. (4.53) Thus, observing that x(t)-x 0 t → [f (x 0 ) -N C (x 0 )] • (recall Theorem 4.
15(a)), and using the lower semi-continuous of V and W, as t ↓ 0 in the last inequality we get

V (x 0 , (f (x 0 ) -N C (x 0 )) • ) = lim inf t↓0 w→(f (x 0 )-N C (x 0 )) • V (x 0 + tw) -V (x 0 ) t ≤ lim inf t↓0 t -1 (V (x(t)) -V (x 0 )) ≤ -aV (x 0 ) -W (x 0 ).
(iv) ⇒ (ii) and (v) ⇒ (iii), when ∂ = ∂ F or ∂ = ∂ P . These implications follow due to the relation ξ, v ≤ V (x, v) for all ξ ∈ ∂ F V (x), x ∈ domV, and v ∈ H.

Differential inclusions with prox-regular sets (iii) ⇒ (i) is an immediate consequence of the first part of the theorem together with (4.43).

Finally, to prove the last statement of the theorem when ∂ = ∂ L in the finitedimensional case, we first check that (i) =⇒ (iii). Assume that (i) holds and take x ∈ domV together with ξ ∈ ∂ L V (x), and let sequences

x k V → x together with ξ k ∈ ∂ P V (x k ) such that ξ k → ξ. Since (iii) holds for ∂ = ∂ P , for each k there exists x * k ∈ N C (x k ) ∩ B(θ, ||f (x k )||) such that ξ k , f (x k ) -x * k + aV (x k ) + W (x k ) ≤ 0.
We may assume that (x * k ) converges to some x * ∈ N C (x) ∩ B(θ, ||f (x)||) (thanks to the r-uniform prox-regularity of C), which then satisfies ξ, f (x) -x * + aV (x) + W (x) ≤ 0 (using the lower semi-continuous of the involved functions), showing that (iii) holds. Thus, since (iii) (with ∂ = ∂ P ) =⇒ (i), we deduce that (i) ⇐⇒ (iii). This suffices to get the conclusion of the theorem.

Because the solution x(•) of differential inclusion (4.1) naturally lives in C, it is immediate that a (lower semi-continuous) function V : H → R is Lyapunov for (4.1) iff the function V + I C is Lyapunov. Hence, Theorem 4.19 also provides the characterization of Lyapunov functions without any restriction on their domains; for instance, accordingly to Theorem 4.19(iii), V is Lyapunov for (4.1) iff for every x ∈ domV ∩ C and ξ ∈ ∂(V + I C )(x) it holds min

x * ∈N C (x)∩B(θ,||f (x)||) ξ, f (x) -x * + aV (x) + W (x) ≤ 0.
The point here is that this condition is not completely written by means exclusively of the subdifferential of V. Nevertheless, this condition becomes more explicit in each time one can decompose the subdifferential set ∂(V + I C )(x). For instance, this is the case, if V is locally Lipschitz and lower regular (particularly convex, see [START_REF] Mordukhovich | Variational Analysis and Generalized Differentiation I: Basic Theory, Grundlehren der Mathematischen Wissenschaften[END_REF]Definition 1.91]). This fact is considered in Corollary 4.21 below. However, the following example shows that we can not get rid of the condition domV ⊂ C, in general. Remark 4.20. We consider the differential inclusion (4.1) in R 2 , with C := B and f (x, y) = (-y, x), whose unique solution such that x(0) = (1, 0) is x(t) = (cos t, sin t). We take V = I S , where ξ, f (x) -x * ≤ ξ, f (x) = (x, y), (0, 1) = y ≤ 0, which shows that condition (iii) of Theorem 4.19 holds. However, it is clear that V is not a Lyapunov function of (4.1).

S := {(1, y) : y ∈ [0, 1]},
Corollary 4.21. Let V , W and a be as in Theorem 4.19. Then the following assertions hold:

(i) If V is Fréchet differentiable on domV ∩ C, then (V, W ) is an a-Lyapunov pair of differential inclusion (4.1) iff for every x ∈ domV ∩ C ∇V (x), (f (x) -N C (x)) • + aV (x) + W (x) ≤ 0. (ii) If V is locally Lipschitz on domV ∩ C, then (V, W ) is an a-Lyapunov pair for differential inclusion (4.1) if for every x ∈ domV ∩ C ξ, (f (x) -N C (x)) • + aV (x) + W (x) ≤ 0 ∀ξ ∈ ∂ L V (x).
(iii) If H is of finite dimension and V is regular and locally Lipschitz on domV ∩ C, then (V, W ) is an a-Lyapunov pair for differential inclusion (4.1) iff for every

x ∈ domV ∩ C, ξ, (f (x) -N C (x)) • + aV (x) + W (x) ≤ 0 ∀ξ ∈ ∂ L V (x).
Proof. (i). Since x(t) ∈ C for every t ≥ 0, we have that (V, W ) forms an a-Lyapunov pair for (4.1) iff the pair (V + I C , W ) does. Thus, since ∂ F (V + I C )(x) = ∇V (x) + N C (x) for every x ∈ domV ∩ C, according to Proposition 1.107 in [START_REF] Mordukhovich | Variational Analysis and Generalized Differentiation I: Basic Theory, Grundlehren der Mathematischen Wissenschaften[END_REF], Theorem 4.19 ensures that (V, W ) is an a-Lyapunov pair of (4.1) iff for every x ∈ domV ∩ C and ξ ∈ N C (x)

∇V (x) + ξ, (f (x) -N C (x)) • + aV (x) + W (x) ≤ 0. (4.54) Because 0 ∈ N C (x) and (f (x) -N C (x)) • ∈ T B C (x) = (N C (x)) * , it follow that this last inequality is equivalent to ∇V (x), (f (x) -N C (x)) • + aV (x) + W (x) ≤ 0.
(ii). Under the current assumption, for every x ∈ V ∩ C we have that ∂ L (V + I C )(x) ⊂ ∂ L V (x) + N C (x), and we argue as in the proof of statement (i).

(iii). In this case, we argue as above but using the relation

∂ L (V + I C )(x) = ∂ L V (x) + N C (x).
It the result below, Theorem 4.19 is rewritten in order to characterize invariant sets associated to differential inclusion (4.1) (see Definition 4.5). Criterion (iii) below is of the same nature as the one used in [START_REF] Colombo | The minimum time function for the controlled Moreau's sweeping process[END_REF].

Theorem 4.22. Given a closed set S ⊂ C we denote by N S either N P S or N F S , and by T S either T B S , T w S , coT w S , or (N S ) * . Then S is an invariant set for (4.1) iff one of the following equivalent statements hold:

(i) (f (x) -N C (x)) • ∈ T S (x) ∀x ∈ S; (ii) [f (x) -N C (x)] ∩ T S (x) ∩ B(θ, ||f (x)||) = ∅ ∀x ∈ S; (iii) inf x * ∈[f (x)-N C (x)]∩B(θ,||f (x)||) ξ, x * ≤ 0 ∀x ∈ S, ∀ξ ∈ N S (x).
Proof. Under the invariance of S we write (recall Theorem 4.15)

(f (x) -N C (x))) • = d + x(0; x) dt = lim t 0 x(t) -x t ∈ T B S (x),
showing that (i) with T S (x) = T B S (x) holds. The rest of the implications follows by applying Theorem 4.19 with the use of the following equalities

T B S (x) ⊂ T w S (x) ⊂ co(T w S (x)) ⊂ (N F S (x)) * ⊂ (N P S (x)) * , x ∈ S,
where the star in the superscript refers to the dual cone.

Stability and observer designs

In this section, we give an application of the results developed in the previous sections, to study the stability and observer design for Lur'e systems involving nonmonotone set-valued nonlinearities. The state of the system is constrained to evolve inside a time-independent prox-regular set. More precisely, let us consider the following problem

ẋ(t) = Ax(t) + Bu(t), a.e. t ∈ [0, ∞), (4.55a
) The following proposition shows that system (4.55), or equivalently (4.56), can be transformed into a differential inclusion of the form (4.1). Proposition 4.24. Let us consider system (4.55). Assume that S is contained in the range space of D and there exists a symmetric positive definite matrix P such that P B = D T . Then every solution of (4.55) is also a solution of the following system ż(t) ∈ f (z(t)) -N S (z(t)), a.e. t ≥ 0, z(0) ∈ S ,

y(t) = Dx(t), ∀t ≥ 0, (4.55b) u(t) ∈ -N S (y(t)) ∀ t ≥ 0, (4.55c) x(0) = x 0 ∈ D -1 (S); (4.55d) where x(t) ∈ R n , A ∈ R n×n , B ∈ R n×l , D ∈ R l×n , l ≤ n,
with z(t) = P 1 2 x(t), f = P 1 2 AP -1 2 and S = (DP -1 2 ) -1 (S).
Proof. We set R := P 1 2 . According to Lemma 4.23, the set S is r -uniformly proxregular with r :=

rδ + DR -1 ||DR -1 || 2 .
Combining this with the basic chain rule (see Theorem 10.6, [START_REF] Rockafellar | Variational Analysis, Grundlehren der Mathematischen Wissenschaften[END_REF]), for any x ∈ R n , one has

(DR -1 ) T N S (DR -1 x) = (DR -1 ) T ∂I S (DR -1 x) ⊂ ∂(I S • (DR -1 ))(x) = ∂I S (x) = N S (x).
By the hypothesis P B = C T , we deduce that DR -1 = (RB) T . From the above inclusion, it is easy to see that for a.e. t ≥ 0, one has

ż(t) ∈ RAR -1 z(t) -RBN S (DR -1 z(t)) = RAR -1 z(t) -(DR -1 ) T N S (DR -1 z(t)) ⊂ RAR -1 z(t) -N S (z(t)). (4.57)
The proof of Proposition 4.24 is thereby completed.

The above Proposition proves that under some assumptions, system (4.55) can be studied within the framework of (4.1). Let us now investigate the asymptotic stability of differential inclusion (4.1)

   ẋ(t) ∈ f (x(t)) -N C (x(t)) a.e. t ≥ 0, x(0; x 0 ) = x 0 ∈ C,
at the equilibrium point 0, with the assumption 0 ∈ C and f (0) = θ. Recall that the set C is an r-uniformaly prox-regular set (r > 0), and that f is a Lipschitz continuous mapping with Lipschitz constant L.

We have the following result which provides a partial extension of [78, Theorem 3.2] (here, we are considering the case where the set C is time-independent).

Theorem 4.25. Assume that 0 ∈ C, f (0) = θ. If there exist ε, δ > 0 such that x, f (x) + δ||x|| 2 ≤ 0 ∀x ∈ C ∩ B(θ, ε). (4.58)
Then lim t→∞ x(t, x 0 ) = 0 for all x 0 ∈ int(B(θ, min{rδl -1 , ε})) ∩ C.

Proof. We shall verify that the (lower semi-continuous proper) function V : H → R ∪ {+∞}, defined by V (x) := 1 2 ||x|| 2 +I C (x), satisfies the assumption of Theorem 4.19 (when W ≡ 0 and a = δ). We fix η ∈ (0, min{rδL -1 , ε}), x ∈ B(θ, η) ∩ C and ξ ∈ ∂

P V (x) ⊂ x + N C (x) ([32, Ch. 1, Proposition 2.11]); hence, since (f (x) - N C (x)) • = Π T C (x) (f (x)) ∈ T C (x) we obtain ξ, (f (x) -N C (x)) • ≤ x, (f (x) -N C (x)) • = x, f (x) -Π N C (x) (f (x)) , so that, by (4.58), ξ, (f (x) -N C (x)) • ≤ -x, Π N C (x) (f (x)) -2δV (x). (4.59)
Moreover, because Π N C (x) (f (x)) ∈ N C (x) and θ ∈ C, from the r-uniformaly proxregularity of the set C we have

Π N C (x) (f (x)), -x ≤ Π N C (x) (f (x)) r V (x) ≤ f (x) r V (x),
and we get, using (4.59),

ξ, f (x) -Π N C (x) (f (x)) + δV (x) = ξ, (f (x) -N C (x)) • + δV (x) ≤ (r -1 f (x) -δ)V (x).
But, by the choice of η we have

f (x) = f (x) -f (0) ≤ l x ≤ Lη ≤ rδ, and so, ξ, f (x) -Π N C (x) (f (x)) + δV (x) ≤ 0. Consequently, observing that Π N C (x) (f (x)) ∈ N C (x) ∩ B(θ, f (x) )
, by Theorem 4.19 we deduce that for every x 0 ∈ C ∩ int(B(θ, η)), there exists t 0 > 0 such that

e δt V (x(t; x 0 )) ≤ V (x 0 ) ∀t ∈ [0, t 0 ]; hence, in particular, 1 2 ||x(t; x 0 )|| 2 ≤ 1 2 ||x 0 || 2 and x(t 0 ; x 0 ) ∈ C ∩ int(B(θ, η)). This proves that t0 := sup {t > 0 | e δt V (x(s; x 0 )) ≤ V (x 0 ) ∀s ∈ [0, t]} = +∞,
and we conclude that e δt V (x(t; x 0 )) ≤ V (x 0 ) ∀t ≥ 0, which leads us to the desired conclusion.

Corollary 4.26. Let us consider system (4.55). Assume that S is uniformly proxregular set such that S is contained in the rank of D. If there exists a symmetric positive definite matrix P and δ > 0 such that

A T P + P A ≤ -δP, P B = D T , (4.60) then lim t→∞ x(t; x 0 ) = 0 for all x 0 ∈ int(B(θ, ρ)) ∩ S,
where ρ := (2||R -1 || ||DR -1 || ||RAR -1 ||) -1 δrδ + DR -1 .
Proof. Firstly we will show that for any x ∈ R n , one has

RAR -1 x, x + δ 2 ||x|| 2 ≤ 0.
Differential inclusions with prox-regular sets Indeed, by the first inequality of (4.60), for every x ∈ R n , one has

(A T P + P A + δP )x, x = (A T R 2 + R 2 A + δR 2 )x, x ≤ 0.
Since R is positive definite, for any z = R -1 x, one has 0 ≥ (A

T P + P A + δP )R -1 x, R -1 x = (A T R + P AR -1 + δR)x, R -1 x = (R -T A T R + RAR -1 + δI n )x, x = 2 RAR -1 x, x + δ||x|| 2 .
for every z 0 ∈ int[B(θ, 1 2 ||R -1 AR|| -1 r δ)] ∩ S .
Combining this with the fact that x(t) = R -1 z(t), the conclusion of Corollary 4.26 follows because

z = Rx ∈ int B(0, 1 2 ||R -1 AR|| -1 r δ) , for any x ∈ int[B(0, ρ)].
Next let us remind the Luenberger-like observer associated to differential inclusion (4.55). Given x 0 ∈ D -1 (S), we assume that the output equation associated with differential inclusion (4.55) is

y(t) = G(x(t; x 0 ))
where G ∈ R p×n with p ≤ n. The Luenberger-like observer associated to differential inclusion (4.55) has the following form

ẋ(t) = (A -LG)x(t) + Ly(t) + B û(t), (4.62a) ŷ(t) = Dx(t), (4.62b) û(t) ∈ -N S (ŷ(t)), (4.62c) x(0) = z 0 , (4.62d)
where L ∈ R n×p is the observer gain. This differential inclusion always has a unique solution, denoted by x(•; z 0 ). We want to find the gain L for the basic similarly to the proof of Proposition 4.24, we have

ż(t) ∈ (RAR -1 -RLG )ẑ(t) + RLG z(t) -I S (ẑ(t)),
where G -1 , ẑ(t) := Rx(t; z 0 ) and z(t) = Rx(t; x 0 ), S -1 ) -1 (S). On the other hand, one has Next, we investigate a general Luenberger-like observer associated to our differential inclusion (4.1). Following the same idea as above, we assume that x 0 ∈ C and the output equation associated with differential inclusion (4.1) is

||R|| -1 ||ẑ(t) -z(t)||≤ ||x(t) -x(t)||≤ ||R -1 || ||ẑ(t) -z(t)
y(t) = G(x(t; x 0 )),
where G : H → H is a Lipschitz mapping. We want to find a Lipschitz mapping L : H → H such that the solution x(•; z 0 ) of the differential inclusion

   ẋ(t) ∈ f (x(t)) -L(G(x(t)) + L(y(t)) -N C (x(t)) a.e. t ≥ 0 x(0) = z 0 ∈ C, (4.66)
satisfies, for some ρ > 0, lim t→∞ ||x(t; z 0 ) -x(t; x 0 )||= 0, for all z 0 ∈ B(x 0 , ρ) ∩ C.

Differential inclusions with prox-regular sets

To solve this problem we consider the Lipschitz mapping f :

H × H → H × H, defined as f (z, x) := f (z) -L(G(z)) + L(G(x)), f (x) , (4.67)
together with the set S := C × C; hence, N P S (x, y) = N C (x) × N C (y), for every (x, y) ∈ S, so that S is also an r-uniformly prox-regular set. Consequently, we easily check that y(t) := (x(t; z 0 ), x(t; x 0 )) is the unique solution of the differential inclusion ẏ(t) ∈ f (y(t)) -N S (y(t)) a.e. t ≥ 0, y(0) = (z 0 , x 0 ) ∈ S.

We have the following result, which extends [START_REF] Tanwani | Stability and observer design for Lur'e systems with multivalued, non-monotone, time-varying nonlinearities and state jumps[END_REF]Proposition 3.5] in the case where the set C does not depend on the time variable. Theorem 4.27. Fix (z 0 , x 0 ) ∈ C×C and assume that the solution of (4.1), x(t; x 0 ), is bounded, say ||x(t; x 0 )||≤ m for all t ≥ 0. If M := sup{||f (x)||, x ∈ B(θ, m)}, we choose a Lipschitz continuous mapping L together with positive numbers δ, ε, η > 0 such that ε < δr -M, η ≤ (6l) -1 ε, and

||x -y||≤ 3η, x, y ∈ H =⇒ L(C(x)) -L(C(y)) ≤ ε, (4.68)
at the same time as, for all x, y ∈ B(θ, m + 3η),

x -y, (f -L • G)(x) -(f -L • C)(y) ≤ -δ||x -y|| 2 . ( 4

.69)

Then for every z 0 ∈ B(x 0 , η) we have that

||x(t; z 0 ) -x(t; x 0 )||≤ e -(δ-M +ε r ) 2 t ||z 0 -x 0 ||,
and, consequently,

||x(t; z 0 ) -x(t; x 0 )||→ 0 as t → +∞. Proof. For every z, y ∈ B(θ, m + 3η) ∩ C such that ||z -y||≤ 3η we have that max{||f (z)||, ||f (y)||} ≤ M + 3ηl ≤ M + ε 2 , ||L(G(z)) -L(G(y))||≤ ε.
We consider the (C 1 -) function V : H × H → R defined as V (z, y) := 1 2 ||z -y|| 2 .

Stability and observer designs

If β := δ -M +ε r , then by definition (4.67), we obtain

V (z, y), ( f (z, y) -N S (z, y)) • + βV (z, y) = z -y, f (z) -L(G(z)) + L(G(y)) -Π N C (z) (f (z) -L(G(z)) + L(G(y))) + y -z, f (y) -Π N C (y) (f (y)) + β 2 ||z -y|| 2 = z -y, f (z) -f (y) -L(G(z)) + L(G(y)) + z -y, Π N C (y) (f (y)) -z -y, Π N C (z) (f (z) -L(G(z)) + L(G(y))) + β 2 ||z -y|| 2 .
Since Π N C (y) (f (y)) ∈ N C (y) and Π N C (y) (f (y)) ≤ f (y) , and similarly for

Π N C (z) (f (z) -L(G(z)) + L(G(y))
), the last equality yields

V (z, y), ( f (z, y) -N S (z, y)) • + βV (z, y) ≤ z -y, f (z) -f (y) -L(G(z)) + L(G(y)) + ||f (y)|| 2r ||z -y|| 2 + ||f (z) -L(G(z)) + L(G(y))|| 2r ||z -y|| 2 + β 2 ||z -y|| 2 ,
which by assumptions (4.68) and (4.69) gives us

V (z, y), ( f (z, y) -N S (z, y)) • + βV (z, y) ≤ z -y, f (z) -f (y) -L(G(z)) + L(G(y)) + ||f (z)||+||f (y)|| 2r ||z -y|| 2 + ||L(G(z)) -L(G(y)|| 2r ||z -y|| 2 + β 2 ||z -y|| 2 ≤ z -y, f (z) -f (y) -L(G(z)) + L(G(y)) + M + ε r ||z -y|| 2 + β 2 ||z -y|| 2 ≤ -δ||z -y|| 2 +( M + ε r + β 2 )||z -y|| 2 ≤ 0. (4.70) Now we choose z 0 ∈ B(x 0 , η) ∩ C, so that B(z 0 , η)×B(x 0 , η) ⊂ [B(θ, m+3η)×B(θ, m+3η)]∩{(z, y) ∈ H ×H : ||z -y||≤ 3η}.
Then, thanks to (4.70), we can apply Corollary 4.21(i) to find some t 0 > 0 such that for every t ∈ [0, t 0 ] e βt V (x(t; z 0 ), x(t; x 0 )) ≤ V (z 0 , x 0 ); Differential inclusions with prox-regular sets that is,

||x(t; z 0 ) -x(t; x 0 )||≤ e -βt 2 ||z 0 -x 0 ||.
Moreover, since ||x(t 0 ; z 0 ) -x(t 0 ; x 0 )||≤ η and x(t 0 ; z 0 ) ∈ B(θ, m + 2η) ∩ C, we can also find t 1 > 0 such that for any t ∈ [0, t 1 ]

||x(t + t 0 ; z 0 ) -x(t + t 0 ; x 0 )|| ≤ e -βt 2 ||x(t 0 ; z 0 ) -x(t 0 ; x 0 )|| ≤ e -βt 2 e -βt 0 2 ||z 0 -x 0 ||= e -β(t+t 0 ) 2 ||z 0 -x 0 ||.
Consequently, we deduce that for every t ≥ 0

||x(t; z 0 ) -x(t; x 0 )||≤ e -βt 2 ||z 0 -x 0 ||,
which completes the proof.

To close this section we consider the special case of linear Luenberger-like , where the assumption of Theorem 4.27 takes a simpler form. In this case (4.66) is written as

   ẋ(t) ∈ (A -LG)x(t) + LGx(t) -N C (x(t)) a.e. t ≥ 0 x(0) = z 0 ∈ C,
where A, L, G : H → H are linear continuous mappings; A * and G * will denote the corresponding adjoints mappings. Assume that x(•) := x(•; x 0 ), x 0 ∈ C, is the solution of (4.1) (corresponding to f = A).

Corollary 4.28. Fix (z 0 , x 0 ) ∈ C × C and assume that the solution of (4.1)

(corresponding to f = A), x(t; x 0 ), is bounded, say ||x(t; x 0 )||≤ m for all t ≥ 0. Let δ, ε, ρ > 0 be such that r -1 (m||f ||+ε) < δ, and 1
2 (A + A * ) -ρG * G ≤ -δid.
If L := ρG * , η := min{(6||A||) -1 ε, (3||LG||) -1 ε},and β := δ -r -1 (m||A||+ε), then for every z 0 ∈ B(x 0 , η) we have that, for all t ≥ 0,

||x(t; z 0 ) -x(t; x 0 )||≤ e -βt 2 ||z 0 -x 0 ||.
Proof. The proof is similar as the one of Theorem 4.27, by observing that for every 4.7. Concluding remarks

x ∈ H, we have

x, (A -LG)x = x, (A -LG)x + x, (A * -G * L * )x 2 .

Concluding remarks

In this paper, we proved that a differential variational inequality involving a proxregular set can be equivalently written as a differential inclusion governed by a maximal monotone operator. Therefore, the existence result and the stability analysis can be conducted in a classical way. We also give a characterization of lower semi-continuous a-Lyapunov pairs and functions. An application to a Luenberger-like observer is proposed. These new results will open new perspectives from both the numerical and applications points of view. An other interesting problem dealing with sweeping processes was introduced by J.J. Moreau in the seventies, which is of a great interest in applications. This problem is obtained by replacing the fixed set C by a moving set C(t), t ∈ [0, T ]. It will be interesting to extend the ideas developed in this current work to the sweeping process involving prox-regular sets. Many other issues require further investigation including the study of numerical methods for problem (4.1) and the extension to second-order dynamical systems. This is out of the scope of the present paper and will be the subject of a future project of research.

Chapter 5

Lyapunov stability of differential inclusions with Lipschitz Cusco perturbations of maximal monotone operators

We characterize weak and strong invariant closed sets with respect to differential inclusions given in R n and governed by Lipschitz Cusco perturbations of maximal monotone operators. Correspondingly, we provide different characterizations for Lyapunov functions and pairs for such differential inclusions. Our criteria of invariance and Lyapunov functions/pairs only depend on the data of the system and the geometry of the involved candidates for invariant sets and Lyapunov functions, and thus, no need to explicit calculus of the solutions, nor to calculus on the semi-group generated by the underlying maximal monotone operator.

Introduction

Our main purpose in this paper is to give explicit characterizations for weak and strong invariant closed sets with respect to the following differential inclusion, given ∈ R n as

ẋ(t) ∈ F (x(t)) -A(x(t)) a.e. t ≥ 0, x(0) = x 0 ∈ domA, (5.1)
where F : R n ⇒ R n is a Lipschitz Cusco multifunction; that is, a Lipschitz setvalued mapping with nonempty, convex and compact values, and A : R n ⇒ R n is a maximal monotone operator. There is no restriction on the initial condition x 0 that can be any point in the closure of the domain of A, possibly not a point of definition of A. We also characterize weak and strong Lyapunov functions and, more generally, a-Lyapunov pairs associated to the differential inclusion above. Our criteria are given by means only of the data of the system; that is, the multifunction F and the operator A, together with first-order approximations of the invariant sets candidates, using Bouligand tangent cones, or, equivalently, Fréchet or proximal normal cones, and first-order (general) derivatives of Lyapunov functions candidates, using directional derivatives, Fréchet or proximal subdifferentials.

Our analysis aims at gathering two different kinds of dynamic systems in one, that were studied separately in the literature, at least in what concerns Lyapunov stability. The first kind of these dynamic systems is governed exclusively by Cusco multifunctions, and gives rise to a natural extension of the classical differential equations, given in the form

ẋ(t) ∈ F (x(t)) a.e. t ≥ 0, x(0) = x 0 ∈ R n .
(5.

2)

The consideration of differential inclusions rather than differential equations allows more useful existence theorems, as revealed by Filippov's theory for differential equation with discontinuous right-hand-sides [START_REF] Filippov | Differential equations with multi-valued discontinuous righthand side,(Russian)[END_REF]. Stability of such systems, namely, the study of Lyapunov functions and invariant sets, has been extensively studied and investigated especially during the nineties by many authors; see, for example, [START_REF] Clarke | Nonsmooth Analysis and Control Theory[END_REF][START_REF] Clarke | Approximate invariance and differential inclusions in Hilbert spaces[END_REF][START_REF] Donchev | Strong invariant and one-sided Lipschitz multifunctions[END_REF], as well as [START_REF] Artstein | Extensions of Lipschitz selections and an application to differential inclusion[END_REF][START_REF] Aubin | Set-Valued Analysis[END_REF][START_REF] Frankowska | A measurable upper semicontinuous viability theorem for tubes[END_REF] (see, also, the references therein). For instance, complete weak and strong invariance characterizations for closed sets can be found in [START_REF] Clarke | Nonsmooth Analysis and Control Theory[END_REF] in the finite-dimensional setting, and in [START_REF] Clarke | Approximate invariance and differential inclusions in Hilbert spaces[END_REF] for Hilbert spaces.

It is worth recalling that only the upper semi-continuity of the Cusco mapping F is required for the weak invariance, while Lipschitzianity is used for the strong invariance (see [START_REF] Clarke | Approximate invariance and differential inclusions in Hilbert spaces[END_REF]). Invariance characterizations of a same nature have been done in [START_REF] Donchev | Strong invariant and one-sided Lipschitz multifunctions[END_REF] for one-side Lipschitz (not necessary Lipschitz) and compact valued multifunctions. These results have been adapted in [START_REF] Colombo | The minimum time function for the controlled Moreau's sweeping process[END_REF] to the following more general differential inclusion (for T ∈ [0, +∞])

ẋ(t) ∈ F (t, x(t)) -N C(t) (x(t)) a.e. t ∈ [0, T ], x(0) = x 0 ∈ C(0), (5.3)
where C(t) is a uniformly prox-regular sets in R n and N C(t) is the associated normal cone. Observe here that the right-hand-side may be unbounded, but however, in the case when T < +∞, the last differential inclusion above is equivalent to the Differential inclusions with Lipschitz Cusco perturbations following one, for some positive constant M > 0,

ẋ(t) ∈ F (t, x(t)) -N C (t)(x(t)) ∩ B(θ, M ) a.e. t ∈ [0, T ], x(0) = x 0 ∈ C(0),
giving rise to a differential inclusion in the form of (5.2).

The other kind of systems that is covered by (5.1) concerns differential inclusions governed by maximal monotone operators, or, more generally, (singlevalued) Lipschitz perturbations of these operators, that we write as

ẋ(t) ∈ f (x(t)) -A(x(t)) a.e. t ≥ 0, x(0) = x 0 ∈ domA.
(5.4)

This system can be seen as perturbations of the ordinary differential equation ẋ(t) = f (x(t)), where A could represent some associated control action. In this single-valued Lipschitzian setting, weak and strong invariance coincide since differential inclusion (5.1) possesses unique solutions. Compared to (5.2) the right-hand-side in this differential inclusion can be unbounded, or even empty. Typical examples of (5.4) involve the Fenchel subdifferential of proper, lower semicontinuous convex functions ( [START_REF] Adly | A stability theory for second order non-smooth dynamical systems with application to friction problems[END_REF]). System (5.4) has been extensively studied, namely, regarding existence, regularity and properties of the solutions [START_REF] Brézis | Operateurs Maximaux Monotones et Semi-Groupes de Contractions Dans Les Espaces de Hilbert[END_REF], while Lyapunov stability of such systems have been initiated in [START_REF] Pazy | The Lyapunov Method for Semigroups of Nonlinear Contractions in Banach Space[END_REF]; see, also, [START_REF] Adly | Invariant sets and Lyapunov pairs for differential inclusions with maximal monotone operator[END_REF][START_REF] Adly | Nonsmooth Lyapunov pairs for infinite-dimensional first order differential inclusions[END_REF][START_REF] Adly | Nonsmooth Lyapunov pairs for differential inclusions governed by operators with nonempty interior domain[END_REF] for recent contributions on the subject. Different criteria using the semi-group generated by the operator A can also be found in [START_REF] Kocan | Lyapunov Functions for Infinite-Dimensional Systems[END_REF], where Lyapunov functions are characterized as viscosity-type solutions of Hamilton-Jacobi equations, and in [START_REF] Carja | Characterization of Lyapunov pairs in the nonlinear case and applications[END_REF], using implicit tangent cones associated to the invariant sets candidates.

It is worth observing that (5.1) is a special case of the following more general differential inclusion

ẋ(t) ∈ F (t, x(t)) -A(t)(x(t)) a.e. t ≥ 0, x(0) = x 0 ∈ domA(0, •), (5.5)
where A and F are also allowed to move in an appropriate way with respect to the time, satisfying some natural continuity and measurability conditions. Existence of solution of (5.5) have been also studied in [START_REF] Aizicovici | Multivalued evolution equations with nonlocal initial conditions in Banach spaces[END_REF][START_REF] Laouir | Existence of Solutions for a First Order Maximal Monotone Differential Inclusion with a Lipschitz Perturbation[END_REF][START_REF] Saïdi | Control problems governed by time-dependent maximal monotone operators[END_REF] among others. In particular, [START_REF] Aizicovici | Multivalued evolution equations with nonlocal initial conditions in Banach spaces[END_REF] considered in a Hilbert setting similar systems as the one in (5.1), but with requiring strong assumptions on the multifunction F. In [START_REF] Saïdi | Control problems governed by time-dependent maximal monotone operators[END_REF] the authors assume that F is a single-valued mapping, that is Lipschitz with respect to the second variable, while the minimal section mapping of the maximal monotone operators A(t) is uniformly bounded.

In this paper, we study and characterize strong and weak invariant closed subsets of the closure of the domain of A, domA, with respect to differential inclusion (5.1). We shall assume in our analysis that the invariant sets candidates S ⊂ R n satisfy the following condition Π S (x) ⊂ S ∩ domA ∀x ∈ domA, (5.6) where Π S refers to the projection operator on S. This condition has been used in many works; see, for instance, [START_REF] Barbu | Flow-invariant closed sets with respect to nonlinear semigroup flows[END_REF], where the author is concerned with flow invariance characterizations for differential equations, with right-hand-sides given by nonlinear semigroup generators in the sense of Crandall-Liggett (see [START_REF] Crandall | Generation of Semi-Groups of Nonlinear Transformations on General Banach Spaces[END_REF]).

It is clear that condition (5.6) holds whenever S ⊂ domA. When dealing with weak invariant closed sets, we shall require some usual boundedness conditions on the invariant set, relying on the minimal norm section of the maximal monotone operator A. The invariance criteria are then used to characterize weak and strong a-Lypaunov pairs of extended-real-valued proper lower semi-continuous functions (V, W ) associated to (5.1), such that domV ⊂domA. These results are specified to differential inclusions involving normal cones to a uniformly prox-regular set C, given in the form

ẋ(t) ∈ F (x(t)) -N C (x(t)) a.e. t ∈ [0, T ], x(0) = x 0 ∈ C,
where we provide existence and properties of the solutions as well as different characterizations for invariant closed subsets and Lyapunov functions/pairs, all written by means of the multifunction F and the set C.

The paper is organized as follows: After Section 2, reserved to give the necessary notations and present the main tools, we make in Section 3 a review of the existence theorems of differential inclusion (5.1), and establish some first properties of the solutions. In Section 4 we characterize weak and strong invariant closed sets with respect to (5.1), while in Section 5 strong and weak Lyapunov pairs are provided. In Section 6 we apply the previous results to study differential inclusions involving normal cones to uniformly prox-regular sets.

Notation and main tools

In this paper, R n is a (real) finite-dimensional Hilbert space with the null vector is denoted by θ, the notations •, • and • are the inner product and the norm, respectively. For each x ∈ R n and ρ ≥ 0, B(x, ρ) is the closed with center x and Differential inclusions with Lipschitz Cusco perturbations radius ρ and B r := B(θ, r). Given a nonempty set S, the notation S is closure of S. We denote S is real positive number define by

S := sup{ v : v ∈ S}.
The distance function to S is defined by

d S (x) := inf{ x -s , s ∈ S},
and orthogonal projection mapping onto S defined as

Π S (x) := {s ∈ S : x -s = d S (x)}.
If S is a closed set then Π S (x) = ∅ for every x ∈ R n , we denote S • := Π S (θ) is minimal norm in S. The indicator function of S is defined as

I S (x) :=    0 if x ∈ S +∞ if x / ∈ S,
and the support function of S is defined as σ S (x) := sup{ x, s : s ∈ S}, with the convention that σ ∅ = -∞. Given a function ϕ : R n → R ∪ {+∞}, its domain and epigraph are defined by

domϕ := {x ∈ R n : ϕ(x) < +∞}; epiϕ := {(x, α) ∈ R n+1 : ϕ(x) ≤ α}.
We say ϕ is proper if domϕ = ∅; lower semi-continuous , if epiϕ is closed. Notation F(R n ) is the set all proper, lower semi-continuous functions. We now introduce some basic concepts of nonsmooth and variational analysis. Let

ϕ ∈ F(R n ) and x ∈ domϕ. We call ξ ∈ R n is a proximal subgradient of ϕ at x, written ξ ∈ ∂ P ϕ(x) if lim inf y→x,y =x ϕ(y) -ϕ(x) -ξ, y -x y -x 2 > -∞. A vector ξ ∈ R n is said to be a Fréhet subgradient of ϕ at x, written ξ ∈ ∂ P ϕ(x) if lim inf y→x,y =x ϕ(y) -ϕ(x) -ξ, y -x y -x ≥ 0.
The limiting subdifferential of ϕ at x is defined as

∂ L ϕ(x) := { lim n→∞ ξ n | ξ n ∈ ∂ P ϕ(x n ), x n → x, f (x n ) → f (x)};
and the singular subdifferential of ϕ at x is defined as

∂ ∞ ϕ(x) := { lim n→∞ α n ξ n | ξ n ∈ ∂ P ϕ(x n ), x n → x, f (x n ) → f (x), α n ↓ 0};
The Clarke subdifferential is defined as

∂ C ϕ(x) := co(∂ L ϕ(x) + ∂ ∞ ϕ(x)).
In the case x / ∈ domϕ, then by convention, we set

∂ P ϕ(x) = ∂ F ϕ(x) = ∂ L ϕ(x) = ∅. We have the classical inclusions ∂ P ϕ(x) ⊂ ∂ F ϕ(x) ⊂ ∂ L ϕ(x). If ϕ is locally Lipschitz around x, then ∂ ∞ ϕ(x) = {θ} and ∂ C ϕ(x) = co(∂ L ϕ(x)).
The generalized directional derivative of ϕ at x in the direction v which defined by ϕ 0 (x; v) := lim sup y→x,t↓0 ϕ(y + tv) -ϕ(y) t .

We have that

ϕ 0 (x; v) = sup ξ∈∂ C ϕ(x) ξ, v ∀v ∈ R n .
We also remind the contingent directional derivative of

ϕ at x ∈ domϕ in the direction v ∈ R n is ϕ (x; v) := lim inf t→0 + ,w→v ϕ(x + tw) -ϕ(x) t .
From definitions of proximal subgradient, Fréhet subgradient, it is easy to prove that σ

∂ P V (x) (•) ≤ σ ∂ F V (x) (•) ≤ V (x; •) ∀x ∈ domV. ( 5.7)

Differential inclusions with Lipschitz Cusco perturbations

The proximal, Fréhet, limiting normal cone are defined, respectively, by

N P S (x) := ∂ P δ S (x), N F S (x) := ∂ F δ S (x), N L S (x) := ∂ L δ S (x).
We also define singular prox-subdifferential ∂ P,∞ ϕ(x) of ϕ at x as follows (ξ, 0) ∈ N P epiϕ (x, ϕ(x)).

The Bouligand tangent cones to S at x is defined as

T B S (x) := v ∈ H | ∃ x k ∈ S, ∃ t k → 0, st t -1 k (x k -x) → v as k → +∞ .
Next we remind some basic concepts and properties of a maximal monotone operator. A multivalued operator A : R n ⇒ R n , the domain and the graph of A are given, respectively, by

domA := {x ∈ R n | A(x) = ∅}, Gr(A) := {(x, y) | y ∈ A(x)};
to simplify, we may identify A to Gr(A). The operator A is said to be monotone if

y 1 -y 2 , x 1 -x 2 ≥ 0 for all (x i , y i ) ∈ Gr(A), i = 1, 2.
If in addition A is not properly included in any other monotone operator then A is said that maximal monotone. In this case, for any x ∈ domA, then A(x) is closed, convex, hence (A(x)) • is singleton. By maximal property, if a sequence

(x n , y n ) n ⊂ A such that (x n , y n ) → (x, y) as n → ∞ then (x, y) ∈ A.
Concerning to evolution equations associated with maximal monotone operator. Let T > 0 and let f :

[0, T ] → R n be a function such that f ∈ L 1 (0, T ; R n ). The differential inclusion ẋ(t) ∈ f (t) -A(x(t)) a.e. t ∈ [0, T ], x(0) = x 0 ∈ domA always has a unique solution x(•) (see [21]). Moreover, for almost t ∈ [0, T ], one has d + x(t) dt := lim t ↓t x(t ) -x(t) t -t = f (t + ) -Π A(x(t)) (f (t + 0)),
where

f (t + ) := lim h→0,h =0 1 h t+h t f (τ )dτ.
Finally, we remind Gronwall's Lemma Lemma 5.1. (Gronwall's Lemma [START_REF] Adly | A convex approach to differential inclusions with prox-regular sets: Stability analysis and observer design[END_REF]) Let T > 0 and a, b

∈ L 1 (t 0 , t 0 + T ; R) such that b(t) ≥ 0 a.e. t ∈ [t 0 , t 0 + T ]. If, for some 0 ≤ α < 1, an absolutely continuous function w : [t 0 , t 0 + T ] → R + satisfies (1 -α)w (t) ≤ a(t)w(t) + b(t)w α (t) a.e. t ∈ [t 0 , t 0 + T ],
then

w 1-α (t) ≤ w 1-α (t 0 )e t t 0 a(τ )dτ + t t 0 e t s a(τ )dτ b(s)ds, ∀t ∈ [t 0 , t 0 + T ].

Solutions of the system

In this section, we investigate and review some properties of the solution of differential inclusion (5.1), that is given by

ẋ(t) ∈ F (x(t)) -A(x(t)), a.e. t ≥ 0, x(0) = x 0 ∈ domA,
where A : H ⇒ H is a maximal monotone operator and F is an L-Lipschitz Cusco mapping.

Definition 5.2. A continuous function x : [0, ∞) → R n is said to be a solution of (5.1) if it is absolutely continuous on every compact subset of (0, +∞) and satisfies ẋ(t) ∈ F (x(t)) -A(x(t)) a.e. t ≥ 0, x(0) = x 0 ∈ domA.

The following characterization will be useful in the sequel.

Proposition 5.3. A continuous function x : [0, ∞) → R n is a solution of (5.1)
iff x(•) is absolutely continuous on every compact subset of (0, +∞), and for every

T > 0 there exists a function f ∈ L ∞ (0, T ; R n ) with f (t) ∈ F (x(t)) a.e. t ∈ [0, T ], such that ẋ(t) ∈ f (t) -A(x(t)) a.e. t ∈ [0, T ], x(0) = x 0 ∈ domA. (5.8)
Proof. The sufficient condition is clear and, so, we only need to justify the necessary part. Suppose that x(•) is any solution of (5.1) and fix T > 0. Since F is Lipschitz and x(•) is continuous, there exists m > 0 such that F (x(t)) ⊂ B(θ, m) for all t ∈ [0, T ]. We define the set-valued mapping G : [0, T ] ⇒ R n as

G(t) := [ ẋ(t) + A(x(t))] ∩ F (x(t)) = [ ẋ(t) + A(x(t))] ∩ B(θ, m) ∩ F (x(t)).

Differential inclusions with Lipschitz Cusco perturbations

We are going to check that G is measurable. Since operator A is maximal monotone, the mappings

x → A n (x) := A(x) ∩ B(θ, n), n ≥ 1,
are upper semi-continuous, and so are the mappings

t → A n (x(t)) := A(x(t)) ∩ B(θ, n), n ≥ 1,
due to the continuity of the solution x(•). Then, due to the relation

A(x(t)) = ∪ n∈N A n (x(t)), we deduce that the multifunction t -→ A(x(t)) is measurable. Since ẋ(t) = lim n→+∞ n(x(t + 1 n ) -x(t)) for ae t ∈ [0, T ], ẋ(•)
is measurable, and we deduce that the multifunction t -→ [ ẋ(t) + A(x(t))] ∩ B(θ, m) is measurable. Similarly, the multifunction t -→ F (x(t)) is measurable too. Consequently, according to [START_REF] Castaing | Convex analysis and measurable multifunctions[END_REF]Proposition III.4], the mapping G is measurable, and we conclude from [START_REF] Castaing | Convex analysis and measurable multifunctions[END_REF]Theorem III.6] that G admits a measurable selection; i.e., a measurable function

f : [0, T ] → R n such that f (t) ∈ G(t) = [ ẋ(t) + A(x(t))] ∩ B(θ, m) ∩ F (x(t)) ⊂ F (x(t)) a.e. t ∈ [0, T ]. Hence, ẋ(t) ∈ f (t) -A(x(t)) and f (t) ≤ F (x(t)) ≤ m, so that f ∈ L ∞ (0, T ; R n ).
The next theorem shows that differential inclusion (5.1) has at least one solution whenever x 0 ∈ domA. We use the following lemma, which is a particular case of [10, Theorem A]. Proof. Fix x 0 ∈ domA and, according to Lemma 5.4, let f be a Lipschitz selection of F. Then the differential inclusion ẋ(t) ∈ f (x(t)) -A(x(t)) a.e. t ≥ 0, x(0) = x 0 , admits a unique solution x(•), which is absolutely continuous on every compact subset of (0, +∞) (see e.g. [START_REF] Barbu | Nonlinear differential equations of monotone types in Banach spaces[END_REF][START_REF] Brézis | Operateurs Maximaux Monotones et Semi-Groupes de Contractions Dans Les Espaces de Hilbert[END_REF]). It follows that x(•) is also a solution of 5.3. Solutions of the system differential inclusion (5.1).

We also need to give some further properties of the solutions of differential inclusion (5.1), which will be used in the sequel.

Given a set S ⊂ H and x ∈ domA we denote

(S -A(x)) • := s∈S (s -A(x)) • = {s -Π Ax (s) | s ∈ S} .
Proposition 5.6. Fix x 0 ∈ domA and let x(•) := x(•; x 0 ) be any solution of (5.1).

Then the following assertions hold :

(i) x(t) ∈ domA, for every t > 0, and for a.e. t ≥ 0,

d + x(t) dt := lim h↓0 x(t + h) -x(t) h ∈ (F (x(t)) -A(x(t))) • .
Conversely, if x 0 ∈ domA, then for any v ∈ (F (x 0 ) -A(x 0 )) • , there exists a solution y(•) of (5.1) such that

y(0) = x 0 , d + y(0) dt = v.
(ii) There exists a real number c > 0 such that for any x 0 ∈ domA and any solutions x(•) := x(•; x 0 ) and y(•) := y(•; x 0 ) of (5.1), one has for all t ≥ 0

x(t) -x 0 ≤ 3( F (x 0 ) + A • (x 0 ) )te ct , x(t) -y(t) ≤ 4( F (x 0 ) + A • (x 0 ) )te ct ,
Consequently, for every T > 0 there exists ρ > 0 such that

x(t) ∈ B(x 0 , ρ) ∀t ∈ [0, T ].
Proof. (i) According to Proposition 5.3, for each T > 0 there exists some

f ∈ L ∞ (0, T ; R n ) with f (t) ∈ F (x(t)) a.e. t ∈ [0, T ], such that x(•)
is the unique solution of (5.8); hence, by [START_REF] Brézis | Operateurs Maximaux Monotones et Semi-Groupes de Contractions Dans Les Espaces de Hilbert[END_REF] x(•) satisfies x(t) ∈ domA for all t ∈ (0, T ) and

d + x(t) dt = f (t + ) -A(x(t)) • a.e. t ∈ (0, T ) , (5.9)
where f (t + ) := lim h→0 h -1 h 0 f (t + τ )dτ. Moreover, given ε > 0 there exists some Differential inclusions with Lipschitz Cusco perturbations h > 0 such that for a.e. τ ∈ (0, h), we have

f (t + τ ) ∈ F (x(t + τ )) ⊂ F (x(t)) + L x(t + τ ) -x(t) B ⊂ F (x(t)) + εLB,
and so lim

h→0 + 1 h h 0 f (t + τ )dτ ∈ F (x(t)
) + εLB (this last set is convex and closed). Hence, as ε goes to 0 we get f (t + ) ∈ F (x(t)), and (i) follows from (5.9).

Conversely, we assume that x 0 ∈ domA and take v ∈ [F (x 0 ) -A(x 0 )] • . We choose w ∈ F (x 0 ) such that v = w -Π A(x 0 ) (w). According to Lemma 5.4, there exists a Lipschitz selection f of F such that f (x 0 ) = w. Then the unique solution y(•) of the following differential inclusion

ẏ(t) ∈ f (y(t)) -A(y(t)), y(0) = x 0 , satisfies d + y(0) dt = f (x 0 ) -Π A(x 0 ) (f (x 0 )) = w -Π A(x 0 ) (w),
and the proof of (i) is complete.

(ii) Let x(•) be a solution of differential inclusion (5.1), with x(0) = x 0 , and fix T > 0. Then by Proposition 5.3 there exist functions k, g ∈ L 1 (0, T ; R n ) such that k(t) ∈ F (x(t)), g(t) ∈ A(x(t)), and

ẋ(t) = k(t) -g(t) a.e t ∈ [0, T ] .
We also choose by Lemma 5.4 a Lipschitz mapping f : R n → R n , with Lipschitz constant c (c ≥ L), and consider the unique solution z(•) of the differential inclusion

ż(t) ∈ f (z(t)) -A(z(t)) a.e. t ≥ 0, z(0) = x 0 .
So, for any t ≥ 0 one has

d + z(t) dt ≤ d + z(0) dt and d + z(0) dt = (f (x 0 ) -A(x 0 )) • ≤ F (x 0 ) + A • (x 0 ) , so z(t) -x 0 ≤ t 0 e cτ d + z(0) dt dτ = e ct -1 c d + z(0) dt ≤ e ct -1 c ( F (x 0 ) + A • (x 0 ) ) (5.10) ≤ te ct ( F (x 0 ) + A • (x 0 ) ) (5.11) 5.4

. Strong and weak invariant sets

By the Lipschitzianity of F we choose a function w(•)

: [0, T ] → R n such that w(t) ∈ F (z(t)), k(t) -w(t) ≤ L x(t) -z(t) ∀t ∈ [0, T ] . (5.12)
Then we obtain

ẋ(t) -ż(t), x(t) -z(t) = k(t) -g(t) -f (z(t)) + Π A(z(t)) (f (z(t))), x(t) -z(t) = k(t) -f (z(t)), x(t) -z(t) + -g(t) + Π A(z(t)) (f (z(t))), x(t) -z(t) ≤0, by the monotonicity of A ≤ k(t) -w(t), x(t) -z(t) + w(t) -f (z(t)), x(t) -z(t) ≤L x(t) -z(t) 2 + 2 F (z(t)) x(t) -z(t) (by (5.12)) ≤L x(t) -z(t) 2 + 2( F (x 0 ) + L z(t) -x 0 ) x(t) -z(t) ≤L x(t) -z(t) 2 + 2 F (x 0 ) + (e ct -1)( F (x 0 ) + A • (x 0 ) ) x(t) -z(t) ≤L x(t) -z(t) 2 + 2( F (x 0 ) + A • (x 0 ) )e ct x(t) -z(t) .
Consequential, from Gronwall Lemma we get, for every t ≥ 0,

x(t) -z(t) ≤ 2( F (x 0 ) + A • (x 0 ) )te ct ,
which together with (5.11) give us

x(t) -x 0 ≤ 3( F (x 0 ) + A • (x 0 ) )te ct ,
and, for every other solution y = y(•; x 0 ),

x(t) -y(t) ≤ x(t) -z(t) + y(t) -z(t) ≤ 4( F (x 0 ) + A • (x 0 ) )te ct ;
that is the conclusion of (ii) follows.

Strong and weak invariant sets

In this section, we give explicit characterizations for a closed set S ⊂ R n to be strong or weak invariant for differential inclusion (5.1),

ẋ(t) ∈ F (x(t)) -A(x(t)) a.e. t ≥ 0, x(0) = x 0 ∈ domA,
where A : H ⇒ H is a maximal monotone operator and F is an L-Lipschitz Cusco mapping. Invariance criteria are written exclusively by means of the data; that is, multifunction F and operator A, and involve the geometry of the set S, using the associated proximal and Fréhet normal cones.

Definition 5.7. Let S be a closed subset of R n .

(i) S is said to be strong invariant if for any x 0 ∈ S ∩ domA and any solution x(•; x 0 ) of (5.1), we have

x(t; x 0 ) ∈ S ∀t ≥ 0.
(ii) S is said to be weak invariant if for any x 0 ∈ S ∩ domA, there exists at least one solution x(•; x 0 ) of (5.1) such that

x(t; x 0 ) ∈ S ∀t ≥ 0.
Since any solution of differential inclusion (5.1) lives in domA (Proposition 5.6), we may assume without loss of generality that S is a closed subset of domA. We shall need the following two lemmas. Lemma 5.8. (e.g. [7, Lemma A.1])Let S ⊂ R n be closed. Then for every x ∈ R n \ S we have

∂ L d S (•)(x) ∈ x -Π S (x) d S (x) and ∂ C d S (•)(x) ∈ co({ x -Π S (x) d S (x) }).
Lemma 5.9. Let ϕ : R n → R be an l-Lipschitz function. Then for every x ∈ R n we have

ϕ(x + v) ≤ ϕ(x) + ϕ 0 (x; v) + o( v ), v ∈ R n .
Proof. We proceed by contradiction and suppose that for some α > 0 and sequence

(v n ) n ⊂ R n \ {θ} converging to θ it holds ϕ(x + v n ) -ϕ(x) > ϕ 0 (x; v n ) + α v n for all n ≥ 1.
(5.13)

Strong and weak invariant sets

Without loss of generality, we can assume that vn vn → v = θ. Then

ϕ(x + v n ) -ϕ(x) =ϕ(x + v n -v n v + v n v) -ϕ(x + v n -v n v) + ϕ(x + v n -v n v) -ϕ(x) ≤ϕ(x + v n -v n v + v n v) -ϕ(x + v n -v n v) + l (v n -v n v) .
Hence, from the inequality (5.13) one gets

ϕ(x + v n -v n v + v n v) -ϕ(x + v n -v n v) v n +l v n v n -v ≥ ϕ 0 (x; v n v n )+α, which as n → ∞ leads us to the contradiction ϕ 0 (x; v) ≥ ϕ 0 (x; v) + α > ϕ 0 (x; v).
Before we state the main strong invariance result we give the following result:

Proposition 5.10. Let S ⊂ domA satisfy condition (5.6), and take x 0 ∈ S. If there is some ρ > 0 such that for any x ∈ B(x 0 , ρ) ∩ S ∩ domA, sup

ξ∈N P S (x) sup v∈F (x) inf x * ∈A(x) ξ, v -x * ≤ 0, (5.14)
then given any solution x(•; x 0 ) of (5.1), there exists T > 0 such that x(t; x 0 ) ∈ S for every t ∈ [0, T ].

Proof. Let x(•) := x(•; x 0 ) be any solution of differential inclusion (5.1), so that for some T 1 > 0 we have

x(t) ∈ B(x 0 , ρ 3 ) ∩ domA, a.e. t ∈ [0, T 1 ], (5.15)
where ρ > 0 is as in the current assumption, and so (by condition (5.6))

Π S (x(t)) ⊂ B(x 0 , 2 3 ρ) ∩ S ∩ domA ⊂ B(x 0 , ρ) ∩ S ∩ domA for a.e. t ∈ (0, T 1 ]. (5.16) We denote the function η : [0, T 1 ] → R as η(t) := d 2 S (x(t)).
Fix ε > 0. Since the function d 2 S (•) is Lipschitz on each bounded set and x(•) is absolutely continuous on [ε, T 1 ], function η is also absolutely continuous on [ε, T 1 ]; hence, differentiable on a set T 0 ⊂ [ε, T 1 ] of full measure (we may also suppose that (5.16) holds for all t ∈ T 0 ). We pick t ∈ T 0 so that, according to Lemma 5.9, for all s > 0

d 2 S (x(t + s)) = d 2 S (x(t) + ẋ(t)s + o(s)) ≤ d 2 S (x(t) + ẋ(t)s) + o(s) ≤ d S (x(t)) + sd 0 S (x(t); ẋ(t)) + o(s) 2 + o(s) ≤ d 2 S (x(t)) + 2d S (x(t))d 0 S (x(t); ẋ(t)s) + o(s), (5.17)
While by Lemma 5.8 we have

d S (x(t))d 0 S (x(t); ẋ(t)) = d S (x(t)) max ξ∈∂ C d(x(t)) ξ, ẋ(t) (5.18)
≤ max u∈Π S (x(t))

x(t) -u, ẋ(t) .

Let us write ẋ(t) as ẋ(t) = v -w for some v ∈ F (x(t)) and w ∈ A(x(t)), and fix u ∈ Π S (x(t)) (⊂ B(x 0 , ρ) ∩ S ∩ domA by (5.16)). By the Lipschitzianity of F we choose some v ∈ F (u) such that

v -v ≤ L x(t) -u = Ld S (x(t)).
Since x(t) -u ∈ N P S (u), by the current hypothesis of the theorem there exist w ∈ A(u) such that x(t) -u, v -w ≤ 0, which in turn yields, due to the monotonicity of A,

x(t) -u, ẋ(t) = x(t) -u, v -w = x(t) -u, v -v + x(t) -u, v -w + x(t) -u, w -w ≤ L x(t) -u 2 = Ld 2 S (x(t)).
Thus, continuing with (5.17) and (5.18) we arrive at

η(t + s) ≤ η(t) + 2Lη(t)s + o( s ),
which implies that η(t) ≤ 2Lη(t). Hence, by Gronwall Lemma, we obtain that η(t) ≤ η(ε)e 2L(t-ε) for all t ∈ T 0 , or, equivalently, η(t) ≤ η(ε)e 2L(t-ε) for all t ∈ [ε, T 1 ]. Then, as ε goes to 0 we conclude that η(t) = 0 for all t ∈ [0, T 1 ], which proves that x(t) ∈ S for all t ∈ [0, T 1 ].

We give the required characterization of strong invariant closed sets with respect to differential inclusion (5.1).

Theorem 5.11. Let S be a closed subset of domA satisfying relation (5.6). Then the following statements are equivalent, provided that N S = N P S or N F S and T S = T B S , or T S = coT B S , (i) S is strong invariant for differential inclusion (5.1).

(ii) For every x ∈ S ∩ domA, one has

v -Π A(x) (v) ∈ T S (x) ∀v ∈ F (x).
(

(iii) For every x ∈ S ∩ domA, one has

[v -A(x)] ∩ T S (x) = ∅ ∀v ∈ F (x). (5.20)
(iv) For every x ∈ S ∩ domA, one has

sup ξ∈N S (x) sup v∈F (x) ξ, v -Π A(x) (v) ≤ 0. (5.21)
(v) For every x ∈ S ∩ domA, one has

sup ξ∈N S (x) sup v∈F (x) inf x * ∈A(x) ξ, v -x * ≤ 0. (5.22)
(vi) For every x ∈ S ∩ domA, one has

sup ξ∈N S (x) sup v∈F (x) inf x * ∈A(x)∩B(θ, F (x) + A • (x) ) ξ, v -x * ≤ 0. (5.23)
Proof. The implication (ii) ⇒ (iii) and (vi) ⇒ (v) are trivial, while the implications (ii) ⇒ (iv) and (iii) ⇒ (v) come from the relation T S (x) ⊂ (N F S (x)) * for all x ∈ S. The implications (v) (with N S = N P S ) ⇒ (i) is a direct consequence of Proposition 5.10.

(i) ⇒ (ii). To prove this implication we suppose that S is strong invariant and take x 0 ∈ S ∩ domA and v ∈ F (x 0 ). According to Lemma 5.4, there exists a Lipschitz selection f of F such that f (x 0 ) = v, and so there is a unique solution x(•) of the following differential inclusion,

ẋ(t) ∈ f (x(t)) -A(x(t)), a.e. t ≥ 0, x(0) = x 0 .
It follows that x(•) is also a solution of differential inclusion (5.1), so that x(t) ∈ S for any t ≥ 0. Then we get

v -Π A(x 0 ) (v) = (f (x 0 ) -A(x 0 )) • = d + x(0) dt = lim t↓0 x(t) -x 0 t ∈ T B S (x 0 ) ⊂ T S (x 0 ).
(iv) ⇒ (vi). This implication holds since for any x ∈ domA and v ∈ F (x) we have that

Π A(x) (v) ≤ Π A(x) (v) -A • (x) + A • (x) = Π A(x) (v) -Π A(x) (θ) + A • (x) ≤ v + A • (x) ≤ F (x) + A • (x) .
The proof of the theorem is complete.

The following proposition, which provides the counterpart of Proposition 5.10 for the weak invariance, is essentially given in [39, Theorem 1]. The specification of the interval on which the solution remains in S also comes from the proof given in that paper. Proposition 5.12. Let S ⊂ domA be closed and take x 0 ∈ S such that, for some r, m > 0,

||A • (x)||≤ m ∀x ∈ S ∩ B(x 0 , r). (5.24)
Assume that for all x ∈ S ∩ B(x 0 , r),

sup ξ∈N P S (x) inf v∈F (x) inf x * ∈A(x)∩B m+ F (x) ξ, v -x * ≤ 0. (5.25)
Then there exists a solution x(•; x 0 ) of (5.1) such that x(t; x 0 ) ∈ S for every t ∈

[0, T ] with T = r 3 m + sup x∈B(x 0 ,r)∩S F (x) -1
.

Consequently, we obtain the desired characterization of weak invariant sets with respect to differential inclusion (5.1). Recall that A • is said to be locally bounded on S if for every x ∈ S we have (ii) For every x ∈ S, one has

∪ v∈F (x) v -A(x) ∩ B m(x)+ F (x) ∩ T S (x) = ∅. (5.27)
(iii) For every x ∈ S, one has

sup ξ∈N S (x) inf v∈F (x) inf x * ∈A(x)∩B m(x)+ F (x) ξ, v -x * ≤ 0.
(5.28)

Proof. (i) ⇒ (ii). Given an x 0 ∈ S we choose a solution x(•) := x(•; x 0 ) of (5.1) that belongs to S. Fix ε > 0. By (5.26) and the current assumption we also choose ρ > 0 such that

A • (x) ≤ m(x 0 ) + ε for all x ∈ B(x 0 , ρ) ∩ S.
Then for any x ∈ B(x 0 , ρ) ∩ S and any v ∈ F (x) we get

Π A(x) (v) ≤ Π A(x) (v) -A • (x) + A • (x) ≤ v + A • (x) ≤ F (x) +m(x 0 )+ε;
Let T > 0 be such that x(t) ∈ B(x 0 , ρ) ∩ S for all t ∈ [0, T ], so that for all v ∈ F (x(t)) and t ∈ [0, T ] we have

Π A(x(t)) (v) ≤ F (x(t)) + m(x 0 ) + ε; hence, by Proposition 5.6(i), ẋ(t) ∈ F (x(t)) -A(x(t)) ∩ B F (x(t)) +m(x 0 )+ε a.e. t ∈ [0, T ], (5.29)
and x(•) is Lipschitz on [0, T ] (observing that B F (x(t)) +m(x 0 )+ε ⊂ B F (x 0 ) +Lρ+m(x 0 )+ε ). Take w ∈ Limsup t↓0 t -1 (x(t) -x 0 ) (this Painleve-Kuratowski upper limit is nonempty, due to the Lipschitzianity of x(•)). Then, since the mappings x → A(x) ∩ B(θ, F (x) + m(x 0 ) + ε) and x → F (x) are upper Differential inclusions with Lipschitz Cusco perturbations semi-continuous, by using (5.29) we get

w ∈ Limsup t↓0 1 t t 0 ẋ(τ )dτ ⊂ Limsup t↓0   co   τ ∈[0,t] F (x(τ )) -A(x(τ )) ∩ B F (x(τ )) +m(x 0 )+ε     ⊂ F (x 0 ) -A(x 0 ) ∩ B F (x 0 ) +m(x 0 )+ε , (5.30)
and we conclude that, as ε goes to 0 (observe that v is independent of ε),

w ∈ F (x 0 ) -A(x 0 ) ∩ B F (x 0 ) +m(x 0 ) .
Thus, (ii) follows, due to the obvious fact that Limsup t↓0 t -1 (x(t) -x 0 ) ⊂ T S (x 0 ).

(iii) ⇒ (i). Fix x 0 ∈ S. By (5.26) we choose r, m > 0 such that m(x) ≤ m for every x ∈ S ∩ B(x 0 , r). It suffices to prove that T := sup{T : ∃ x(•; x 0 ) a solution of (5.1) such that x(t; x 0 ) ∈ S ∀t ∈ [0, T ]} = +∞.

According to Proposition 5.12, there exist some T 1 > 0 and a solution x 1 (•; x 0 ) of differential inclusion (5.1) such that x 1 (t; x 0 ) ∈ S for all t ∈ [0, T 1 ]; hence, T ≥ T 1 > 0.

We proceed by contradiction and assume that T < +∞. By Proposition 5.6, we let r 1 > 0 be such that for every solution x(•; x 0 ) of (5.1) we have

x(t; x 0 ) ∈ B(x 0 , r 1 ) ∀t ∈ [0, T ]. We set k := sup x∈B(x 0 ,r 1 +1) F (x) + sup x∈B(x 0 ,r 1 +1)∩S A • (x) ,
so that k < +∞, due to (5.26) and the compactness of the set B(x 0 , r 1 + 1) ∩ S. By definition of T , for 0 < ε < min 1 3k , T we choose a solution x ε (•; x 0 ) of (5.1) such that x ε (t; x 0 ) ∈ S for all t ∈ [0, T -ε]. We put y 0 := x ε ( T -ε; x 0 ) ∈ B(x 0 , r 1 ) ∩ S, so that B(y 0 , 1) ⊂ B(x 0 , r 1 + 1) and the following relations follows easily

||A • (y)||≤ sup u∈B(x 0 ,r 1 +1)∩S A • (u) =: m 1 ∀y ∈ S ∩ B(y 0 , 1), sup ξ∈N S (y) inf v∈F (y) inf x * ∈A(y)∩B m 1 + F (y)
ξ, v -x * ≤ 0 for all y ∈ S ∩ B(y 0 , 1).

Then, according to Proposition 5.12, there exists a solution x 2 (•; y 0 ) of (5.1) such that x 2 (t; y 0 ) ∈ S for all t ∈ [0, 1 3k ]. Consequently, the function z(•; x 0 ) defined as

z(t; x 0 ) :=    x ε (t; x 0 ) if s ∈ [0, T -ε] x 2 (t -T + ε; y 0 ) if s ∈ [ T -ε, +∞[,
is a solution of (5.1) and satisfies z(t; x 0 ) ∈ S for all t ∈ [0, T ] with T := T + 1 3k -ε > T , which contradicts the definition of T . Hence T = ∞, and S is weak invariant.

Strong a-Lyapunov pairs

In this section, we use the invariance results of the previous section to characterize strong a-Lyapunov pairs with respect to differential inclusion (5.1),

ẋ(t) ∈ F (x(t)) -A(x(t)), a.e. t ≥ 0, x(0) = x 0 ∈ domA,
where A : H ⇒ H is a maximal monotone operator and F is an L-Lipschitz Cusco mapping.

Definition 5.14. Let V, W : R n → R ∪ {+∞} be lower semi-continuous functions such that W ≥ 0 and let a ≥ 0. We say that (V, W ) is a strong a-Lyapunov pair for (5.1) if for any x 0 ∈ domA we have

e at V (x(t; x 0 )) + t 0 W (x(τ ; x 0 ))dτ ≤ V (x 0 ) ∀t ≥ 0, (5.31)
for every solution x(•; x 0 ) of (5.1).

The following lemma shows that the non-regularity of the functions V, W candidates to form a-Lyapunov pairs is mainly carried by the function V. (ii) If there are solutions x k (•; x 0 ), k ≥ 1, of (5.1) such that sup k≥1 ess sup T ẋk (•; x 0 ) < +∞ and

e at V (x k (t; x 0 )) + t 0 W k (x k (τ ; x 0 ))dτ ≤ V (x 0 ) ∀t ∈ [0, T ] and ∀k ≥ 1,
then there exists a solution x(•; x 0 ) of (5.1) such that

e at V (x(t; x 0 )) + t 0 W (x(τ ; x 0 ))dτ ≤ V (x 0 ) ∀t ∈ [0, T ].
Proof. The first statement of the lemma is known (see, e.g., [START_REF] Clarke | Nonsmooth Analysis and Control Theory[END_REF]). Statement (i) follows easily from Fatou's lemma. To prove statement (ii) we observe that the current assumption, that sup k≥1 ess sup T ẋk (•; x 0 ) ≤ m, yields the existence of some r > 0 such that x k (t; x 0 ) ∈ B(x 0 ; r) for all t ∈ [0, T ] and all k ≥ 1. Hence, there exists M > 2m such that sup k≥1 sup t∈[0,T ] F (x k (t; x 0 )) ≤ M 2 ; in other words, for all k ≥ 1 it holds ẋk (t;

x 0 ) ∈ F (x k (t; x 0 )) -A(x k (t; x 0 )) ∩ B M a.e. t ∈ [0, T ], x k (0; x 0 ) = x 0 . (5.33)
We may assume without loss of generality that ẋk (•; x 0 ) converges weakly in L 2 (0, T ; R n ) to some v(•) ∈ L 2 (0, T ; R n ), and, due to Arzela-Ascoli's Theorem, that x k (•; (5.34) that is, x(•) is a solution of differential inclusion (5.1). Finally, by taking the limits as k, p → +∞ for k ≥ p in the inequalities for all t ∈ [0, T ]

e at V (x k (t; x 0 ))+ t 0 W p (x k (τ ; x 0 ))dτ ≤ e at V (x k (t; x 0 ))+ t 0 W k (x k (τ ; x 0 ))dτ ≤ V (x 0 ),
we obtain that

e at V (x(t)) + t 0 W (x(τ ))dτ ≤ V (x 0 ) ∀t ∈ [0, T ],
as we wanted to prove. together with the mappings Fk : R n+3 → R n+3 , k ≥ 1, given by (recall (5.32))

Fk (x, α, β, γ) := (F (x), W k (x), 1, 0).

Then  is maximal monotone with dom  = domA × R 3 , Fk is Lipschitz with constant (L 2 +k 2 ) 1 2
, and consequently, the following differential inclusion possesses solutions,

ż(t) ∈ Fk (z(t)) -Â(z(t)) a.e. t ≥ 0, z(0) = z 0 = (x 0 , y 0 , z 0 , w 0 ) ∈ domA × R 3 ,
(5.36) and every solutions is written as

z(t; z 0 ) = (x(t; x 0 ), y 0 + t 0 W k (x(τ ; x 0 ))dτ, z 0 + t, w 0 ),
for a solution x(•; x 0 ) of (5.1). We need the following result which provides us with a local criterion for strong a-Lyapunov pairs. Proposition 5.17. Let V, W : R n → R ∪ {+∞} be two proper lower semicontinuous functions such that domV ⊂ domA, W ≥ 0 and let a ≥ 0. Fix x 0 ∈ domV and assume that for some ρ > 0 we have, for all x ∈ B(x 0 , ρ),

sup ξ∈∂ P V (x) sup v∈F (x) inf x * ∈A(x) ξ, v -x * + aV (x) + W (x) ≤ 0, (5.37) sup ξ∈∂ P,∞ V (x) sup v∈F (x) inf x * ∈A(x) ξ, v -x * ≤ 0. (5.38)
Then there exists some T > 0 such that for every solution x(•; x 0 ) of differential inclusion (5.1) one has

e at V (x(t; x 0 )) + t 0 W (x(τ ; x 0 ))dτ ≤ V (x 0 ) ∀t ∈ [0, T ].
Proof. First, by Proposition 5.6(ii) we let c > 0 be such that for any solutions

x(•) := x(•; x 0 ) of (5.1) it holds

x(t) -x 0 ≤ 3( F (x 0 ) + A • (x 0 ) )te ct for all t ≥ 0, and choose T > 0 such that 3( F (x 0 ) + A • (x 0 ) )T e cT ≤ ρ.

(5.39)

As in Lemma 5.16, we define the proper and lower semi-continuous function Ṽ :

R n+1 × R + → R ∪ {+∞} as Ṽ (x, α, β) := e aβ V (x) + α, so that epi Ṽ is closed and satisfies epi Ṽ ⊂ domV × R 3 ⊂ domA × R 3 = dom Â,
where  is also defined as in Lemma 5.16; hence, condition (5.6) is obviously satisfied for epi Ṽ .

Claim. We claim that for any given z := (x 1 , y 1 , z 1 , w 1 ) ∈ epi Ṽ with

x 1 -x 0 < ρ, there exists small enough ε > 0 such that for each (x, y, z, w) ∈ B(z, ε) ∩ epi Ṽ , ( ξ, -κ) ∈ N P epi Ṽ (x, y, z, w), and (v, W k (x), 1, 0) ∈ Fk (x, y, z, w) there exists x * ∈ A(x) such that ( ξ, -κ), (v -x * , W k (x), 1, 0) ≤ 0.

(5.40)

Indeed, with z as in the claim let us choose ε > 0 such that (x, y, z, w) ∈ B(z, ε) ∩ epi Ṽ ⇒ x ∈ B(x 0 , ρ).

Let (x, y, z, w), ( ξ, -κ), and (v, W k (x), 1, 0) be as in the claim, so that x ∈ B(x 0 , ρ) ∩ domV and v ∈ F (x), as well as κ ≥ 0 (see [START_REF] Clarke | Nonsmooth Analysis and Control Theory[END_REF]Exercise 2.1]). We may distinguish two cases:

(i) If κ > 0, then w = Ṽ (x, y, z) and, without loss of generality, we may suppose that κ = 1. Hence, ξ = (e az ξ, 1, ae az V (x)) ∈ ∂ P Ṽ (x, y, z) for some ξ ∈ ∂ P V (x). Consequently, by the current hypothesis there exists x * ∈ A(x) such that ξ, v -x * + aV (x) + W k (x) ≤ ξ, v -x * + aV (x) + W (x) ≤ 0.

Strong a-Lyapunov pairs

In other words, we have (v -x * , W k (x), 1, 0) ∈ Fk (x, y, z, w) -Â(x, y, z, w) and ( ξ, -1), (v -x * , W k (x), 1, 0) = (e az ξ, 1, ae az V (x), -1), (v -x * , W k (x), 1, 0) = e az ξ, v -x * + W k (x) + ae az V (x) = e az ( ξ, v -x * + aV (x) + W k (x)) +(1 -e az )W k (x) ≤ 0, (5.41) and (5.40) follows.

(ii) If κ = 0, then ξ ∈ ∂ P,∞ Ṽ (x, y, z) and, so, ( ξ, -κ) = (ξ, θ R 3 ) for some ξ ∈ ∂ P,∞ V (x). Then, by arguing as in the paragraph above, the current hypothesis yields some x * ∈ A(x) such that ξ, v -x * ≤ 0. Hence, (v -x * , W k (x), 1, 0) ∈ Fk (x, y, z, w) -Â(x, y, z, w) and ( ξ, 0), (v -x * , W k (x), 1, 0) = ξ, v -x * ≤ 0;

(5.42) that is, (5.40) follows in this case too. The claim is proved. Now, we take a solution x(•; x 0 ) of (5.1), so that z(•; z 0 ) := (x(•; x 0 ),

• 0 W k (x(τ ; x 0 ))dτ, •, V (x 0 )), with z 0 := (x 0 , 0, 0, V (x 0 )), becomes a solution of (5.36). Then, from the claim (with z := z 0 ) above and Proposition 5.10, there exists some t > 0 such that z(t; z 0 ) ∈ epi Ṽ ∀t ∈ [0, t];

(5.43)

that is, T := sup{t ≥ 0 : such that z(s; z 0 ) ∈ epi Ṽ ∀s ∈ [0, t]} > 0.

(5.44)

Let us show that T ≥ T, where T is defined in (5.39). We proceed by contradiction and assume that T < T . Then, because (by Proposition 5.6(ii))

x( T ; x 0 ) -x 0 ≤ 3( F (x 0 ) + A • (x 0 ) ) T e c T < ρ,
and z( T ; z 0 ) = (x( T ; x 0 ), T 0 W k (x(τ ; x 0 ))dτ, T , V (x 0 )) ∈ epi Ṽ , from the claim above (with z := z( T ; z 0 )) and Proposition 5.10, there exists some t 1 > 0 such that z(t; z( T ; z 0 )) ∈ epi Ṽ for all t ∈ [0, t 1 ]. Thus, z(t + T ; z 0 ) = z(t; z( T ; z 0 )) ∈ epi Ṽ for every t ∈ [0, t 1 ], and we get a contradiction to the definition of T .

Finally, from (5.44) we get

e at V (x(t; x 0 )) + t 0 W k (x(τ ; x 0 ))dτ ≤ V (x 0 ) ∀t ∈ [0, T ].
Moreover, because T is independent of k, by taking the limit as k → ∞ we arrive at (as W k (x) W (x), by Lemma 5.15)

e at V (x(t; x 0 )) + t 0 W (x(τ ; x 0 ))dτ ≤ V (x 0 ) ∀t ∈ [0, T ],
which is the desired inequality.

We give now the desired characterization of strong a-Lyapunov pairs.

Theorem 5.18. Let V, W, and a be as in Proposition 5.17 Proof. To prove the sufficiency part, we take x 0 ∈ domV and a solution x(•; x 0 ) of differential inclusion (5.1). By Proposition 5.17 there exists some T > 0 such that e at V (x(t; x 0 )) + Otherwise, if T is finite, then x(T ; x 0 ) ∈ domV (because V is lower semicontinuous), and again from Proposition 5.17 we find η > 0 such that for all t ∈ [0, η], using the semi-group property of x(•; x 0 ),

e a(t+T ) V (x(t + T ; x 0 )) + t+T 0 W (x(τ ; x 0 ))dτ ≤ e aT e at V (x(t + T ; x 0 )) + t+T T W (x(τ ; x 0 ))dτ + T 0 W (x(τ ; x 0 ))dτ ≤ e aT V (x(T ; x 0 )) + T 0 W (x(τ ; x 0 ))dτ ≤ V (x 0 ),
and we get the contradiction T ≥ T + η. Hence, T = +∞ and (5.47) holds for all t ≥ 0, showing that (V, W ) forms a strong Lyapunov pair for differential inclusion (5.1).

Strong a-Lyapunov pairs

To prove the necessity of the current conditions, we start by verifying (5.45) with ∂ = ∂ F . We fix x 0 ∈ domV (⊂ domA) and v ∈ F (x 0 ), and, according to Proposition 5.6, we choose a solution x(•; x 0 ) of differential inclusion (5.1) such that d + x(0;x 0 ) dt = v -Π A(x 0 ) (v). Thus, since (V, W ) is assumed to be a strong a-Lyapunov pair for (5.1), we obtain for every t > 0

V (x(t; x 0 )) -V (x 0 ) t + e at -1 t V (x(t; x 0 )) + 1 t t 0 W (x(τ ; x 0 ))dτ ≤ 0,
which give us, as t ↓ 0,

σ ∂ F V (x 0 ) (v -Π A(x 0 ) (v)) ≤ V (x 0 ; v -Π A(x 0 ) (v)) ≤ lim inf t↓0 V (x(t; x 0 )) -V (x 0 ) t ≤ -aV (x 0 ) -W (x 0 ).
Hence, (5.45) follows with either ∂ = ∂ F or ∂ = ∂ P . To verify (5.46) we fix x 0 ∈ domV , v ∈ F (x 0 ) and ξ ∈ ∂ P,∞ V (x 0 ); that is, (ξ, 0) ∈ N P epiV (x 0 , V (x 0 )). According to Proposition 5.6, we choose a solution x(•; x 0 ) of differential inclusion (5.1) such that d + x(0;x 0 ) dt = v -Π A(x 0 ) (v). Since (V, W ) is strong a-Lyapunov for differential inclusion (5.1), one has that (x(t; x 0 ), e -at V (x 0 )) ∈epiV for all t ≥ 0. Then, by the definition of the proximal normal cone, there exists η > 0 such that for all small t ≥ 0 (ξ, 0), (x(t; x 0 ), e -at V (x 0 ))-(x 0 , V (x 0 )) ≤ η( x(t; x 0 ) -x 0 2 +|e -at V (x 0 )-V (x 0 )|) 2 , and so ξ, x(t; x 0 ) -x 0 ≤ η( x(t; x 0 ) -x 0 2 + (e -at -1)

2 |V (x 0 )| 2 ).
Hence, by dividing on t > 0 and taking limits as t ↓ 0, we obtain that ξ, v -Π A(x 0 ) (v) ≤ 0, as we wanted to prove.

We give in the following corollary other criteria for strong a-Lyapunov pairs for (5.1), Recall that A • is said to be locally bounded on domV if condition (5.26) holds for all x ∈ domV ; that is, for every x ∈ domV we have

m(x) = lim sup y→x,y∈domV A • (y) < +∞.
We also observe that the function m is upper semi-continuous at every x ∈ R n such that m(x) < +∞; that is, lim sup y→x,y∈domV m(y) = m(x).

(5.48)

Corollary 5.19. Let V, W, and a be as in Proposition 5.17 (ii) For any x ∈ domV ,

sup v∈F (x) V (x; v -Π A(x) (v)) + aV (x) + W (x) ≤ 0.
(iii) For any x ∈ domV ,

sup v∈F (x) inf x * ∈A(x)∩B F (x) +m(x) V (x; v -x * ) + aV (x) + W (x) ≤ 0.
Proof. (ii) ⇒ (iii). This implication follows since that for any x ∈ domV (⊂ domA) any v ∈ F (x)

Π A(x) (v) ≤ A • (x) + Π A(x) (v) -A • (x) ≤ A • (x) + v ≤ m(x) + F (x) .
(iii) ⇒ (i). When ∂ stands for either ∂ P or ∂ F this implication follows from the relation σ

∂ P V (x) (•) ≤ σ ∂ F V (x) (•) ≤ V (x; •). If ∂ = ∂ L , we take ξ ∈ ∂ L V (x) and v ∈ F (x)
, and choose sequences (x i ) and (ξ i ) such that

x i V → x, ξ i ∈ ∂ P V (x i ), ξ i → ξ as i → ∞;
moreover, due to the upper semi-continuity of m at x and m(x) < +∞, by assumption, we may assume up to a subsequence that

m(x i ) ≤ m(x) + 1 i ∀i ∈ N. (5.49)
By the Lipschitzianity of F we also choose a sequence (v i ) i≥1 such that v i ∈ F (x i ) and v i → v. Since (i) holds with ∂ = ∂ P , for each i there exists

x * i ∈ A(x i ) ∩ B F (x i ) +m(x i ) such that ξ i , v i -x * i + aV (x i ) + W (x i ) ≤ 0. (5.50)
Then, since the maximal monotone operator A has a closed graph, and (x * i ) i is bounded, we assume w.l.o.g. that

x * i → x * ∈ A(x) ∩ B(θ, m(x)) as i → ∞.
So, by passing to the limit in (5.50) as i → ∞, and using the lower semicontinuity of W, we obtain that ξ, v -x * + aV (x) + W (x) ≤ 0, (5.51) which shows that (i) holds when ∂ = ∂ L .

(i) ⇒ (V, W ) is a strong a-Lyapunov pair for (5.1). According to Theorem 5.18 we only need to show that (5.46) holds. We fix x ∈ domV, ξ ∈ ∂ P,∞ V (x) and v ∈ F (x). There exist sequences (x i ) i , (ξ i ) i , and (α i ) i such that

x i V → x, ξ i ∈ ∂ P V (x i ), α i ↓ 0, α i ξ i → ξ as i → ∞.
By arguing as in the last paragraph above there also exists a sequence (v i ) i such that v i ∈ F (x i ) and v i → v as i → ∞. Moreover, using the current assumption on A • , there exists m > 0 such that sup i m(x i ) ≤ m. Now, by assumption (ii), for each i ∈ N there exists a sequences

x * i ∈ A(x i ) ∩ B F (x i ) +m(x i ) ⊂ A(x i ) ∩ B F (x i ) +m and ξ i , v i -x * i + aV (x i ) + W (x i ) ≤ 0.
(5.52)

By using again that A has a closed graph, and that x * i → x * ∈ A(x), By multiplying the last inequality above (5.52) by α i and next taking limits as i → ∞, we arrive at (5.46). The proof of the corollary is finished since (ii) is a necessary condition for strong a-Lyapunov pairs, as we have shown in the proof of Theorem 5.18.

Weak a-Lyapunov pairs

In this section, we characterize weak a-Lyapunov pairs with respect to differential inclusion (5.1), ẋ(t) ∈ F (x(t)) -A(x(t)), a.e. t ≥ 0, x(0) = x 0 ∈ domA, where A : H ⇒ H is a maximal monotone operator and F is an L-Lipschitz Cusco mapping.

Definition 5.20. Let V, W : R n → R ∪ {+∞} be lower semi-continuous functions such that W ≥ 0 and let a ≥ 0. We say that (V, W ) is a weak a-Lyapunov pair for (5.1) if for any x 0 ∈ domA, there exists at least one solution x(•; x 0 ) of (5.1) such that

e at V (x(t; x 0 )) + t 0 W (x(τ ; x 0 ))dτ ≤ V (x 0 ) ∀t ≥ 0.
Definition 5.21. Let c > 0 be as in Proposition 5.6(ii), and take x 0 ∈ domA and ρ > 0. We denote by T c,ρ (x 0 ) the positive number that satisfies the following equation in t, 3( F (x 0 ) + A • (x 0 ) )te ct = ρ 2 .

Proposition 5.22. Let V, W : R n → R ∪ {+∞} be two proper lower semicontinuous functions such that domV ⊂ domA, W ≥ 0 and let a ≥ 0. Fix x 0 ∈ domV and assume that for some m, ρ > 0 we have, for all x ∈ B(x 0 , ρ)∩domV

sup ξ∈∂ P V (x) inf v∈F (x) inf x * ∈A(x)∩B m+ F (x) ξ, v -x * + aV (x) + W (x) ≤ 0.
Then there is a solution x(•; x 0 ) of differential inclusion (5.1) such that

e at V (x(t; x 0 )) + t 0 W (x(τ ; x 0 ))dτ ≤ V (x 0 ) ∀t ∈ [0, T c,ρ (x 0 )],
where T c,ρ (x 0 ) > 0 is given in Definition 5.21.

Proof. We fix k ∈ N, and let Â, Ṽ , and F k be as in Lemma 5.16. First, since {x ∈ R n | ∂ P V (x) = ∅} is dense in domV (see, e.g., [START_REF] Clarke | Nonsmooth Analysis and Control Theory[END_REF]Theorem 1.3.1]), and the maximal monotone operator has closed graph, from the current assumption and the Cusco property of F, there is some m 1 > 0 that, for all (x, y, z, w) ∈epi Ṽ such Differential inclusions with Lipschitz Cusco perturbations ξ, v -x * ≤ 0; that is, ( ξ, 0), (v -x * , W k (x), 1, 0) = ξ, v -x * ≤ 0, and we get (5.55).

Step 2). Given z := (x, ỹ, ũ, w) ∈ epi Ṽ such that x -x 0 < ρ, we prove that there exists z(•; z) solution of differential inclusion (5.36) such that

z(t; z) ∈ epi Ṽ ∀t ∈ 0, ρ -x -x 0 6(m 1 + β k ) ,
where

β k := ( F (x 0 ) + Lρ) 2 + ( W (x 0 ) + kρ) 2 + 1 1 2 .
(5.58a) Indeed, let z be as in the claim. Then for every z := (x, y, u, w) ∈epi Ṽ ∩ B(z, 1 2 (ρ -x -x 0 )) we have

x -x 0 ≤ x -x 0 + x -x ≤ x -x 0 + 1 2 (ρ-x -x 0 ) = 1 2 (ρ+ x -x 0 ) < ρ, (5.59
) and, so, • (z) ≤ m + F (x) ≤ m 1 (recall (5.53)). In other words, according to the first step, for every z ∈epi Ṽ ∩ B(z, 1 2 (ρ -x -x 0 )) and ( ξ, -κ) ∈ N P epi Ṽ (z), there exist (v, W k (x), 1, 0) ∈ Fk (z) and (x * , θ R 3 ) ∈ Â(z) ∩ B m+ F (x) ⊂ Â(z) ∩ B m 1 + F k (z) such that (5.55) holds. Consequently, taking into account that (using (5.59))

sup p∈epi Ṽ ∩B(z, 1 2 (ρ-x-x 0 )) Fk (p) = sup x∈B(x 0 ,ρ) (F (x), W k (x), 1, 0) ≤ β k , (5.60)
by Proposition 5.12 there exists a solution of differential inclusion (5.36) as required.

Step 3). We put z 0 := (x 0 , 0, 0, V (x 0 )) ∈ epi Ṽ . Then T := sup{t ≥ 0 : there exists a solution z k (•; z 0 ) of (5.36) st. z k (s;

z 0 ) ∈ epi Ṽ ∀s ∈ [0, t]} ≥ T c,ρ (x 0 ), (5.61)
where T c,ρ (x 0 ) is given in Definition 5.21; hence, T c,ρ (x 0 ) satisfies 3( F (x 0 ) + A • (x 0 ) )T c,ρ (x 0 )e cTc,ρ(x 0 ) < ρ, (5.62) 5.6. Weak a-Lyapunov pairs with c > 0 being such that (see Proposition 5.6(ii)) for any solutions x(•) := x(•; x 0 ) of (5.1) it holds

x(t) -x 0 ≤ 3( F (x 0 ) + A • (x 0 ) )te ct for all t ≥ 0.

(5.63) Indeed, from Step 2 we have that T ≥ ρ 6(m 1 +β k ) , where β k is defined in (5.58a).

To prove (5.61) we proceed by contradiction and assume that T < T c,ρ (x 0 ). By definition of T , for every 0 < ε < min{ T , ρ-3( F (x 0 ) + A • (x 0 ) )Tc,ρ(x 0 )e cTc,ρ(x 0 )

6(m 1 +β k )
} there exists a solution z k (•; z 0 ) of (5.36), and a solution x k (•; x 0 ) of (5.1), such that

z k (t; z 0 ) = (x k (t; x 0 ), t 0 W k (x k (τ ; x 0 ))dτ, t, V (x 0 )) ∈ epi Ṽ ∀t ∈ [0, T -ε].
But by (5.63) and (5.62) we have that (5.64) and, so, by Step 2 there exists zk (•; z k ( T -ε; z 0 )) a solution of differential inclusion (5.36) such that zk (t;

x k ( T -ε; x 0 ) -x 0 ≤ 3( F (x 0 ) + A • (x 0 ) )( T -ε)e c( T -ε) ≤ 3( F (x 0 ) + A • (x 0 ) )T c,ρ (x 0 )e cTc,ρ(x 0 ) < ρ,
z k ( T -ε; z 0 )) ∈ epi Ṽ ∀t ∈ 0, ρ -x k ( T -ε; z 0 ) -x 0 6(m 1 + β k ) .
We denote zk (t; z 0 ) :=

   z k (t; z 0 ) if t ∈ [0, T -ε] zk (t -T + ε; z k ( T -ε; z 0 )) if t ∈ [ T -ε, ∞).
Then zk (•; z 0 ) is a solution of (5.36) and we have that

zk (t; z 0 ) ∈ epi Ṽ ∀t ∈ 0, T + ρ -x k ( T -ε; z 0 ) -x 0 6(m 1 + β k ) -ε .
Thus, since (recall (5.64))

ρ -x k ( T -ε; z 0 ) -x 0 6(m 1 + β k ) ≥ ρ -3( F (x 0 ) + A • (x 0 ) )T c,ρ (x 0 )e cTc,ρ(x 0 ) 6(m 1 + β k ) > ε, we get the contradiction T ≥ T + ρ-x k ( T -ε;z 0 )-x 0 6(m 1 +β k ) -ε > T .
Step 3 is now proved.

Differential inclusions with Lipschitz Cusco perturbations

Step 4). In this last step we get the conclusion of the proposition. From Step 3 there is a solution x k (•; x 0 ) of (5.1) such that

z k (t; z 0 ) = (x k (t; x 0 ), t 0 W k (x k (τ ; x 0 ))dτ, t, V (x 0 )) ∈ epi Ṽ for all t ∈ [0, T c,ρ (x 0 )];
that is,

e at V (x k (t; x 0 )) + t 0 W k (x k (τ ; x 0 ))dτ ≤ V (x 0 ) for all t ∈ [0, T c,ρ (x 0 )].
Moreover, since that x k (•; x 0 ) -x 0 ≤ 3( F (x 0 ) + A • (x 0 ) )T c,ρ (x 0 )e cTc,ρ(x 0 ) < ρ (by Proposition 5.6(ii)), by using the Lipschitz property of F and (5.53) from Proposition 5.6(i) we obtain that ẋk (t;

x 0 ) ≤ F (x k (t; x 0 )) + A • (x k (t; x 0 )) ≤ F (x 0 ) + Lρ + m 1 .
Consequently, by Lemma 5.15 there exists a solution x(•; x 0 ) of (5.1) such that

e at V (x(t; x 0 )) + t 0 W (x(τ ; x 0 ))dτ ≤ V (x 0 ) ∀t ∈ [0, T c,ρ (x 0 )],
which yields the conclusion of the proposition.

The assumption of Proposition 5.22 easily implies that (see (5.53)) A • is locally bounded on domV ; that is,

m(x) = lim sup y domV -→ x A • (y) < +∞ for all x ∈ domV .
(5.65)

The following theorem characterizes weak a-Lyapunov pairs for (5.1) under the condition above. V (x; v -x * ) + aV (x) + W (x) ≤ 0.

Proof. (i) ⇒ ((V, W ) is a weak a-Lyapunov pair for (5.1)).

As in Lemma 5.16 we let the maximal monotone operator  : R n × R 3 → R n+3 , the proper lower semi-continuous function Ṽ : R n+1 × R + → R ∪ {+∞}, and the Cusco mappings Fk : R n+3 → R n+3 , k ≥ 1, defined by Â(x, α, β, γ) := (A(x), θ R 3 ), Ṽ (x, α, β) := e aβ V (x)+α, Fk (x, α, β, γ) := (F (x), W k (x), 1, 0).

Let us fix z 0 = (x 0 , 0, 0, V (x 0 )) ∈ epi Ṽ and k ∈ N. We set T := sup{s : there exists a solution z k (•; z 0 )) of (5.36)

s.t. z k (•; z 0 ) ∈ epi Ṽ ∀t ∈ [0, s]}
(5.66) By current hypothesis (i) and the locally boundedness of A • , there exists m > 0 and ρ > 0 such that for all x ∈ B(x 0 , ρ)

sup ξ∈∂ P V (x) inf v∈F (x) inf x * ∈A(x)∩B m+ F (x) ξ, v -x * + aV (x) + W k (x) ≤ 0.
Then, according to Proposition 5.22, there exists s > 0 and a solution x(•; x 0 ) of (5.1) such that

e at V (x(t; x 0 )) + t 0 W k (x(τ ; x 0 ))dτ ≤ V (x 0 ) ∀t ∈ [0, s]; that is, z k (•; z 0 ) := (x(•; x 0 ), • 0 W k (x(τ ; x 0 ))dτ, •, V (x 0 )
) is a solution of (5.36) such that z k (t; z 0 ) ∈epi Ṽ for all t ∈ [0, s]. Hence, T > 0. If T is finite, then by Proposition 5.6 there would exist r > 0 such that for any solution z k (•; z 0 ) of (5.36) we have

z k (•; z 0 ) ∈ B(z 0 , r) ∀t ∈ [0, T ].
Also, since the set B(z 0 , r + 2) ∩ epi Ṽ and its projection on

R n E = x ∈ R n : there exist (y, u, v) ∈ R 3 s.t. (x, y, u, v) ∈ B(z 0 , r + 2) ∩ epi Ṽ ,
are compact, by the current assumptions there exists M > 0 such that

( Â) • (z) ≤ M ∀z ∈ B(z 0 , r + 2) ∩ epi Ṽ , m(x) ≤ M ∀x ∈ E, (5.67)
5.6. Weak a-Lyapunov pairs

v i → v ∈ F (x) and x * i → x * ∈ A(x) ∩ B M + F (x) as i → ∞.
Then, by multiplying both sides of the inequality above by α i and taking limits as i → ∞, we obtain ξ, v -x * ≤ 0; that is, ( ξ, 0), (v -x * , W k (x), 1, 0) = ξ, v -x * ≤ 0, and (5.68) also follows in this case.

Step 2). We show that T = ∞. From Step 1 and Proposition 5.12, for every z ∈ B(z 0 , r) ∩ epi Ṽ , there exists a solution z k (•; z) such that

z k (t; z) ∈ epi Ṽ ∀t ∈ [0, t],
where t :=

1 3 (M + sup z ∈B(z 0 ,r+1)∩epi Ṽ F k (z ) ) -1 . Let us fix ε ∈ (0, t).
From the definition of T , there exists a solution zk (•; z 0 ) of (5.36) such that zk (t, z 0 ) ∈ epi Ṽ for all t ∈ [0, T -ε]. By the result above, it is easy to find a solution zk (•; z 0 ) of (5.36) such that zk (t; z 0 ) ∈ epi Ṽ for all t ∈ [0, T + t -ε] which contradicts the definition of T , hence we get T = +∞.

Step 3). In this step, we get (V, W ) is a weak a-Lyapunov pair for (5.1). By the result of Step 2, then for every T and k ∈ N, there exists a solution x k (•; x 0 ) of (5.1) such that

z k (t; z 0 ) := (x k (t; x 0 ), t 0 W k (x k (τ ; x 0 ))dτ, t, V (x 0 )) ∈ epi Ṽ ∀ t ∈ [0, T ],
or, equivalently,

e at V (x k (t; x 0 )) + t 0 W k (x k (τ ; x 0 ))dτ ≤ V (x 0 ) ∀t ∈ [0, T ],
that is, using Proposition 5.6(ii),

(x k (t; x 0 )) k ⊂ D := [V ≤ V (x 0 )] ∩ B(x 0 , 3 F (x 0 ) + A • (x 0 ) T e cT )
where c > 0 is defined in Proposition 5.6(ii). Thus, by the current assumption, and the lower semi-continuity of the function V, A • is bounded on the compact set D, so that by Proposition 5.6(i) we obtain some M > 0 such that ẋk (t; x 0 ) ≤ M for all k ≥ 1. Therefore, by Lemma 5.15 there exists a solution x(•; x 0 ) of (5.1)

Differential inclusions with Lipschitz Cusco perturbations such that (ii) ⇒ (i). This implication follows when ∂ = ∂ P or ∂ = ∂ F , due to the relations

e at V (x(t; x 0 )) + t 0 W (x(τ ; x 0 ))dτ ≤ V (x 0 ) ∀t ∈ [0, T ]. ( 5
σ ∂ P V (x) (•) ≤ σ ∂ F V (x) (•) ≤ V (x; •).
It remains to check the case ∂ = ∂ L . We take ξ ∈ ∂ L V (x) and, be definition, let the sequences (x k ) and (ξ k ) converge to x and ξ, respectively, such that ξ k ∈ ∂ P V (x k ) for all k. Since (i) already holds for ∂ = ∂ P , for each k ≥ 1 there exist

v k ∈ F (x k ) and x * k ∈ A(x k ) ∩ B m(x k )+ F (x k ) such that ξ k , v k -x * k + aV (x k ) + W (x k ) ≤ 0.
We may suppose that x * k → x * ∈ A(x) ∩ B m(x)+ F (x) and v k → v ∈ F (x) as k → ∞. Hence, by taking the limit as k → ∞ in the last inequality we get ξ, v -x * + aV (x) + W (x) ≤ 0, as we wanted to prove.

((V, W ) is a weak a-Lyapunov pair for (5.1)) ⇒ (ii). Take x 0 ∈ domV and let x(•; x 0 ) be any solution of (5.1) that satisfies

e at V (x(t; x 0 )) + t 0 W (x(τ ; x 0 ))dτ ≤ V (x 0 ) ∀t ≥ 0.
(5.71)

Then, as in (5.30) for some t n ↓ 0 we have that

v := lim tn↓0 x(t n ; x 0 ) -x 0 t n ∈ F (x 0 ) -(A(x 0 ) ∩ B F (x 0 ) +m(x 0 ) ), Differential inclusions with Lipschitz Cusco perturbations x ∈ C and v ∈ B κ N C (x) ∩ B(θ, κ) + κ r x ⊂ A C,κ (x) ⊂ N C (x) + κ r x, (5.73)
(v -N C (x)) • = (v + κ r x -A C,κ (x)) • .
(5.74)

Proof. We refer [START_REF] Adly | A convex approach to differential inclusions with prox-regular sets: Stability analysis and observer design[END_REF] for (5.73). To verify (5.74) we observe that (by (5.73))

v -N C (x) ∩ B(θ, κ) ⊂ v + κ r x -A C,κ (x) ⊂ v -N C (x).
Then, since v ≤ κ and, so, (v

-N C (x) ∩ B(θ, κ)) • = (v -N C (x)) • , we get (v -N C (x) ∩ B(θ, κ)) • = (v + κ r x -A C,κ (x)) • = (v -N C (x)) • .
The following theorem reviews the main properties of (5.72) (see [START_REF] Adly | A convex approach to differential inclusions with prox-regular sets: Stability analysis and observer design[END_REF]).

Theorem 5.26. Fix x 0 ∈ C, w ∈ (F (x 0 )-N C (x 0 )) • , T > 0 and m > e LT F (x 0 ) .

Then:

(i) There exists a solution x(•; x 0 ) of (5.72) such that d + x(0) dt = w, and the function t → d + x(t) dt is right-continuous.

(ii) For every solution x(•; x 0 ) of (5.72) we have, for ae t ≥ 0,

x(t) -x 0 ≤ F (x 0 ) L (e Lt -1), ẋ(t) ≤ F (x 0 ) e Lt . (5.75)
(iii) Differential inclusion (5.72) has the same solutions set on [0, T ] as the differential inclusion

ẋ(t) ∈ (F + m r Id)(x(t)) -A C,m (x(t)) a.e. t ∈ [0, T ]; x(0) = x 0 ∈ C, ( 5
ẋ(t) ∈ (F (x(t)) -N C (x(t))) • a.e. t ≥ 0. Proof. (i) Let v ∈ F (x 0 ) be such that w = v -Π N C (x 0 ) (v)
, and, according to Lemma 5.4, let f be a Lipschitz selection of F such that f (x 0 ) = v. Then the 5.7. Differential inclusions with prox-regular sets following differential inclusion ẋ(t) ∈ f (x(t)) -N C (x(t)), x(0) = x 0 , ae t ≥ 0, has a unique solution, which satisfies the conditions of statement (i) (see [START_REF] Adly | A convex approach to differential inclusions with prox-regular sets: Stability analysis and observer design[END_REF]).

(ii) Let x(•; x 0 ) be any solution of (5.72) and fix T 1 > 0. By Proposition 5.3, we choose a function g ∈ L 1 ([0, T 1 ]; R n ) such that g(t) ∈ F (x(t)) and (see [START_REF] Adly | A convex approach to differential inclusions with prox-regular sets: Stability analysis and observer design[END_REF]Lemma 9])

g(t) -ẋ(t), ẋ(t) = 0 and ẋ(t) ∈ g(t) -N C (x(t)) a.e. t ∈ [0, T 1 ]; hence, ẋ(t) ≤ g(t) ≤ F (x(t)) ≤ F (x 0 ) + L x(t) -x 0 . ( 5
(t) = 2 ẋ(t), x(t) -x 0 ≤ 2 ẋ(t) x(t) -x 0 ≤ 2 F (x 0 ) x(t) -x 0 + 2L x(t) -x 0 2 = 2Lη(t) + 2 F (x 0 ) η 1 2 (t).
So, on the one hand, by Lemma 5.1 we obtain η

1 2 (t) ≤ F (x 0 )
L (e Lt -1) for all t ∈ [0, T 1 ], and on the other hand, this last inequality together with (5.77) give us for ae

t ∈ [0, T 1 ] ẋ(t) ≤ F (x(t)) ≤ F (x 0 ) + L x(t) -x 0 ≤ F (x 0 ) e Lt .
This proves (5.75).

(iii) If x(•) := x(•; x 0 ) is a solution of of (5.72) on [0, T ], then by (5.75) we get for ae t ∈

[0, T ] ẋ(t) ≤ m, F (x(t)) ≤ F (x 0 ) + L x(t) -x 0 ≤ m;
(5.78) that is, using Lemma 5.25, and we get, using again Lemma 5.25 and combining with (5.78),

ẋ(t) ∈ F (x(t)) -N C (x(t)) ∩ B 2m ⊂ F (x(t)) + 2m r x (
ẋ(t) ∈ (F (x(t)) -N C (x(t))) • = (F (x(t)) -N C (x(t)) ∩ B m ) • ⊂ F (x(t)) -N C (x(t)) ∩ B m ⊂ F (x(t)) + m r x(t) -A C,m (x(t)),
and x(•) is a solution of (5.76).

Conversely, let x(•) := x(•; x 0 ) be a solution of (5.76) on [0, T ] . So, according to Lemma 5.25, we only need to verify that x(t) ∈ C for all t ∈ [0, T ] . For this aim, we take ε > 0 such that F (x 0 ) e LT + Lε < m. Next, given y ∈ C ∩ B(x 0 , F (x 0 ) (e LT -1)) L + ε) and v ∈ F (y), we have v < m and, so by Lemma 5.25 it follows that

(v + m r y -A C,m (y)) • = (v -N C (y)) • = v -Π N C (y) (v) ∈ T C (y) = (N C (y)) * ; (5.80) that is, sup ξ∈N C (y) sup v∈F (y)+ m r x inf x * ∈A C,m (y)
ξ, v -x * ≤ 0.

Then, according to Theorem 5.11, there exists t ∈ (0, T ] such that for every solution y(•) = y(•; x 0 ) of (5.76) we have that y(t) ∈ C for all t ∈ [0, t]. Hence, by Lemma 5.25, for ae t ∈ [0, t]

ẋ(t) ∈ (F (x(t)) + m r x(t) -A C,m (x(t))) • = (F (x(t)) -N C (x(t))) • . (5.81)
In particular, ẋ(t) ≤ F (x(t)) for ae t ∈ [0, t].

(5.82)

In order to prove that we can take t = T we consider the nonempty set

S := {s ∈ [0, T ] | x(t) ∈ C for all t ∈ [0, s]}, (5.83)
which is obviously closed, due to the continuity of x(•) and the closedness of C.

Let the function η 2 be defined on [0, t] as η 2 (t) := x(t) -x 0 .

Then, as we did with function η above, by (5.82) we have for ae t ∈ [0, t],

η2 (t) ≤ ẋ(t) ≤ F (x(t)) ≤ F (x 0 ) + L x(t) -x 0 = F (x 0 ) +Lη 2 (t),
so that (by Gronwall's Lemma),

x(t) -x 0 = η 2 (t) ≤ F (x 0 ) (e Lt -1) L < F (x 0 ) (e Lt -1) L + ε;
that is, in particular, x(t) ∈ C ∩ B(x 0 , F (x 0 ) (e LT -1)) L + ε). So, by arguing as in the paragraph above (to get (5.76)), we find t 1 > 0 such that x(t + t) = x(t; x( t)) ∈ C for all t ∈ [0, t 1 ]. Hence, the set S is also open and so S = [0, T ] . Consequently, x(t) ∈ C for all t ∈ [0, T ] , as we wanted to prove. Now, we give the characterizations of a-Lypaunov pairs for (5.72).

Theorem 5.27. Let V, W : H → R ∪ {+∞} be two proper lower semi-continuous functions such that domV ⊂ C, W ≥ 0 and let a ≥ 0, and let ∂V stand for either ∂ P V or ∂ F V. Then the following are equivalent:

(i) (V, W ) is a strong a-Lyapunov pair for differential inclusion (5.72).

(ii) For every x ∈ domV ,

sup ξ∈∂V (x) sup v∈F (x) ξ, v -Π N C (x) (v) + aV (x) + W (x) ≤ 0. (iii) For every x ∈ domV , sup ξ∈∂V (x) sup v∈F (x) inf x * ∈N C (x)∩B F (x) ξ, v -x * + aV (x) + W (x) ≤ 0. (iv) For every x ∈ domV , sup v∈F (x) V (x; v -Π N C (x) (v)) + aV (x) + W (x) ≤ 0. Differential inclusions with Lipschitz Cusco perturbations (v) For every x ∈ domV , sup v∈F (x) inf x * ∈N C (x)∩B F (x) V (x; v -x * ) + aV (x) + W (x) ≤ 0.
Proof. The implications (ii) ⇒ (iii) and (iv) ⇒ (v) follow from the fact that, for all x ∈ C and v ∈ F (x),

Π N C (x) (v) = v -v -Π N C (x) (v) ≤ v ≤ F (x) .
The implications (iv) ⇒ (ii) and (v) ⇒ (iii) follow from the fact that

σ ∂ P V (x) (•) ≤ σ ∂ F V (x) (•) ≤ V (x; •).
(i) ⇒ (iv). Let us fix x 0 ∈ C and v ∈ F (x). According to Theorem 5.26(i), there exists a solution x(•; x 0 ) of differential inclusion (5.72) with x(0) = x 0 such that

d + x(0) dt = v -Π N C (x) (v)
, and the proof follows the same way as the one of (i) ⇒ (iv) in Corollary 5.19.

(iii) ⇒ (i). Let us fix T > 0, x 0 ∈ C and m > e LT F (x 0 ) , and denote by A C,m an arbitrary maximal monotone extension of the monotone operator N C (•) ∩ B m + m r Id (see Lemma 5.25), so that by Theorem 5.72(iii) every solution x(•; x 0 ) of (5.72) on [0, T ] is a solution of (5.76). Due to the Lipschitzianity of F we choose k ∈ (0, m) and ρ > 0 such that F (x) ∈ B k for all x ∈ x ∈ B(x 0 , ρ). So, the current hypothesis reads, for all x ∈ B(x 0 , ρ) ∩ domV sup

ξ∈∂V (x) sup v∈F (x) inf x * ∈N C (x)∩B k ξ, v -x * + aV (x) + W (x) ≤ 0,
which can also be written as (see Lemma 5.25)

sup ξ∈∂V (x) sup v∈F (x)+ m r x inf x * ∈A C (x)∩B k ξ, v -x * + aV (x) + W (x) ≤ 0,
where k := k + sup Let us first show that for any y ∈ B(x 0 , F (x 0 )

L (e LT -1)) ∩ domV, there exists a solution x(•; y) such that e at V (x(t; y))

+ t 0 W (x(τ ; y))dτ ≤ V (y) ∀t ∈ [0, T ]. (5.84)
where T is the positive number that satisfies the following equation in t te ct sup x∈B(x 0 ,

F (x 0 ) L (e LT -1)) ( F (x) + m r x ) = ρ 6 .
Indeed, fix y ∈ B(x 0 , F (x 0 )

L (e Lt -1)) ∩ domV . Then for every z ∈ B(y, ρ) ∩ domV , we have z ∈ B(x 0 , F (x 0 ) L (e Lt -1) + ρ), so that by the current hypothesis (ii)

sup ξ∈∂V (z) inf v∈F (z)+ m r z inf z * ∈N C (z)∩B F (z) + m r z ξ, v -z * + aV (z) + W (z) ≤ 0. But F (z) ≤ F (x 0 ) +L( F (x 0 )
L (e Lt -1) + ρ) < m, and so sup L (e LT -1)).

ξ∈∂V (z) inf v∈F (z)+ m r z inf z * ∈A C,m (z)∩B m + F (z)+ m r z ξ, v -z * + aV (z) + W (z) ≤ 0,

Differential inclusions with Lipschitz Cusco perturbations

Now, by Theorem 5.26 we know that every solution x(•; x 0 ) of (5.76) satisfies

x(t; x 0 ) ∈ B(x 0 , F (x 0 ) L (e LT -1)) ∀t ∈ [0, T ],
and, since T defined above does not depend on the points in B(x 0 , F (x 0 )

L (e LT -1)) ∩ domV , we prove as before the existence of a solution x(•; x 0 ) of (5.76) such that

e at V (x(t; x 0 )) + t 0 W (x(τ ; x 0 ))dτ ≤ V (x 0 ) ∀t ∈ [0, T ].
(i) ⇒ (iii). Let us fix x 0 ∈ domV ⊂ C and T > 0. By the current hypothesis (i), there exists a solution x(•; x 0 ), t ∈ [0, T ] of differential inclusion (5.72) such that

e at V (x(t; x 0 )) + t 0 W (x(τ ; x 0 ))dτ ≤ V (x 0 ).
(5.85)

But by Theorem 5.26(ii), x(•; x 0 ) is also a solution of differential inclusion

ẋ(t) ∈ F (x(t)) -N C (x(t)) ∩ B F (x(t)) a.e. t ≥ 0, x(0) = x 0 ∈ C.
Hence, x(•; x 0 ) is Lipschitz on [0, T ], and we may take v ∈ limsup t↓0 x(t;x 0 )-x 0 t

. So, by the Lipschitzian property of F , and the upper semicontinuity of the mapping x → N C (x)∩B F (x) , we get v ∈ F (x 0 )-N C (x 0 )∩B F (x 0 ) . Then, if v = lim tn↓0 x(tn;x 0 )-x 0 tn , by using (5.85) we get

V (x 0 ; v) ≤ lim inf n→∞ V (x(t n ; x 0 )) -V (x 0 ) t n ≤ lim inf n→∞ - e atn -1 t n V (x(t n ; x 0 )) - 1 t n tn 0 W (x(τ ; x 0 )dτ ≤ -aV (x 0 ) -W (x 0 ),
which shows that (iii) holds.

Examples

In this last section we consider a couple of examples, one on selector-linear differential inclusions, and the other one on the minimal time function.

Examples

First, we consider the following differential inclusion

   ẋ(t) ∈ F (x(t)) -N C (x(t)) a.e. t ≥ 0, x(0) = x 0 ∈ C;
(5.86)

where A i : R n → R n , i = 1, ..., k are linear mappings,

F (x) = co{A i x : i = 1, ..., k} = {Ax : A ∈ co(A 1 , ..., A k )},
and C is an r-uniformly-prox-regular set of R n such that . It is easy to see that F is an L-Lipschitz mapping with L := max{ A i , i = 1, ..., k}.

We apply the results of the previous section to study θ ∈ C the stability of differential inclusion (5.86).

Proposition 5.29. Let δ > 0 and β ∈ (0, rδ L ) be given.

(i) If for every x ∈ C, there exists i ∈ {1, ..., k} such that

x, (A i + A T i )x ≤ -δ x 2 ,
then for every x 0 ∈ C ∩ int(B β ), there exists a solution x(•; x 0 ) of (5.86) such that x(t; x 0 ) ≤ e -1 2 (δ-βL r )t x 0 ∀t ≥ 0.

(ii) If for every i ∈ {1, ..., k} A i + A T i ≤ -δId, then for every x 0 ∈ C ∩ int(B β ) and any solution x(•; x 0 ) of (5.86), one has

x(t; x 0 ) ≤ e -1 2 (δ-βL r )t x 0 ∀t ≥ 0.
Consequently, for any ε > 0, x 0 ∈ int(B β ) ∩ C, and solution of (5.86), there exists t > 0 such that x(t; x 0 ) ≤ ε ∀t ≥ t.

Proof. Let us consider the function

V (x) := 1 2 x 2 + I C (x).
(5.87)

Differential inclusions with Lipschitz Cusco perturbations

It is easy to see that V is lower semi-continuous and for every x ∈ C, one has According to Theorem 5.26, for any y ∈ int(B β )∩C differential inclusion (5.86) is equivalent to the following differential inclusion

∂ F V (x) = x + N C (x). ( 5
   ẋ(t) ∈ F (x(t)) -A C,m (x(t)) a.e t ∈ [0, 1], x(0) = y ∈ int(B β ) ∩ C; where A C,m is a maximal monotone extension of the mapping x → N C (x)∩B m + m r x (see Lemma 5.25). (i) We take x ∈ C and let i ∈ {1, • • • , k} be such that x, (A i +A T i )x ≤ -δ x 2 . Fix ξ ∈ ∂ F V (x) (= x + N C (x)). Since A i x -Π N C (x) (A i x) ∈ T B C (x)
, from the definition of r-uniformly-prox-regularity, and the fact that θ ∈ C, one gets

-x, Π N C (x) (A i x) ≤ Π N C (x) (A i x) 2r x 2 ≤ A i x 2r x 2 , (5.89)
and so, using (5.88),

ξ, A i x -Π N C (x) (A i x) ≤ x, A i x -Π N C (x) (A i x) ≤ 1 2 x, (A i + A T i )x + -x, Π N C (x) (A i x) ≤ - δ 2 x 2 + A i x 2r x 2 ≤ 1 2 L x r -δ x 2 ; that is, for every x ∈ int(B β ) ∩ domV and ξ ∈ ∂ P V (x) ξ, A i x -Π N C (x) (A i x) + (δ - Lβ r )V (x) ≤ 0;
(5.90) moreover, since A i x ≤ Lβ < m, and so (by Lemma 5.25) (ii). We take x ∈ C, v ∈ F (x), and ξ ∈ ∂ P V (x), where V is defined in (5.87); hence, v = k i=1 α i A i x for some α i such that k i=1 α i = 1, and v ≤ L x . As in the proof of statement (i) above we get

A i x -Π N C (x) (A i x) = (A i x + m r x -A C,m (x)) • ∈ A i x + m r x -A C,m (x) ∩ B k , 5
ξ, v -Π N C (x) (v) ≤ x, v -Π N C (x) (v) ≤ k i=1 1 2 α i x, (A i + A T i )x + -x, Π N C (x) (v) ≤ - δ 2 x 2 + Π N C (x) (v) 2r x 2 ≤ 1 2 Π N C (x) (v) r -δ x 2 ≤ 1 2 v r -δ x 2 ≤ 1 2 ( L x r -δ) x 2 ,
and we conclude as above.

We recall differential inclusion (5.72),

   ẋ(t) ∈ F (x(t)) -N C (x(t)) a.e. t ≥ 0, x(0) = x 0 ∈ C.
where as before F : R n ⇒ R n is an L-Lipschitz Cusco mapping and C is an r-uniformly-prox-regular set of R n . Given a closed set S ⊂ C, we consider the minimum time function T : C → R ∪ {+∞} defined as T (z) := inf{t ≥ 0 : ∃ solution x(•) of (5.72) st. x(0) = z, x(t) ∈ S}.

(5.92)

The following lemma gathers some easy facts on the function T, which we call a minimal time function. (ii) If T (z), z ∈ C, is finite, then there exists a solution x(•; z) of (5.72) such that x(T (α); x 0 ) ∈ S.

(iii) For every solution x(•; x 0 ) of (5.72) we have T (x(s; x 0 )) + s ≤ T (x(t; x 0 )) + t, for all 0 ≤ s ≤ t, and the equality holds for optimal trajectories.

(iv) If t > 0 is such that t < T (z) < +∞, z ∈ C, then there exists ε > 0 such that [S + εB] ∩ {x(s; x 0 ), s ∈ [0, t], x(•; x 0 ) solution of (5.72)} = ∅.
Proof. Statements (i) and (iii) are clear and follow easily from the definition of the function T.

To prove (ii) we assume that T (z) < +∞, and let x n (•; z) be a sequence of solutions of (5.72) such that x n (t n ; z) ∈ S for some sequence t n ↓ T (z). Then, as in the proof of Lemma 5.15(ii), we may suppose that x n (•; z) uniformly converges to a solution x(•; z) on [0, T (z) + 1]. Hence, from the closedness of the set S we obtain that x(T (z); z) ∈ S.

To prove (iv) we proceed by contradiction and assume that there are sequence of solutions (x n (•; z)) n of (5.72) and sequence (t n ) n such that t n ≤ t and d(x n (t n ; z), S) → 0. Without lost of generally, we can suppose that (x n (•; z)) n uniformly converges to a solution x(•; z) on [0, t] (see the proof of Lemma 5.15 (ii)). It follows that x(s; z) ∈ S for some s ∈ [0, t], which contradicts the fact that t < T (x 0 ).

We now consider the differential inclusion

   ( ẋ(t), ẏ(t)) ∈ (F (x(t)) -N C (x(t)), 1) a.e. t ≥ 0; (x(0), y(0)) = (x 0 , α) ∈ C × R,
(5.93) 5.8. Examples so that from Lemma 5.30 it follows that the function (x, y) → T (x) + y is a weak Lyapunov function for (5.93), while the function (x, y) → -T (x) -y, where

T (x) :=    -T (x) if x ∈ C +∞ if x / ∈ C, (5.94)
is a strong Lyapunov function for differential inclusion (5.93).

We get the following result (also see [START_REF] Colombo | The minimum time function for the controlled Moreau's sweeping process[END_REF]).

Proposition 5.31. Suppose that the minimum time function T as defined in (5.92) is continuous on C. Then T is the unique continuous function such that

T (x) = 0 for all x ∈ S, T (x) > 0 for all x ∈ C \ S, (5.95)
and, for any x ∈ C \ S,

sup ξ∈∂ P T (x) sup v∈F (x) inf x * ∈N C (x)∩B F (x) ξ, v -x * -1 ≤ 0 (5.96) sup ξ∈∂ P T (x) inf v∈F (x) inf x * ∈N C (x)∩B F (x) ξ, v -x * + 1 ≤ 0. ( 5.97)
Proof. By Lemma 5.30 and the paragraph before the current proposition, the minimum time function T as defined in (5.92) satisfies T (x) = 0 for x ∈ S and T (x) > 0 for x ∈ C \S, and the functions (x, y) → T (x)+y and (x, y) → -T (x)-y (see (5.94)) are respectively weak and strong Lyapunov functions for (5.93). Then, T and T also satisfy (5.96) and (5.97), thanks to Theorems 5.27 and 5.28. Now, let V a continuous that satisfies (5.95), (5.96) and (5.97). We proceed by steps:

Step [START_REF] Acary | Nonsmooth modeling and simulation for switched circuits[END_REF]. We prove in this step that V (x) ≥ T (x) for all x ∈ C \ S. We fix x 0 ∈ C \ S and denote t := sup{t ≥ 0 : ∃x(•; x 0 ) solution of (5.72) st. V (x(s; x 0 ))+s ≤ V (x 0 ) ∀s ∈ [0, t]}.

Then, due to (5.97), by Theorem 5.28 there exist t > 0 and solution x(•; x 0 ) of (5.72) such that V (x(s;

x 0 )) + s ≤ V (z) ∀s ∈ [0, t],
so that t > 0. Moreover, if sequences t n t and (x n (•; x 0 )) n are such x n (•; x 0 ) is a Chapter 6

Boundary of maximal monotone operators values

We characterize the boundary of the values of maximal monotone operators defined in Hilbert spaces, by means only of the values at nearby points, which are closed enough to the reference point but distinct of it. This allows to write the values of such operators using finite convex (2-)combinations of the values at such nearby points. We also provide similar characterizations for the normal cone to proxregular sets. for compact space T and continuous functions a and b on T . The characterization above was the main ingredient in [START_REF] Cánovas | Boundary of subdifferentials and calmness moduli in linear semi-infinite optimization[END_REF][START_REF] Cánovas | Outer limit of subdifferentials and calmness moduli in linear and nonlinear programming[END_REF][START_REF] Cánovas | Calmness of the feasible set mapping for linear inequality systems[END_REF] to derive point-based explicit expressions for the so-called calmness moduli of the associated feasible and optimal solutions set-valued mappings; we refer to [START_REF] Henrion | On the calmness of a class of multifunctions[END_REF][START_REF] Henrion | Calmness of constraint systems with applications[END_REF][START_REF] Klatte | Nonsmooth equations in optimization: regularity, calculus, methods and applications[END_REF] for more details on this calmness property. For instance, if and whose subdifferential mapping can be easily estimated by means only of the data vectors a and b. From a qualitative point of view, the calmness of the mapping F a , say clmF a ( b, x) > 0, is equivalent to the fact that the function s has an (global) error bound at x (see [START_REF] Kruger | Stability of error bounds for semiinfinite convex constraint systems[END_REF][START_REF] Kruger | Stability of error bounds for convex constraint systems in Banach spaces[END_REF]).

Introduction

At this stage, if, in addition, the set F a ( b) turns to be the singleton {x}, in which case s(x) > 0 iff x = x, then formula (6.1) goes into the play and entails a point-based expression of the calmness modulus of the mapping F a , that is given by clmF a ( b, x) = (d * (0, bd(∂s(x)))) -1 .

It is worth observing that in the framework of semi-infinite linear programming problems, this singleton's assumption is required for the solutions set-valued mapping and not for the feasible set-valued mapping (see [START_REF] Cánovas | Boundary of subdifferentials and calmness moduli in linear semi-infinite optimization[END_REF][START_REF] Cánovas | Outer limit of subdifferentials and calmness moduli in linear and nonlinear programming[END_REF][START_REF] Cánovas | Calmness of the feasible set mapping for linear inequality systems[END_REF] for more details).

For the aim of adapting this kind of analysis in a further research to more general semi-infinite linear programming problems with a non-necessarily compact index set T, so that the function s above lacks to be continuous, we extend in this paper formula (6.1) to the class of proper and lower semi-continuous convex functions. More generally, we establish similar characterizations for maximal monotone operators in the setting of Hilbert spaces. The first result given in Boundary of maximal monotone operators Theorem 6.5 asserts that given a maximal monotone operator A : H ⇒ H, for all x ∈ H we have that bd(A(x)) =Limsup where the Limsup is taken with respect to the norm. As a consequence, we prove that the value of A at x can be expressed using only different nearby points in the sense that for every x ∈ H such that bd(Ax) = ∅ it holds (Theorem 6.12)

A(x) = N cl(domA) (x) + co 2 Limsup y→ = x A(y) ,
where co 2 is the set of all the segments generated by the elements of the underlying set, and N cl(domA) (x) is the normal cone in the sense of convex analysis to the closure of the domain of the operator A. Characterizations of similar type are given for the faces of the values of A, see Theorem 6.9. Extensions to nonconvex objects, as prox-regular sets and functions, is also considered in Theorems 6.15 and 6.18. This paper is organized as follows: After Section 6.2, dedicated to present the necessary notations and the preliminary tools, we give the main result in Section 6.3: Theorem 6.5 characterizes the boundary of the values of maximal monotone operators, while Theorem 6.12 recovers the values of such operators using these boundary points. Theorem 6.9 specifies such characterizations to the faces of the values of maximal monotone operators. In Section 6.4 we extend this analysis to non-convex objects, which are the normal cone to prox-regular sets (Theorem 6.15) and the subdifferential of prox-regular functions with uniform parameters (Theorem 6.18).

Notations and preliminary results

In this paper, H is a Hilbert space endowed with inner product •, • and associated norm ||•||. The weak topology on H is denoted by ω, while the strong and weak convergences in H are denoted by → and , resp. We denote by B(x, ρ) the closed ball with center x ∈ H and radius ρ > 0; in particular, we write B ρ := B(θ, ρ). The null vector in H is denoted θ. Given a set S ⊂ H, co(S) and co 2 (S) are respectively the convex hull of S and the set

Boundary of maximal monotone operators

In this section, we give the desired property which expresses a given maximal monotone operator A : H ⇒ H, defined on a Hilbert space H, by means of its values at nearby points. Since A(x), x ∈ domA, is convex and closed, A(x; •) coincides with the subdifferential mapping of the proper, convex and lsc support function oe A(x) . As a consequence, the following remark resumes some easy properties of the set A(x; v). Remark 6.2. Given x ∈ domA and v ∈ H, we have: (i) A(x; v) is convex and closed (possibly empty), and nonempty whenever the set A(x) is bounded.

(ii) A(x; θ) = A(x), and if v = θ then A(x; v) is a subset of bd(A(x)). In the last case, we refer to A(x; v) as the face of A(x) with respect to the direction v.

(iii) A(x; αv) = A(x; v) for any v = θ and α > 0; thus, the face A(x; v) depends only on the direction v.

We shall need the following lemma. Thus, taking into account Theorem 6.5, we may suppose that y * n → x * ∈ bd(A(x)); that is, d(θ, bd(A(x))) ≤ x * ≤ α.

We get the desired inequality " ≤ " when α goes to lim inf To prove the last statement, we observe that under the current assumption, we have that A • (x) = d(θ, A(x)) = d(θ, bd(A(x))), and so it suffices to use the first statement of the theorem. We assume now that H is finite-dimensional, so that according to the first statement we only need to prove that A(x) ≥ lim sup To see the first inclusion, we take x * ∈ Limsup w v, t↓0

A(x + tw), so that x * = lim n x * n for some sequences (x * n ), (w n ) ∈ H, (t n ) ⊂ R + , such that x * n ∈ A(x + t n w n ), w n v, and t n ↓ 0. It follows by the maximal monotonicity of A that x * ∈ A(x), and for all ξ ∈ A(x)

x * n -ξ, w n = 1 t n x * n -ξ, x + t n w n -x ≥ 0. So, by taking the limit as n → +∞ we obtain that x * , v ≥ sup ξ∈A(x) ξ, v ≥ x * , v , which shows that x * ∈ A(x; v), and the first inclusion in (6.5) follows. We conclude the proof of the theorem because the second inclusion in (6.5) can be obtained using the same arguments as in the first inclusion.

The following example shows the necessity of moving the vector v in the expression of Theorem 6.9. Boundary of maximal monotone operators deduce that x * -x * 0 ∈ N cl(domA) (x), and so we get

x * ∈ x * 0 + N cl(domA) (x) ⊂ N cl(domA) (x) + co 2 {bd(A(x))}, which yields (6.7). On the other hand, if S ∩ bd(A(x)) = ∅, then there exists some t > 1 such that z * = x * 0 + t(x * -x * 0 ) ∈ bd(A(x)). Thus, we get

x * = 1 t z * + (1 -1 t )x * 0 ∈ co 2 {bd(A(x))} ⊂ N cl(domA) (x) + co 2 {bd(A(x))}, and this completes the proof of the theorem.

Prox-regular analysis

In this section, we extend the results of the previous section to two classes of operators of nonsmooth analysis, the normal cone to uniformly r-prox-regular sets, and the class of prox-regular extended-real-valued functions with uniform parameters. As before, we work in the setting of a given Hilbert space H.

We start by giving the definition of the proximal normal cone. Definition 6.13.

( [START_REF] Clarke | Nonsmooth Analysis and Control Theory[END_REF]) Given a set C ⊂ H and x ∈ C, the proximal normal cone to C at x, denoted by N P C (x), is the set of vectors x * ∈ H for which there exists m > 0 such that

x * , y -x ≤ m y -x 2 for all y ∈ C. Definition 6.14. ( [START_REF] Mazade | Regularization of differential variational inequalities with locally prox-regular sets[END_REF]) For positive numbers r and α, a closed set C is said to be (r, α)-prox-regular at x ∈ C provided that one has x = Π C (x + v), for all x ∈ C ∩ B(x, α) and all v ∈ N P C (x) such that ||v||< r. The set C is r-prox-regular (resp., prox-regular) at x when it is (r, α)-prox-regular at x for some real α > 0 (resp., for some numbers r, α > 0). The set C is said to be r-uniformly prox-regular when α = +∞.

The following theorem describes the boundary set of the normal cone of a uniformly r-prox-regular set, by means of its values at nearby points, which are different from the reference point. We also characterize such normal cone by means of their boundaries points. Recall that the Bouligand tangent cone of a • for all t ∈ [0, 1[ (see [START_REF] Adly | A convex approach to differential inclusions with prox-regular sets: Stability analysis and observer design[END_REF]Theorem 4.6] for more details).

We are going to prove the converse inclusions of (6.10). We take ξ ∈ Limsup y→ = x N P C (y), and let the sequences (y n ) and (ξ n ) be such that ξ n ∈ N P C (y n ), y n → x, ξ n → ξ as n → +∞; hence, we may suppose that for some M > 0 we have that ξ n ∈ N P C (y n ) ∩ B M for all n ∈ N. Next, using the r-uniform prox-regularity of the set C, we obtain that ξ ∈ N P C (x) ( [START_REF] Mazade | Regularization of differential variational inequalities with locally prox-regular sets[END_REF]). We claim that ξ ∈ bd(N P C (x)). Proceeding by contradiction, we assume that for some positive number ρ such that ρ < M it holds ξ + B ρ ⊂ N P C (x); that is, ξ + ρ y n -x y n -x ∈ N P C (x) ∀n ∈ N. Now, using the monotonicity of the mapping x → N P C (x) ∩ B 2M + 2M r x (see [START_REF] Mazade | Regularization of differential variational inequalities with locally prox-regular sets[END_REF]), 6.4. Prox-regular analysis of f at x, written x * ∈ ∂ P f (x), if there are ρ, δ > 0 such that f (y) ≥ f (x) + x * , y -x -δ||y -x|| 2 , ∀y ∈ B(x, ρ).

A vector x * ∈ H is called limiting subgradient of f at x, written ξ ∈ ∂ L f (x), if there are sequence (x k ), (x * k ) ⊂ H such that

x * = ω -lim k→∞ x * k , x k -→ x, f (x k ) -→ f (x), x * k ∈ ∂ P f (x k ).
Definition 6.17 It is worth observing that for prox-regular functions with uniform parameters f at x ∈ domf , we have that ∂ P f (x) = ∂ L f (x), and, in particular, if f is convex, then ∂ P f (x) = ∂f (x). In the following result, we give the counterpart of Theorem 6.5 to the proximal subdifferential mapping of prox-regular functions. To prove the second statement we observe that domf ∩ U = domg ∩ U , which yields N domf (x) = N domg (x). Thus, sine bd(∂g(x)) = bd(∂ P f (x)) + rx = ∅ due to the current assumption, by applying Theorem 6.12 and taking into account (6.12) we get where we used the fact that cl(dom∂g) = cl(domg) (see, e.g. [START_REF] Zȃlinescu | Convex Analysis in General Vector Spaces[END_REF]). In this PhD thesis, we make some contributions to nonsmooth Lyapunov stability of first-order differential inclusions with maximal monotone operators, in the setting of infinite-dimensional Hilbert spaces. We provide primal and dual explicit characterizations for parameterized weak and strong Lyapunov pairs of lower semicontinuous extended-real-valued functions, referred to as a-Lyapunov pairs, associated to differential inclusions with right-hand-sides governed by Lipschitz or Cusco perturbations F of maximal monotone operators A, ẋ(t) ∈ F (x(t)) -A(x(t)) t ≥ 0, x(0) ∈ domA.

∂ P f (x) =
Equivalently, we study the weak and strong invariance of sets with respect to such differential inclusions. As in the classical Lyapunov approach to the stability of differential equations, the presented results make use of only the data of the differential system; that is, the operator A and the multifunction F , and so no need to know about the solutions, nor the semi-groups generated by the monotone operators. Because our Lyapunov pairs and invariant sets candidates are just lower semicontinuous and closed, respectively, we make use of nonsmooth analysis to provide first-order-like criteria using general subdifferentials and normal cones.

We provide similar analysis to non-convex differential inclusions governed by proximal normal cones to prox-regular sets. Our analysis above allowed to prove that such apparently more general systems can be easily coined into our convex setting. We also use our results to study the geometry of maximal monotone operators, and specifically, the characterization of the boundary of the values of such operators by means only of the values at nearby points, which are distinct of the reference point. This result has its application in the stability of semiinfinite programming problems. We also use our results on Lyapunov pairs and invariant sets to provide a systematic study of Luenberger-like observers design for differential inclusions with normal cones to prox-regular sets. The thesis is organized as follows: In chapter 1, we explain the main objectives of the thesis, the methodology that we follow, and we give a preview of the main results. We also make in this chapter a general overview of Lyapunov's theory, and present the main previous achievements on the subject. In Chapter 2, we present the main tools and preliminary results that we need in our analysis. In Chapter 3, we give the desired characterizations of Lyapunov pairs and invariant sets for differential inclusions with Lipschitz perturbations of maximal monotone operators, while in Chapter 4, we investigate differential inclusions with Lipschitz perturbations of proximal normal cones. This chapter includes the application to Luenberger-like observers design. In Chapter 5, we study differential inclusions with Lipschitz Cusco perturbations of maximal monotone operators. In Chapter 6, we give a result on the geometry of maximal monotone operators, and describe the boundary of their values. Finally, we give in Chapter 7 a resume of the results we obtained.

  (y)∩B(θ,m)

1. 1 .

 1 General introduction where T C (x) is the tangent cone in the sense of convex analysis. These conditions have a clear geometrical meaning (see the figure below) (ii) For every x ∈ S ∩ domA sup ξ∈N S (x)

  (i ) For any x ∈ domV ∩ domA sup ξ∈∂V (x) ξ, (f (x) -A(x)) • + aV (x) + W (x) ≤ 0, where ∂ stands for either the proximal subdifferential ∂ P or the Fréchet subdifferential ∂ F . (ii ) For any x ∈ domV ∩ domA sup ξ∈∂V (x) inf x * ∈A(x) ξ, f (x) -x * + aV (x) + W (x) ≤ 0. (iii ) For any x ∈ domV ∩ domA inf v∈A(x)

Definition 2 . 4 .

 24 (i) A subset S is called convex if for any two points x, y ∈ S and any α ∈ [0, 1], one has αx + (1 -α)y ∈ S.

Definition 2 . 22 .

 222 A function x : [a, b] → H is said to be absolutely continuous if

  ii) x(•) is differentiable almost everywhere and the right derivative d + x(

  2r), and, since S = S ∩ domA, for every y ∈ B(x, r) ∩ S m , N Sm (y) = N Sm∩B(x,2r) (y) = N S∩domA∩B(x,2r) (y) = N S∩domA (y) = N S (y).

3. 3 .

 3 Invariant sets So (iv) gives us, for every y ∈ B(x, r) ∩ S m , sup ξ∈N Sm (y) inf x * ∈(f (y)-A(y))∩B(θ,m)

. 22 )

 22 Take y := (y 1 , µ 1 ) ∈ B(y 0 , r) ∩ S m , with y 1 := (x 1 , α 1 , β 1 , γ 1 ), and pick (ξ, -κ) ∈ N P Sm (y). Due to (3.21) and [30, Exercise 1.2.1],

Example 4 . 2 .

 42 (Parabolic Variational Inequalities). Let Ω ⊂ R N be an open bounded subset with a smooth boundary ∂Ω. Let us consider the following boundary value problem, with Signorini conditions, of finding a function (t, x) → u = u(t, x) such that (P )

Example 4 . 3 .

 43 (Nonlinear Differential Complementarity Systems). Let us consider the following ordinary differential equation, coupled with a complementarity condition, (N DCS)

Theorem 4 .

 4 7 provides a criterion for the invariance of closed sets satisfying the relation S = S ∩ domA. (4.5)

Proof.

  The equivalence (a) ⇐⇒ (b) is given in [73, Theorem 4.1], while the implication (c) =⇒ (b) is immediate. Then we only have to prove that (b) =⇒ (c). If (b) holds, we choose a maximal monotone operator A, which extends the monotone mapping N P C ∩ B(θ, m) + m

. 24 )

 24 Indeed, the first inclusion is obvious. If x ∈ C and ξ ∈ A(x), then for any y ∈ C we have m r y ∈ A(y) (since 0 ∈ N P C (y) ∩ B(θ, m)) and, so, ξ -m r y, x -y ≥ 0. This implies ξ -m r x, y -x ≤ m r ||y -x|| 2 , which proves that ξ -m r x ∈ N C (x), for every ξ ∈ A(x). Hence, A(x) ⊂ N P C (x) + m r x.

Theorem 4 . 14 .

 414 System (4.1) has a unique solution x(•, x 0 ) starting at x 0 ∈ C, which is Lipschitz on every bounded interval.

4. 5 .

 5 Lyapunov stability analysis (iii) for every x ∈ domV and ξ ∈ ∂V (x); min x * ∈N C (x)∩B(θ,||f (x)||)

  2 max{||f (x 0 )||e lT , ||f (x 0 )||+lT e lT ||f (x 0 )||+l + 1}; m := k + k r (||x 0 ||+ρ). Thanks to Lemma 4.18 we shall assume in what follows that W is Lipschitz continuous on B(x 0 , ρ). As before we denote x(•) the solution of (4.1) on [0, T ] satisfying x(0) = x 0 . According to Theorem 4.15, for a.e. t ∈ [0, T ] we have || ẋ(t)||≤ ||f (x(t))|| and, due to the l-Lipschitzianity of f,

  so that domV ∩ C = {(1, 0)}. For x := (1, 0) and ξ := (x, y) ∈ ∂ P V (x) = {(x, y)| y ≤ 0} we have that min x * ∈N C (x)∩B(θ,||f (x||)

Lemma 4 .

 4 23 ([78]). Consider a nonempty, closed, r-prox-regular set S such that S is contained in the range space of a linear mapping D : R n → R l . Then the set D -1 (S) is r -uniformly prox-regular with r := rδ + D ||D|| 2 , where δ + D denote the least positive singular value of the matrix D.

(4. 61 )

 61 Applying Theorem 4.25 to system (4.57) with f = RAR -1 , C = S , r = r , we get lim t→∞ z(t; z 0 ) = 0,

4. 6 .

 6 Stability and observer designs observer such that lim t→∞ ||x(t; z 0 )-x(t; x 0 )||= 0, for all z 0 ∈ B(x 0 , ρ)∩D -1 (S) for some ρ > 0. (4.[START_REF] Mazade | Regularization of differential variational inequalities with locally prox-regular sets[END_REF] We see that if x(•) := x(•; z 0 ) is the solution of (4.62), then it is also the solution of the differential inclusion ẋ(t) ∈ (A -LG)x(t) + Ly(t) -BN S (Dx(t)) a.e. t ≥ 0, x(0) = z 0 .(4.64)Under the hypothesis ∃ P symmetric positive definite, such that P B = D T , (4.65)

Lemma 5 . 4 .Theorem 5 . 5 .

 5455 Let G : R n ⇒ R n be a Lipschitz multifunction with nonempty, convex and compact values, and let x ∈ R n , v ∈ G(x). Then there exists a Lipschitz selection f of G such that f (x) = v. Differential inclusion (5.1) has at least one solution.

Theorem 5 . 13 . 5 . 4 .

 51354 Let S ⊂ domA be a closed set such that A • is locally bounded on S. Then the following statements are equivalent provided that T S and N S are the same as the ones in Theorem 5.11: Strong and weak invariant sets (i) S is weak invariant for differential inclusion (5.1).

. 32 ) 5 . 15 .

 32515 For k ≥ 1 we denote W k (x) := inf z∈R n {W (z) + k x -z }. (5Lemma Given a function W : R n → R + ∪{ + ∞}, W k defined in (5.32) is k-Lipschitz, and we have W k (x) W (x) for all x ∈ R n . Moreover, the following assertions hold true for every T > 0 and x 0 ∈ domV : (i) If x(•; x 0 ) is a solution of differential inclusion (5.1), then W satisfies inequality (5.31) iff W k does for all k ≥ 1.

  x 0 ) converges uniformly to a continuous function x(•) on [0, T ]. It follows that ẋ(t) = v(t) a.e. t ∈ [0, T ], and so, since the mapping F -A ∩ B M has closed graph, as k → +∞ in (5.33) we get ẋ(t) ∈ F (x(t)) -A(x(t)) ∩ B M a.e. t ∈ [0, T ];

Lemma 5 . 16 .

 516 Consider the operator  : R n × R 3 → R n+3 and the function 5.5. Strong a-Lyapunov pairs Ṽ : R n+1 × R + → R ∪ {+∞} defined as Â(x, α, β, γ) := (A(x), θ R 3 ), Ṽ (x, α, β) := e aβ V (x) + α,(5.35)

  , and let ∂ stand for either ∂ P or ∂ F . Then the pair (V, W ) is a strong a-Lyapunov pair for (5.1) iff for all x ∈ domV sup ξ∈∂V (x)sup v∈F (x) inf x * ∈A(x) ξ, v -x * + aV (x) + W (x) ≤ 0,(5.45)sup ξ∈∂ P,∞ V (x) sup v∈F (x) inf x * ∈A(x)ξ, v -x * ≤ 0.(5.46)

t 0 W

 0 (x(τ ; x 0 ))dτ ≤ V (x 0 ) ∀t ∈ [0, T ]. (5.47) It suffices to prove that the following quantity is +∞, T := sup{s ≥ 0 : (5.47) holds ∀t ∈ [0, s]}.

  , and let ∂ stand for either ∂ P , ∂ F , or ∂ L . If A • is locally bounded on domV , then (V, W ) is a strong a-Lyapunov pair for (5.1) iff one of the following statements holds. (i) For any x ∈ domV , sup ξ∈∂V (x) sup v∈F (x) inf x * ∈A(x)∩B F (x) +m(x) ξ, v -x * + aV (x) + W (x) ≤ 0.

Theorem 5 . 23 .

 523 Let V, W, and a ≥ 0 be as in Proposition 5.22, and let ∂ stand for either ∂ P , ∂ F , or ∂ L . Under the local boundedness of A • on domV , (V, W ) is a weak a-Lyapunov pair for (5.1) iff one of the following assertions holds:(i) For every x ∈ domV , sup ξ∈∂V (x) inf v∈F (x) inf x * ∈A(x)∩B m(x)+ F (x) ξ, v -x * + aV (x) + W (x) ≤ 0.5.6. Weak a-Lyapunov pairs (ii) For every x ∈ domV , inf v∈F (x) inf x * ∈A(x)∩B F (x) +m(x)

. 70 )

 70 As we proceeded many times we can show that T := sup{s ≥ 0 | ∃ x(•; x 0 ) solution of (5.1) st. (5.70) holds ∀t ∈ [0, s]} = +∞, which ensures that (V, W ) is a weak a-Lyapunov pair for (5.1).

. 76 )

 76 where A C,m : R n ⇒ R n is any maximal monotone extension of N C (•) ∩ B m + m r Id. Consequently, every solution x(•; x 0 ) of (5.72) on [0, T ] satisfies

x∈B(x 0

 0 ,ρ) m r x . Consequently, (i) follows from Corollary 5.19. Theorem 5.28. Let V, W : H → R ∪ {+∞} be two proper lower semi-continuous functions such that domV ⊂ C, W ≥ 0 and let a ≥ 0. Let ∂V stand for either ∂ P V or ∂ F V. Then the following statements are equivalent:(i) (V, W ) is a weak a-Lyapunov pair for differential inclusion (5.72).5.7. Differential inclusions with prox-regular sets(ii) For every x ∈ domV , sup ξ∈∂V (x) inf v∈F (x) inf x * ∈N C (x)∩B F (x) (iii) For every x ∈ domV , inf v∈F (x) inf x * ∈N C (x)∩B F (x) V (x; v -x * ) + aV (x) + W (x) ≤ 0.Proof. (iii) ⇒ (ii) follows from inequality (5.7). (ii) ⇒ (i). Let us fix x 0 ∈ domV ⊂ C and T > 0. We choose m, ρ > 0 such that m > e LT F (x 0 ) + Lρ and a maximal monotone extension A C,m of the monotone operator N C (•)∩B m + m r Id; hence, according to Theorem 5.26, differential inclusion (5.72) is equivalent to differential inclusion (5.76) on [0, T ].

  Hence,(5.84) follows from Proposition 5.22 by taking into account that T ≤ T c,ρ (y) for all y ∈ B(x 0 , F (x 0 )

F

  .88) Now we chose m > rδe L , so that m > rδe L ≥ e L sup x∈B β (x) .

. 8 . 1 2 1 2

 811 Exampleswith k := 2Lβ + m r β, we also have that infv∈F (x)+ m r x inf x * ∈A C,m (x)∩B k ξ, v -x * + (δ -Lβ r )V (x) ≤ 0. (5.91) Now, we choose T > 0 such that 3( F (x 0 ) + A C (x 0 ) )T e cT < 3β(L + m r )T e cT < β -x 0 .So, according to Proposition 5.22, by (5.91) there exists a Lipschitz solution x(•; x 0 ) of (5.76) on [0, 1] such thate (δ-Lβ r )t x(t; x 0 ) ≤ x 0 ∀t ∈ [0, T ]. Also, since x(T ; x 0 ) ∈ B x 0 ∩ C ⊂ int(B β ) ∩ C, we can find (by extending the current solution) a solution x(•; x 0 ) of (5.76) such that e (δ-Lβ r )t x(t; x 0 ) ≤ x 0 ∀t ≥ 0.

Lemma 5 .

 5 30. (i) T (z) = 0 if z ∈ S, and T (z) > 0 if z ∈ C \ S.

  Given a continuous convex function ϕ : R n → R, according to[START_REF] Cánovas | Outer limit of subdifferentials and calmness moduli in linear and nonlinear programming[END_REF] Theorem 3.1] the topological boundary of the Fenchel subdifferential of ϕ is completely characterized by means of the values of such subdifferential mapping at points, which are closed enough to the reference point but distinct of it. More specifically, for every x ∈ R n we have that bd(∂ϕ(x)) =Limsupy-→ = x ∂ϕ(y). (6.1)This characterization has been shown useful for many stability purposes of parametrized semi-infinite linear programming problems, given in R n as ([START_REF] Goberna | Linear semi-infinite optimization[END_REF])P (c, a, b) : minimize c x subject to a t x ≤ b t , t ∈ T,

1 ,

 1 F a : C(T, R) → R n denotes the feasible set-valued mapping, F a (b) := {x ∈ R n : a t x ≤ b t ∀t ∈ T }, then the calmness modulus of F a at a point ( b, x) in its graph, given implicitly as clmF a ( b, x) := lim sup x→x, b→ b x∈Fa(b) d(x, F a ( b)) d(b, b) , is rewritten in the more explicit form (using the convention 1 0 = +∞) clmF a ( b, x) = lim inf x→x, s(x)>0 d * (0, ∂s(x)) where s : R n → R is the convex continuous function given by s(x) := max t∈T {a t x -b t },

co 2 (FF

 2 S) := {αs 1 + (1 -α)s 2 : α ∈ [0, 1], s 1 , s 2 ∈ S}.Boundary of maximal monotone operators ofAx; that is, A • (x) := {x * ∈ A(x) : x * = min z * ∈A(x) z * },is well-defined and unique. Finally, given multifunction F : H ⇒ H we denote Limsup y→x (y) := {x * ∈ H : ∃ y n → x, y * n → x * , s.t. y * n ∈ F (y n ) ∀n ≥ 1}, Limsup y x (y) := {x * ∈ H : ∃ y n x, y * n → x * , s.t. y * n ∈ F (y n ) ∀n ≥ 1}, ω-Limsup y→x F (y) := {x * ∈ H : ∃ y n → x, y * n x * , s.t. y * n ∈ F (y n ) ∀n ≥ 1}.

Definition 6 . 1 .

 61 Given x ∈ domA and v ∈ H, we define the set A(x; v) ⊂ H asA(x; v) := x * ∈ A(x) : x * , v = σ A(x) (v) ,with the convention that A(x, v) = ∅ when σ A(x) (v) = +∞.

  that y n → = x, y * n ∈ A(y n ), and lim n→∞ y * n < α.

  y→ = x d(θ, A(y)), and this completes the proof of the first statement.

Corollary 6 . 8 .

 68 For every x ∈ H such that A(x) is a nonempty bounded set, we have A(x) ≤ lim supy→ = x A(y) ,and, when H is finite-dimensional,A(x) = lim sup y→ = xA(y) .Proof. Let x ∈ H be as in the corollary. Then for any ε > 0 there exists x * ∈ bd(A(x)) such that x * ≥ A(x) -ε. According to Theorem 6.5, there exist sequences y n → x and y * n ∈ A(y n ) such that y n = x and y * n → x * as n → +∞. Thus, lim supy→ = x A(y) ≥ lim sup n→+∞ A(y n ) ≥ lim n→∞ y * n = x * ≥ A(x) -ε,and the desired inequality follows when ε goes to 0.

  lim sup y→ = x A(y) = +∞, then since A is locally bounded in int(cl(domA)) (when this set is nonempty), it follows that x ∈ bd(cl(domA)). Hence, N cl(domA) (x) = {θ} and the equality A(x) = A(x) + N cl(domA) (x), which comes from the maximality of the operator A, entail the contradiction A(x) = +∞. Boundary of maximal monotone operators showing that A(x; v) ⊂ Limsup w→v, t↓0 A(x + tw) ⊂ Limsup w v, t↓0 A(x + tw). Thus, since A(x; v) ⊂ Limsup w→v, t↓0 A(x + tw) ⊂ ω -Limsup w→v, t↓0 A(x + tw), we only need to verify that Limsup w v, t↓0 A(x + tw) ⊂ A(x; v) and ω -Limsup w→v, t↓0A(x + tw) ⊂ A(x; v).(6.5)

Example 6 . 10 .

 610 Consider the maximal monotone operator A defined on H as A(x) := x + N B(θ,1) (x), and let x, v ∈ H \ {θ} be such that x = 1 and v, x = 0.Then one can easily check that A(x) = [1, +∞[ x, and soA(x; v) = x * ∈ A(x) : x * , v = sup ξ∈A(x) ξ, v = sup α∈[1,+∞[ αx, v = 0 = A(x).But for any t > 0 we have that A(x + tv) = ∅, which shows that ω -Limsup t↓0 A(x + tv) = Limsup t↓0 A(x + tv) = ∅.

6. 4 . 8 )

 48 Prox-regular analysis prox-regular closed set C at x ∈ C is given by T C (x) := (N P C (x)) * . Theorem 6.15. Let C ⊂ H be a uniformly r-prox-regular set. Then for everyx ∈ C we have thatIf int(T C (x)) = ∅, then N P C (x) = co 2 bd(N P C (x)) = co 2 Limsup y→ = x N P C (y) . (6.9)Proof. First, we observe that the inclusions bd(N P C (x)) ⊂ Limsup y→ = x bd(N P C (y)) ⊂ Limsup y→ = x N P C (y),(6.10)follow as in the the proof of Theorem 6.5, since the following differential inclusion,ż(t) ∈ f (z(t)) -N P C (z(t)) t ∈ [0, 1], z(0) = x ∈ C,for a given Lipschitz function f : H → H, also possesses a unique solution z(•) such that the function d + z(•) dt is right-continuous on [0, 1[ and d + z(t) dt = f (z(t)) -N P C (z(t))

. [ 19 ,

 19 Definition 3.1. ] A function f : H → R ∪ {+∞} is said to be prox-regular at x ∈ domf with uniform parameters if there exist ε, r > 0 such that for any v ∈ ∂ L f (x), one has, for all (x, v) ∈ Gr(∂ L f ) satisfying x -x < ε, |f (x) -f (x)|< ε and v -v < ε, f (x ) ≥ f (x) + v, x -x -r2x -x 2 ∀x ∈ B(x, ε).

Theorem 6 . 18 .

 618 Let f : H → R ∪ {+∞} be a lower semi-continuous function and let x ∈ domf. If f is prox-regular with uniform parameters on a neighborhood of x with the same parameter r > 0, then bd(∂ P f (x)) = Limsupy→ = x ∂ P f (y),and, provided that bd(∂ P f (x)) = ∅,∂ P f (x) = N domf (x) + co 2 Limsup y→ = x ∂ P f (y) .Proof. According to [19, Proposition 3.6], the current prox-regularity assumption entails the existence of an open convex neighborhood U of x and a lsc convex function g such that f (y) = g(y) -r 2 y 2 ∀y ∈ U ; (6.12) hence, ∂ P f (y) = ∂g(y) -ry for all y ∈ U. Thus, since ∂g is a maximal monotone operator [75], by applying Theorem 6.5 we get bd(∂ P f (x)) = bd(∂g(x) -rx) = bd(∂g(x)) -rx = Limsup y→ = x ∂g(y) -rx = Limsup y→ = x (∂g(y) -ry) = Limsup y→ = x (∂ P f (y)), which yields the first conclusion.

  ∂g(x) -rx = N cl(dom∂g) (x) + co 2 Limsup y→ = x (∂g(y) -ry) = N domf (x) + co 2 Limsup y→ = x (∂ P f (y)) ,

  The motivations of Pazy came from the investigation of some regularity properties of partial differential equations. Pazy's criteria for Lyapunov (a = 0) pairs are given by means of directional-like derivatives of the candidate functions, using the Moreau-Yoshida approximation of operator A. Namely, (V, W ) is a Lyapunov pair for (1.1) if the following relation holds uniformly on bounded sets of domA,

	lim sup
	λ↓0

1.2. Previous results from the literature by maximal monotone operators; that is, F ≡ -A.

  Proposition 2.19. [71, 72] Let A : H ⇒ H be a maximal monotone operator. A is locally bounded at x if and only if x ∈ int(domA).

	(ii) The operator A is called maximal monotone if A is monotone and there exists no monotone operator that contains it strictly. Example 2.16. Let ϕ ∈ F(H) be a convex function. Then the subdifferential ∂ϕ is maximal monotone. In particular, the normal cone N S (•) is maximal monotone whenever S is a closed convex set. Proposition 2.17. [16] Let A : H ⇒ H be a maximal monotone operator. The following assertions hold: (i) domA is convex. (ii) A(x) is closed and convex for every x ∈ H and A(x) = A(x) + N domA (x). (2.13) (iii) Suppose that x * Definition 2.18. Let A : H ⇒ H be a maximal monotone and let S be a subset of H. (i) A is called locally bounded at x if there exist r, m > 0 such that y * ≤ m ∀y * ∈ A(y), ∀y ∈ B(x, r). (ii) A is called locally bounded at x respect to S if there exist r, m > 0 such that y * ≤ m ∀y * ∈ A(y), ∀y ∈ B(x, r) ∩ domA ∩ S. From (2.13), we obtain the following proposition We now remind Minty's Theorem Theorem 2.20. Let A : H ⇒ H be monotone. Then A is a maximal monotone operator if and only if n ∈ A(x 2.2. Maximal monotone operators rank(I + A) = H.

Definition 2.15. Let A : H ⇒ H be a operator.

(i) The operator A is called monotone if for any two points (x, x * ), (y, y * ) ∈ A, one has x * -y * , x -y ≥ 0. n ) for all n ∈ N and x n → x, x * n x * as n → ∞, then x ∈ domA and x * ∈ A(x).

  S (x)}. The Bouligand tangent cone to S at x is defined as

	a closed set
	and s ∈ S, we define the proximal normal cone to S at s as N P S (s) = ∂ P I S (s), the
	Fréchet normal cone to S at s as N F S (s) = ∂ F I S (s), the limiting normal cone to S at
	s as N L S (s) = ∂ L I S (s), and the Clarke normal cone to S at s as N C S (s) = co(N L S (s)).
	Equivalently, we have that N P S (s) = cone(Π -1 S (s) -s), where Π -1 S (s) := {x ∈ H |
	s ∈ Π

  by (4.12)) + 2(t j+1 -t j ) s * j , z j -ŝj -(y j -s j )

		+ 2(t j+1 -t j ) s * j , y j -s j
	≤2δM (by (4.14) and (4.16))	≤0 (by (4.15))

  .22) and(4.19) follows. As for relation(4.20), it follows from the following inequalities, 4.3. Differential inclusions involving maximal monotone operators for i = 0, ..., N -1 and t ∈ [t i , t i+1 ],

  and 4.15 (namely, (4.38)), differential inclusion (101) has a unique (absolutely continuous) solution x(t) := x(t; x 0 ) which satisfies || d + x(t) dt ||≤ ||f (x 0 )||e lT for a.e. t ∈ [0, T ]. Then, we find an m > 0 such that

  Ax(t) -BN S (Dx(t)) a.e. t ∈ [0, ∞), x(0) = x 0 ∈ D -1 (S).(4.56) It is well-known that if D : R n → R m is a linear mapping and S is a convex subset of R m , then the setD -1 (S) := {x ∈ R n : D(x) ∈ S}is always convex. This fails when S is prox-regular (see Example 2 in[START_REF] Adly | Preservation of prox-regularity of sets and application to constrained optimization[END_REF] for a counterexample). The following lemma provides a sufficient condition to ensure that D -1 (S) is still prox-regular.

	4.6. Stability and observer designs
	Using (4.55b) and (4.55c), and putting the resulting equation in (4.55a), we get
	the following differential inclusion
	ẋ(t) ∈

and S ⊂ R l is a uniformlyprox-regular set.

  Titre thèse français: Contribution à la stabilité de Lyapunov non-régulière des inclusions différentielles avec opérateurs monotones maximaux Dans cette thèse de doctorat, nous apportons quelques contributions à la stabilité de Lyapunov non-régulière des inclusions différentielles de premier ordre avec opérateurs monotones maximaux, dans un cadre Hilbertien de dimension infini. Nous fournissons des caractérisations explicites, primales et/ou duales, des paires de Lyapunov faibles et fortes, dont les fonctions sont semi-continues inférieurement à valeurs réelles étendues, et associées à des inclusions différentielles dont la partie de droite est gouvernée par des perturbations Lipschitziennes des operateurs dits Cusco F , ou des opérateurs monotones maximaux A, ou les deux à la fois ẋ(t) ∈ F (x(t)) -A(x(t)) t ≥ 0, x(0) ∈ domA. De manière équivalente, nous étudions l'invariance faible et forte des ensembles fermés pour ces inclusions différentielles. Comme dans L'approche classique de Lyapunov à la stabilité des équations différentielles, les résultats présentés dans cette thèse n'utilisent que les données du système différentiel; c'est-à-dire, l'opérateur A et la multifonction F , et donc pas besoin de connaître les solutions, ni les semi-groupes générés par les opérateurs monotones en question. Parce que les paires de Lyapunov sont formées pars des fonctions qui sont simplement semi-continues nférieurement, et les ensembles invariants ne sont que ensembles fermés, nous faisons usage dans cette thèse à des outils de l'analyse non-lisse, afin de fournir des critères du premier ordre, utilisant des sous-différentiels généraux et des cônes normaux. Nous fournissons une analyse similaire pour les inclusions différentielles gouvernées par le cône normal proximal à des ensembles prox-réguliers. Notre analyse cidessus, nous a permis de présenter ces systèmes prox-réguliers d'apparence plus générale, comme des inclusions différentielles avec opérateurs monotones maximaux. Nous utilisons aussi nos résultats pour étudier la géométrie des opérateurs monotones maximaux, et plus précisément, la caractérisation de la frontière des valeurs de ces opérateurs seulement au moyen des valeurs situées à proximité, distinctes du point de référence. Ce résultat a des applications dans la stabilité des problèmes de la programmation semi-infinie. Nous utilisons également nos résultats sur les paires de Lyapunov et les ensembles invariants pour établir une étude systématique des observateurs de type Luenberger pour des inclusions différentielles avec des cônes normaux à des ensembles prox-réguliers. La thèse est organisée comme suit: Au chapitre 1, nous expliquons les principaux objectifs de la thèse, la méthodologie que nous suivons et nous donnons un aperçu des principaux résultats. Nous faisons aussi dans ce chapitre un aperçu général de la théorie de Lyapunov, et nous présentons les principales réalisations et les différents résultats que nous avons trouvé dans littérature et qui ont, en quelques sortes, guidé les travaux de cette thèse. Au chapitre 2, nous présentons les principaux outils et résultats préliminaires dont nous avons besoin dans notre analyse. Au chapitre 3, nous donnons les caractérisations souhaitées des paires de Lyapunov et des ensembles invariants pour des inclusions différentielles avec des perturbations Lipschitzienne des opérateurs monotones maximaux, Quant au Chapitre 4, nous étudions les inclusions différentielles avec des perturbations Lipschitzienne des cônes normaux proximaux. Ce chapitre comprend l'application à la conception des observateurs de type Luenberger. Au chapitre 5, nous étudions les inclusions différentielles avec des perturbations Lipschitziennes de type Cusco des opérateurs monotones maximaux. Au Chapitre 6, nous donnons un résultat sur la géométrie des opérateurs monotones maximaux, et nous décrivons la limite de leurs valeurs. Enfin, nous donnons au chapitre 7 un résumé des résultats obtenus. Mots clés : Inclusions différentielles, operateurs monotones maximaux, fonctions de Lyapunov, ensembles invariants, ensembles prox-réguliers, opérateurs de type Cusco. Titre thèse anglais: Contribution to Nonsmooth Lyapunov Stability of Differential Inclusions with Maximal Monotone Operators

3.5. Conclusion and further research

6.4. Prox-regular analysis

Acknowledgements

Firstly, I would like to thank God Almighty for giving me the health, knowledge, and all courage to complete my thesis. I also would like to thank Mother Virgin Mary who has helped me overcome difficult times during my life.

I would like to express my gratitude to my advisors: Prof. Abderrahim Hantoute and Prof. Samir Adly for the continuous support during my Ph.D study and related research, for their patience, motivation, and immense knowledge. Their guidance has helped me in all the time of study and writing of the thesis. Without their unconditional support, this thesis could not have been finished. I could not have imagined having a better advisor and mentor for my Ph.D study. I feel very

committee. I would like to thank Prof. Rafael Correa Fontecilla who supported me for the opportunity of studying in Chile

that x -x 0 < ρ, • (x, y, z, w) ≤ m + F (x) ≤ m 1 .

(5.53)

We proceed by steps:

Step 1). We prove that for every (x, y, z, w) ∈epi Ṽ such that x -x 0 < ρ and ( ξ, -κ) ∈ N P epi Ṽ (x, y, z, w), there exist v ∈ F (x) and x * ∈ A(x) ∩ B m+ F (x) such that (v, W k (x), 1, 0) ∈ Fk (x, y, z, w), (x * , θ R 3 ) ∈ Â(x, y, z, w) ∩ B m+ F (x) and (5.54) ( ξ, -κ), (v -x * , W k (x), 1, 0) ≤ 0.

(5.55) Indeed, let (x, y, z, w) ∈epi Ṽ and ( ξ, -κ) be as in the claim, so that κ ≥ . If κ > 0, say κ = 1, we get ξ = (e az ξ, 1, ae az V (x)) ∈ ∂ P Ṽ (x, y, z) for some ξ ∈ ∂ P V (x). Then, by the current hypothesis, there exist v ∈ F (x) and x * ∈ A(x) ∩ B m+ F (x) , that satisfy (5.54), such that ξ, v -x * + aV (x) + W k (x) ≤ ξ, v -x * + aV (x) + W (x) ≤ 0.

Hence, ( ξ, -1), (v -x * , W k (x), 1, 0) = e az ξ, v -x * + W k (x) + ae az V (x) = e az ( ξ, v -x * + aV (x) + W k (x))

+ (1 -e az )W k (x) ≤ 0, (5.56) and (5.55) follows. If κ = 0, then ξ = (ξ, 0, 0) ∈ ∂ P,∞ Ṽ (x, y, z) for some ξ ∈ ∂ P,∞ V (x). Then, taking into account that x -x 0 < ρ, there are sequences (x i ) ⊂ B(x 0 , ρ) ∩ domV, (ξ i ) ⊂ R n , and (α i ) ⊂ R such that x i V → x, ξ i ∈ ∂ P V (x i ), α i ↓ 0, and α i ξ i → ξ as i → ∞. Hence, for each i ∈ N, by the current hypothesis, there exist v i ∈ F (x i ) and x * i ∈ A(x i ) ∩ B m+ F (x i ) such that

Because F is of Cusco and A is maximal monotone, we may suppose w.l.o.g. that v i → v ∈ F (x) and x * i → x * ∈ A(x) ∩ B m+ F (x) as i → ∞. Then, by multiplying both sides of the inequality above by α i and taking limits as i → ∞, we obtain Differential inclusions with Lipschitz Cusco perturbations and so m(z) := lim sup

To continue we shall proceed by steps:

Step 1). We show that for any z ∈ B(z 0 , r + 1) ∩ epi Ṽ we have sup

Indeed, let z := (x, y, u, w) ∈ B(z 0 , r + 1) ∩ epi Ṽ and pick ( ξ, -κ) ∈ N P epi Ṽ (z); hence, κ ≥ 0 and x ∈ E. If κ > 0, say κ = 1 (w.l.o.g.), then we get ξ = (e au ξ, 1, ae au V (x)) for some ξ ∈ ∂ P V (x). Thus, by the current hypothesis (i) and (5.67) there exist v ∈ F (x) and

In other words, we have that (x * , 0, 0, 0

+(1 -e au )W k (x) ≤ 0, which entails (5.68). f κ = 0, then ξ = (ξ, 0, 0) ∈ ∂ P,∞ Ṽ (x, y, z) for some ξ ∈ ∂ P,∞ V (x). Hence, there are sequences (x i ), (ξ i ) ⊂ R n , and

Because F is of Cusco and A is maximal monotone, we may suppose w.l.o.g. that and so, using (5.71),

and we get (ii).

Differential inclusions with prox-regular sets

In this section, we use the previous results to characterize Lyapunov pairs associated to the following differential inclusion

where F : R n ⇒ R n is an L-Lipschitz Cusco mapping and C is an r-uniformlyprox-regular set of R n .

Definition 5.24. [START_REF] Mazade | Regularization of differential variational inequalities with locally prox-regular sets[END_REF][START_REF] Poliquin | Local differentiability of distance functions[END_REF] For positive numbers r and α, a closed set S is said to be (r, α)-prox-regular at x ∈ S provided that one has x = Π S (x + v), for all x ∈ S ∩ B(x, α) and all v ∈ N P S (x) such that v < r. The set S is r-prox-regular (resp., prox-regular) at x when it is (r, α)-prox-regular at x for some real α > 0 (resp., for some numbers r, α > 0). The set S is said to be r-uniformly prox-regular when α = +∞.

When S is r-uniformly prox-regular, the set-valued mapping defined by x → N P S (x) ∩ B is 1 r -hypo-monotone, and for every x ∈ S we have ( [START_REF] Poliquin | Local differentiability of distance functions[END_REF])

so that in the sequel we simply write N S (x) to refer to any one of theses cones. We shall use the following property of r-uniformly-prox-regular sets.

Lemma 5.25. Given κ > 0 and a maximal monotone extension A C,κ of the

where Id is the identity mapping, we have for every

Differential inclusions with Lipschitz Cusco perturbations solution (5.72) and

then we may assume, without lost of generally, that (x n (•; x 0 )) n uniformly converges to a solution x(•; x 0 ) on [0, t], so that, using the continuity of V,

), and so, by applying again Theorem 5.28, there exist number δ > 0 and solution x(•; x( t; x 0 )) of ( 5.72) such that V ( x(s; x( t;

It follows that the following solution of (5.72)

which is a contradiction with the definition of t. Hence, we have t = T (x 0 ) so that, by (5.98) and the fact that V equals 0 on S,

Step [START_REF] Adly | A stability theory for second order non-smooth dynamical systems with application to friction problems[END_REF]. We prove that V (x) ≤ T (x), for all x ∈ C \ S. We fix x 0 ∈ C \ S and let x(•; x 0 ) be any solution of (5.72). Since x(t; x 0 ) ∈ C \ S for any t < T (x 0 ), by (5.96) Theorem 5.27 gives us

In particular, if x(•; x 0 ) is an optimal trajectory we get

), as we wanted to prove.

Notations and preliminary results

Observe that co 2 (S) coincides with co(S) when H = R, but the two sets may be different in general. By int(S), bd(S) and cl(S) (or, indistinctly, S), we denote the interior, the boundary and the closure of S, respectively. The indicator, the support and the distance functions to the set S are respectively given by I S (x) := 0 if x ∈ S; +∞ otherwise, σ S (x) := sup{ x, s : s ∈ S}, d S (x) := inf{||x -y||: y ∈ S} (in the sequel we shall adopt the convention inf ∅ = +∞). We shall write S for the convergence when restricted to the set S, and y -→ = x when y → x with y = x.

We denote Π S the (orthogonal) projection mapping onto S defined as

Next, we review some classical facts about convex functions and monotone operators; we refer to [START_REF] Brézis | Operateurs Maximaux Monotones et Semi-Groupes de Contractions Dans Les Espaces de Hilbert[END_REF][START_REF] Zȃlinescu | Convex Analysis in General Vector Spaces[END_REF] for more details. Given a function ϕ : H → R∪{+∞}, we say that ϕ is proper if its domain domϕ : Given a set-valued operator A : H ⇒ H, the domain and the graph of A are given by

The operator A is said to be monotone if

and maximal monotone if, in addition, A coincides with every monotone operator containing its graph. In such a case, it is known that cl(domA) is convex, and that Ax is convex and closed for every x ∈ H. Hence, the minimal norm element Lemma 6.3. (see, e.g., [START_REF] Clarke | Nonsmooth Analysis and Control Theory[END_REF]) For any nonempty closed convex set S ⊂ H, the set of points s ∈ bd(S) such that N S (s) = {θ} is dense in bd(S).

Proposition 6.4. Let x ∈ domA and v = θ be given. Then we have that

Proof. The inclusion " ⊃ " being obvious, due to the definition of the set A(x; v), we only need to prove the inclusion " ⊂ ". Take an arbitrary vector ξ ∈ bd(A(x)).

According to Lemma 6.3, there exists a sequence (ξ n ) n ⊂ bd(A(x)) such that ξ n → ξ and N A(x) (ξ n ) = {θ}. Hence, for each n there exists First, we observe that when x ∈ domA, these inclusions follows since that, using the norm-weak upper semicontinuity of the (maximal monotone) operator A,

So, we may assume that x ∈ domA. Also, if bd(A(x)) = ∅, then we would have that A(x) = H, so that domA = {x} and this leads to

that is, the conclusion of the first statement is also true in this case. From the observation above we assume now that bd(A(x)) = ∅. Take x * ∈ bd(A(x)) (⊂ A(x)). According to Lemma 6.3, for each n ≥ 1 there exists

Boundary of maximal monotone operators some v n ∈ H \ A(x). We fix n ≥ 1 and consider the following differential inclusion

which (see, e.g., [START_REF] Brézis | Operateurs Maximaux Monotones et Semi-Groupes de Contractions Dans Les Espaces de Hilbert[END_REF]) possesses a unique solution z n (•) that satisfies z n (t) ∈ domA for all t ∈ [0, 1], and such that the function

is right-continuous on [0, 1). In particular, one has

hence, since v n -x * n = θ, we get z n (t) = x for all small t ∈ [0, 1). Then, from the right-continuity of d + zn(•) dt and the expressions in (6.3), there exists a sequence

)) for all k in a cofinite set K, and this would lead to v n ∈ A(z n (t k )) for all k ∈ K. Consequently, as z n (t k ) → x when k goes to +∞, the maximal monotonicity of A would give us v n ∈ A(x), which is a contradiction. Now, we may choose a diagonal sequence (z * n,kn ) n such that z * n,kn → x * as n → +∞, and this shows that x * ∈Limsup y→ = x bd(A(y)), which yields the first inclusion in (6.2).

We take now x * ∈Limsup y→ = x A(y), so that x * = lim n→∞ x * n for some x * n ∈ A(x n ) with x n → x and x n = x. Then by the norm-weak upper semi-continuity of the operator A, we deduce that x * ∈ A(x). Thus, it suffices to prove that x * ∈ H \ int(A(x)). Proceeding by contradiction, we assume that x * + rB ⊂ A(x) for some r > 0. Then, using the monotonicity of A, for every n ≥ 1 one has that

that is, x * n -x * ≥ r for every n ≥ 1, and this contradicts the convergence of (x * n ) to x * . Hence, x * ∈ bd(A(x)) and we conclude the proof of (6.2).

Boundary of maximal monotone operators

It easily follows from Theorem 6.5 that bd(A(x)) ⊂ Limsup

but the last inclusion may be strict, as the following example shows. We observe that the sequence ( en n ) n∈N strongly converges to θ, and

We give an interesting corollary of Theorem 6.5.

Corollary 6.7. For every x ∈ H we have

Consequently, if x is such that θ ∈ int(A(x)), then

Proof. It suffices to consider the case when x ∈ domA, because otherwise both sides of the equality are equal to +∞.

We may distinguish two cases: If θ / ∈ A(x), then d(θ, bd(A(x))) = d(θ, A(x)) = A • (x) . Thus, according to Theorem 6.5 there are sequences (y n ), (y

Hence,

and so

+∞, then the first equality of the corollary obviously. Otherwise, we suppose that lim inf y→ = x d(θ, A(y)) < α for some α ∈ R, and let sequences (y n ), (y * n ) ⊂ H be such Consequently, we may suppose that lim sup

A(y) < +∞. We let a sequence (y n , y * n ) n ⊂ Gr(A) be such that y n → x, y n = x and lim sup

We may also assume that the sequence (y * n ) n converges to some x * ∈ A(x). Then

as we wanted to prove.

The following result concerns the faces of the values of maximal monotone operators. Theorem 6.9. For every x ∈ domA and v = θ we have

Proof. We fix x ∈ domA and v = θ, and take x * ∈ A(x; v). From Definition 6.1, we have that v ∈ (∂oe

As in the proof of Theorem 6.5, this differential inclusion has a unique solution z(•) such that

; hence, (6.4) ensures that

and so

A(x + tw),

In Theorem 6.12 we give the expression of the values of maximal monotone operators by using the values at nearby points. We need first to check the following lemma. Lemma 6.11. Given x ∈ domA, for every x * ∈ A(x) it holds

(6.6)

Proof. Since the operator A+N cl(domA) is monotone and Gr(A) ⊂ Gr(A+N cl(domA) ), the maximality of A ensures that A(x)

), so that x * + tv ∈ A(x) for all t ≥ 0. Then, by the monotonicity of A we get y * -(x * + tv), y -x ≥ 0 ∀y * ∈ A(y), ∀t ≥ 0, which in turn leads to

Hence, v, y-x ≤ 0 for every y ∈ domA, and we deduce that v ∈ N cl(domA) (x).

Theorem 6.12. For every x ∈ domA such that bd(A(x)) = ∅ we have that

A(y) .

Proof. First, according to Theorem 6.5, ensuring that bd(A(x)) = Limsup

and to the maximal monotonicity of the operator A, ensuring that A = A + N cl(domA) , we only need to prove the following inclusion when int(A(x)) = ∅,

Given x * ∈ int(A(x)), we fix x * 0 ∈ bd(A(x)) and introduce the set

On the one hand, if S ∩ bd(A(x)) = ∅, then S ⊂ A(x) and, due to the convexity of A(x), we obtain x * 0 + R + (x * -x * 0 ) ⊂ A(x). Hence, thanks to Lemma 6.11 we Boundary of maximal monotone operators we get

and, dividing by y n -x ,

which is a contradiction. Hence, ξ ∈ bd(N P C (x)) and (6.10) holds as equalities. In this last part of the proof, we assume that int(T C (x)) = ∅; that is, there exist v ∈ H and η > 0 such that v + B η ⊂ int(T C (x)). According to the first statement of the theorem we only need to prove that int(N

We take ξ ∈ int(N P C (x)) \ {θ}, so that -ξ / ∈ N P C (x) by [76, Exercise 9.42] (the proof of [START_REF] Rockafellar | Variational Analysis, Grundlehren der Mathematischen Wissenschaften[END_REF]Exercise 9.42] can be easily extended to the current infinite-dimensional setting), and hence we can choose z * ∈ bd(N P C (x)) \ {θ}. Let us show that for some t 0 > 0 we have that ξ + t 0 (ξ -t 0 z * ) / ∈ N P C (x). Otherwise, ξ + t(ξ -tz * ) ∈ N P C (x) for all t ≥ 0, and we get 1 + t t 2 ξ -z * ∈ N P C (x) ∀t > 0, which as t → +∞ gives us -z * ∈ N P C (x), which contradicts the nonemptyness of the set int(T C (x)) (again by [START_REF] Rockafellar | Variational Analysis, Grundlehren der Mathematischen Wissenschaften[END_REF]Exercise 9.42]). Then, there exists some β ∈ (0, 1) such that w * := ξ + βt 0 (ξ -t 0 z * ) ∈ bd(N P C (x)), and hence ξ = 1 1+βt 0 w * + βt 0 1+βt 0 (t 0 z * ) ∈ co 2 bd(N P C (x)) .

In this last part of the paper, we extend the results of Section 6.3 to the proximal subdifferential mapping of lower semi-continuous functions.

Future work

We are interested in the Lyapunov stability of the following differential inclusion ẋ(t) ∈ -A(t)(x(t)) + f (x(t)), t ≥ 0, x(0) = x 0 ∈ domA(0), (7.1) where f : H → H is a Lipschitz mapping, and for each t ≥ 0, A(t) : H ⇒ H is maximal monotone operator and A(•) is absolutely continuous. Existence and unicity solutions of (7.1) have been already studied by S. Saïdi and M. Yarou [START_REF] Saïdi | Control problems governed by time-dependent maximal monotone operators[END_REF].

Recently, Colombo and Palladino [START_REF] Colombo | The minimum time function for the controlled Moreau's sweeping process[END_REF] provided strong and weak invariant characterizations for the following differential inclusion which is called sweeping process ẋ(t) ∈ -N C(t) (x(t)) + f (t, x(t)) a.e. t ≥ 0, x(0) = x 0 ∈ C(0), where C(t) is uniformly prox-regular. We see that if all C(t) are closed convex sets, then it becomes a special case of (7.1). If C(t) = C for all t ≥ 0 and f does not depend on time t, then the results of [START_REF] Colombo | The minimum time function for the controlled Moreau's sweeping process[END_REF] and the results of Section 6, Chapter 5 coincide.