Convex sweeping process in the framework of measure differential inclusions and evolution variational inequalities

In this paper, we analyze and discuss the well-posedness of two new variants of the so-called sweeping process, introduced by Moreau in the early 70s (Moreau in Sém Anal Convexe Montpellier, 1971) with motivation in plasticity theory. The first new variant is concerned with the perturbation of the normal cone to the moving convex subset C(t)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C(t)$$\end{document}, supposed to have a bounded variation, by a Lipschitz mapping. Under some assumptions on the data, we show that the perturbed differential measure inclusion has one and only one right continuous solution with bounded variation. The second variant, for which a large analysis is made, concerns a first order sweeping process with velocity in the moving set C(t)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C(t)$$\end{document}. This class of problems subsumes as a particular case, the evolution variational inequalities [widely used in applied mathematics and unilateral mechanics (Duvaut and Lions in Inequalities in mechanics and physics. Springer, Berlin, 1976]. Assuming that the moving subset C(t)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C(t)$$\end{document} has a continuous variation for every t∈[0,T]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t\in [0,T]$$\end{document} with C(0)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C(0)$$\end{document} bounded, we show that the problem has at least a Lipschitz continuous solution. The well-posedness of this class of sweeping process is obtained under the coercivity assumption of the involved operator. We also discuss some applications of the sweeping process to the study of vector hysteresis operators in the elastoplastic model (Krejčı in Eur J Appl Math 2:281–292, 1991), to the planning procedure in mathematical economy (Henry in J Math Anal Appl 41:179–186, 1973 and Cornet in J. Math. Anal. Appl. 96:130–147, 1983), and to nonregular electrical circuits containing nonsmooth electronic devices like diodes (Acary et al. Nonsmooth modeling and simulation for switched circuits. Lecture notes in electrical engineering. Springer, New York 2011). The theoretical results are supported by some numerical simulations to prove the efficiency of the algorithm used in the existence proof. Our methodology is based only on tools from convex analysis. Like other papers in this collection, we show in this presentation how elegant modern convex analysis was influenced by Moreau’s seminal work.


Introduction
In the seventies Moreau introduced and thoroughly studied the sweeping process, which is a particular differential inclusion. As a partial viewpoint, consider a timemoving closed convex set C(t) which drags a point u(t), so this point must stay in C(t) at every time t, and the opposite of its velocity, say − du dt (t), has to be normal to the set C(t). To take into account the nonsmoothness of the boundary of the convex set C(t), the law of motion is formulated as ⎧ ⎨ ⎩ − du dt (t) ∈ N (C(t); u(t)) u(0) = u 0 ∈ C(0) u(t) ∈ C(t) ∀t, (1.1) where N (C(t); u(t)) is the (outward) normal cone to the set C(t) at the point u(t) in the sense of Modern Convex Analysis. The following interpretation arises (see [45]) for the way how the point u(t) is "sweept": as long as the point u(t) happens to be in the interior of C(t), the normal cone N (C(t); u(t)) is reduced to zero, so u(t) does not move. When the point is "caught up with" by the boundary of C(t) it moves, subject to an inward normal direction, as if pushed by this boundary. Concrete original motivations of the sweeping process by Moreau are: quasi-static evolution in elastoplasticity, contact dynamics, friction dynamics, granular material (see [43,47] and the references therein). The sweeping process model is also of great interest in nonsmooth mechanics, convex optimization, mathematical economics and more recently in the modeling and simulation of switched electrical circuits [1,2,4,6,12]. Existence and uniqueness results when the convex sets C(t) are absolutely continuous or have bounded retraction are provided in [45]. Those results will be discussed in Sect. 3.1. Moreau [45] also introduced the second order sweeping process for the study of Lagrangian mechanical systems subject to frictionless unilateral constraints. For such systems the velocity may be discontinuous at the impact time. In this case, the acceleration can be defined as a measure. This kind of problems fall within the formalism of measure differential inclusions. For the sweeping process with nonconvex sets C(t), we refer the reader to [7,9,13,16,[18][19][20]26,[58][59][60] and the references therein.
From a numerical point of view, the time-integration (also known as time-stepping) schemes have been applied to find an approximation of the solution to the sweeping process. The so called "catching-up" algorithm was introduced by Moreau [40,45,46] to prove the existence of a solution to (1.1) and is defined by where u k stands for an approximation of u at the time t k . Using the fact that [I + N (C ; ·)] −1 = proj C (the metric projection operator onto C), one sees that (1.3) is equivalent to u 0 ∈ H, u k+1 = proj C(t k+1 ) (u k ). (1.4) When the time step goes to zero, under various assumptions on the variation of C(t), the approximation constructed via (u k ) k in (1.4) contains a subsequence which converge weakly in H to some u satisfying (1.1) a.e. (see [34,45]). Another interesting mathematical formalism, called Differential Variational Inequalities (DVI), was introduced by Pang and Stewart [50]. It is a combination of an ordinary differential equation with a variational inequality or a complementarity constraint. A DVI consists to find trajectories t → x(t) and t → u(t) such that dx where K is a closed convex subset of H , f and F are given mappings. The DVI formalism unifies several known mathematical problems such as: ordinary differential equations with discontinuous right-hand term, differential algebraic equations, dynamic complementarity problems etc . . . (see [50] for more details). The DVI formalism was proved to be powerful for the treatment of many problems in science and engineering such as: unilateral contact problems in mechanics, finance, traffic networks, electrical circuits etc . . .
The main aim of the present paper is to analyze two variants of the sweeping process and to establish existence results for them. The first new variant is concerned with the case where the sweeping process (1.1) is perturbed by a Lipschitz mapping and where the moving convex set C(t) has a bounded variation. The mathematical formulation is then a measure differential inclusion (see Sect. 4).
The second variant is of the form , v −u(t)) + a 1 (u(t), v −u(t)) ≥ l(t), v −u(t) for all v ∈ C(t).
The remainder of this manuscript is organized as follows. Section 2 is devoted to several results of convex analysis which are used throughout the paper; in particular, Rockafellar's theorem on the Legendre-Fenchel conjugate of a convex integral functional on a functional space is stated in Sect. 2.2. In Sect. 3 we review the significance of the differential measure formulation of (1.1) where C(t) has a bounded variation and state Moreau's theorem on existence and uniqueness of solution; various known variants in the literature with applications to hysteresis phenomena, planning procedures and electrical circuits are also briefly described. The first new variant presented above of the perturbation with a Lipschiz mapping of the sweeping process involving convex set C(t) with bounded variation is studied in great detail in Sect. 4; a theorem of existence and uniqueness is established. Section 5 is dedicated to the second variant (1.5), for which a large analysis is made and an existence theorem is provided; under the coercivity of the linear operator A 0 the uniqueness is also obtained. Section 6 is devoted to some illustrative numerical simulations.

Notation and preliminaries
This section is devoted to concepts and preliminary results which will be used in the paper.

Subdifferential, normal cone, conjugate
Given a normed space X with topological dual X * and a convex function ϕ : X → R ∪ {−∞, +∞}, the subdifferential of ϕ at a point x ∈ X with |ϕ(x)| < +∞ is defined as and the effective domain of ϕ is dom ϕ := {x ∈ X : ϕ(x ) < +∞}; the function ϕ is said to be proper whenever dom ϕ = ∅ and ϕ does not take on the value −∞. The subdifferential is related to the directional derivative ϕ (x; ·) in the sense that This characterization shows, for any convex function ϕ 0 Gâteaux differentiable at x, Through the directional derivative and the Hahn-Banach theorem, one also sees that, whenever ϕ is finite and continuous at x, the set ∂ϕ(x) is nonempty and weakly * compact in X * and Concerning the continuity, we recall (see, e.g., [51,56]) that a lower semicontinuous convex function on a Banach space is continuous on the interior of its effective domain. Three particular convex functions arise in general in many problems involving Modern Convex Analysis (see [8,30,48,54,56,57,62]). Given a nonempty closed convex set C of X , those functions correspond to the indicator and support functions ı C and σ C [or σ (C, ·)] of C respectively, and to the distance function d C from the set C, defined by From the definition of σ C , we see that σ C coincides with the Legendre-Fenchel conjugate of ı C , that is, σ C = (ı C ) * where, for the above function ϕ, its Legendre-Fenchel conjugate is defined as The Legendre-Fenchel conjugate is also related to the subdifferential. Indeed, for ϕ(x) finite, one has so, provided the convex function ϕ is proper and lower semicontinuous, the set-valued operator ∂ϕ * : X * ⇒ X is the inverse of the set-valued operator ∂ϕ; this ensures, in particular when the Banach space X is reflexive, that the set-valued operator ∂ϕ : X ⇒ X * is surjective if and only dom ϕ * = X * . Hence in particular ∂ϕ is surjective whenever dom ϕ is bounded and ϕ is bounded from below. (2.3) Indeed, these boundedness properties taken together clearly imply that ϕ * is finite on X * , so ϕ * is continuous on X * (by the continuity property recalled above) and hence Dom ∂ϕ * = X * according to (2.2), which is equivalent (by a property recalled above) to the surjectivity of ∂ϕ. (Above, Dom ∂ϕ * denotes the effective domain of ∂ϕ * , where Dom M := {s ∈ S : M(s) = ∅} for any set-valued mapping M : S ⇒ Y between two sets S and Y ). For ϕ = ı C and x ∈ C, it is readily seen that of outward normals of the convex set C at the point x ∈ C; the latter inequality characterization also says that When X is a Hilbert space H , it is also clear from the inequality characterization above that y − proj C (y) ∈ N (C; proj C (y)) for all y ∈ H, (2.5) where proj C (y) denotes the nearest point of y in C, hence proj C is the metric projection onto C. For the normed space X , it is known and not difficult to see, for x ∈ C, that where B X * := {x * ∈ X : x * ≤ 1} (resp. B X := {x ∈ X : x ≤ 1}) denotes the closed unit ball of X * (resp. X ) centered at the origin. The fundamental concepts of subdifferential or normal cone, directional derivative and Legendre-Fenchel conjugate will be at the heart of our present paper. From the definitions it directly follows the monotonicity property of the subdifferential of the convex function ϕ (resp. normal cone of the convex set C) (property crucial for the paper), say It is worth mentioning that the converse (which is not obvious) also holds true, that is, a lower semicontinuous function ψ on a Banach space X is convex if and only if ∂ 0 ψ is monotone, where ∂ 0 is any subdifferential with appropriate fuzzy sum rule (see [22,52] Then, the subdifferential of a proper lower semicontinuous convex function ϕ on X is maximal monotone in the sense that there is no monotone set-valued operator from X into X * whose graph is larger than the graph of ∂ϕ.

Normal convex integrand
Assume that the normed space X is separable and complete; a set-valued mapping M from a measurable space (S, S) into closed subsets of X is S-measurable provided that, for each open set U of X , one has M −1 (U ) ∈ S. So, following Rockafellar [55], an extended real-valued function ϕ : S × X → R ∪ {+∞} is called a normal integrand whenever ϕ(s, ·) is proper and lower semicontinuous for all s ∈ S and the (epigraphical) set-valued mapping s → epi ϕ(s, ·) (from S into X × R) is S measurable. As usual, B(X ) denotes the Borel σ -field of X . When, in addition ϕ(s, ·) is convex for all s ∈ S, one says that ϕ is a normal convex integrand. For a setvalued mapping M : S ⇒ X with Dom M = ∅, it is readily seen that the function (s, x) → ı M(s) (x) is a normal (convex) integrand if and only if the set-valued mapping M is measurable and takes on closed (convex) values. For a normal integrand, it is known (see, for example, [17,55]) that, for any measurable mapping u(·) : S → X , the function s → ϕ(s, u(s)) is measurable. Furthermore, (s, x * ) → ϕ * (s, x * ) is a normal convex integrand, where by convenience of notation Suppose that the separable Banach space X is reflexive and μ is a σ -finite measure on S. For any p ∈ [1, +∞] denote by L we recall that, for a measurable function ψ : S → R ∪ {−∞, +∞}, the extended real S ψ(s) dμ(s) is the infimum of integrals S ρ(s) dμ(s) of integrable real-valued functions ρ : S → R such that ψ(s) ≤ ρ(s) for μ-almost every s ∈ S (with the standard convention inf ∅ = +∞). One of the key results concerning normal convex integrand is the following theorem due to Rockafellar (see [55,56] Although the above concepts and results are recalled in the context of normed spaces for completeness of their statements, the framework of the rest of the paper is that of a Hilbert space H .

Convex sweeping process
In 1971, Moreau [41,42] introduced the "sweeping process" (in the absolutely continuous framework) as the evolution differential inclusion where 0 ≤ T 0 < T < +∞; for convenience, we will write sometimes, as usual,u(t) in place of du dt (t). In an earlier paper [40], Moreau showed how such an evolution equation arises in the theory of elastic mechanical systems submitted to nonsmooth efforts as dry friction; note that the velocity in such cases may present discontinuity in time. He also provided later in a 1973 paper [43] more details on applications to elasticity and other fields of mechanics.
The paper [41] is concerned with the situation where the discontinuity of the velocity is exhibited by an absolute continuity property of the state of the system. The main result of that paper [41] can be stated as follows.
Then, the evolution equation (3.1) admits one and only one absolutely continuous solution.
To take into account the more general situation where there are jumps, Moreau transformed the above model into a measure differential inclusion and proved in [45] an existence result that we give in the following form.
Then, the measure differential evolution inclusion admits one and only one right continuous solution with bounded variation.
A mapping u(·) : [T 0 , T ] → H is a solution of the measure differential inclusion in the theorem provided that it is right continuous with bounded variation with u(T 0 ) = u 0 and u(t) ∈ C(t) for all t ∈ [T 0 , T ] and the differential measure du associated with u admits the derivative measure du dμ (see the next section for the meaning) as a density relative to μ and In [26] it is shown that u(·) is a solution if and only if the latter inclusion is fulfilled with some positive Radon measure ν on [T 0 , T ] in place of μ.

An elasto-plastic model and hysteresis
Many problems from thermo-plasticity, phase transition (etc) in the literature lead to variational inequalities in the form below. Consider, for example, the following elasto-plastic one (see, e.g. [31]). Let Z be a closed convex set of the 1 2 Assume that the interior of Z is nonempty, so int Z = ∅ corresponds to the elasticity domain and bdry Z to the plasticity. Write the strain tensor ε = (ε) i, j (depending on time t) as ε := ε e + ε p , where ε e is the elastic strain and ε p the plastic strain. The elastic strain ε e is related to the stress tensor σ = (σ ) i, j linearly, that is, ε e = A 2 σ , where A is a (constant) symmetric positive definite matrix. The system is then subjected to the variational inequality: , ∀z ∈ Z : principle of maximal dissipation and to the region constraint σ (t) ∈ Z for all t ∈ [0, T ]; in this system, the tensor strain ε is supposed to be given as an absolutely continuous mapping and the initial tensor stress σ 0 is given in Z . Observing that the above inequality can be written as By setting, Clearly, we have This provides according to Theorem 3.1 above (besides to [31, Proposition 2.2]) another proof of existence and uniqueness of solution for that system. This defines a mapping Φ : W 1,1 ([0, T ], E) assigning to each absolutely continuous mapping ε ∈ W 1,1 ([0, T ], E) the solution Φ(ε) := σ ε of the system. This mapping Φ enjoys two particular properties: • Rate independence Denoting by σ ε the solution associated with ε and taking any absolutely continuous increasing bijection θ : , from which it can be obtained, for almost every t ∈ [0, T ], The uniqueness property guarantees that σ ε • θ is the solution associated with ε • θ , otherwise stated, Φ(ε • θ) = Φ(ε) • θ . The latter equality is known in the literature as the rate independence property (see, e.g., [11,31,61]).
Both rate independence and causality properties translate that Φ is an hysteresis operator according to [11,31,61] where those properties are brought to light with various physical examples with hysteresis phenomena.
For several other models, we refer the reader to [11]. Of course, by Theorem 3.1 the mathematical features and properties above still hold in the context of a Hilbert space H with any closed convex set Z (without any condition on its interior) and any coercive bijective bounded symmetric linear operator A : H → H .

Planning procedure
In mathematical economy, Henry [29] introduced, as mathematical model for the planning procedure, the differential inclusioṅ where K is a closed convex set of R N , F : R N ⇒ R N is an upper semi-continuous setvalued mapping with nonempty compact convex values, and T K (y) denotes the tangent cone of K at y. This differential inclusion is known (see [21,29]) to be completely linked to the following systeṁ which enters in the following class of perturbed sweeping processes where C(t) is, as in Sect. 3.1, a closed convex set moving in an absolutely continuous way. Existence results for such perturbed sweeping process are established in finite  dimensions in [14,37,58], and in [9,37] for the Hilbert setting under compactness growth conditions for the set-valued mapping F. Under compactness growth assumptions on F, existence of solution has been proved in [26] when the set C(t) moves with a bounded variation, and also in [37] when the set C(t) has a bounded retraction and F is weakly-norm upper semicontinuous.

Non-regular electrical circuits
The aim of this section is to illustrate the sweeping process in the theory of non-regular electrical circuits. Electrical devices like diodes are described in terms of Ampere-Volt characteristic (I, V) which is (possibly) a multifunction expressing the difference of potential V D across the device as a function of current i going through the device. The diode is a device that constitutes a rectifier which permits the easy flow of charges in one direction but restrains the flow in the opposite direction. Figure 1 illustrates the ampere-volt characteristic of an ideal diode model.
Let us consider the left circuit depicted in Fig. 2 involving a load resistance R > 0, an inductor L > 0, a diode (assumed to be ideal) and a current source c(t). Using Kirchhoff's laws, we have We have

Therefore, the inclusion (3.4) is equivalent to
which is of the form (1.2).
The right circuit depicted in Fig. 2 involves a load resistance R ≥ 0, a capacitor C > 0, a diode (assumed to be ideal) and a current source c(t). Using Kirchhoff's laws, we have Therefore, If the charge on the capacitor is q and the current flowing in the circuit is x, then which is of the form (1.5). Let us consider now the electrical system shown in Fig. 3 that is composed of two resistors R 1 ≥ 0, R 2 ≥ 0 with voltage/current laws V R k = R k x k (k = 1, 2), three Therefore the dynamics of this circuit is given by which is an ordinary differential equation (see Remark 1, Sect. 5 for more detail). If The same analysis holds for the dynamics (3.7) while R > 0.
In the same way, we can show that the dynamical behavior of the circuit depicted in Fig. 4 is given by the following sweeping process Some other circuits containing Zener diodes, transistors, rectifier-stabilizer circuits, DC-DC Buck and Boost converters can be analyzed in the same way [1][2][3][4][5][6]. The usage of tools from Modern Convex Analysis (and particularly the notion of Moreau's convex superpotential) in electronics for the study of electrical circuits is fairly recent. It is a quite promising topic of research which may help engineers for the simulation of complicated electrical circuits. Due to the lack of smoothness in some circuits, most used softwares like Simulation Program with Integrated Circuits Emphasis (SPICE) can not simulate non-regular circuits without approximation of i-v characteristic of the involved nonlinear electrical devices (Fig. 5).
In the next sections we study and prove existence of solutions of two new variants of Moreau's sweeping process.

Lipschitz single-valued perturbation variant of BV sweeping process
In this section we are concerned with the differential inclusion where f : I × H → H is a Carathéodory mapping and where the variation of C(t) is expressed by a given positive Radon measure μ on I as in the line of Theorem 3.2. The case of a set-valued mapping F : I × H ⇒ H (in place of f ) has been studied in [15,16] in the finite dimensional setting and in [26] under the assumption Our aim here is to study in the Hilbert setting the new variant where f satisfies a Lipschitz condition and no compactness condition is assumed.
Before defining the concept of solution of the measure differential inclusion (4.1), some preliminaries are necessary. Throughout the rest of this section, all the measures on a compact interval I = [T 0 , T ] of R will be Radon measures.
We start this section by recalling some results from vector measures. For two positive Radon measures ν andν on I and for I (t, r ) := I ∩ [t − r, t + r ], it is known (see, e.g., [35,Theorem 2.12]) that the limit (with the convention 0 0 = 0) exists and is finite for ν-almost every t ∈ I and it defines a Borel function of t, called the derivative ofν with respect to ν. Furthermore, the measureν is absolutely continuous with respect to ν if and only if dν dν (·) is a density ofν relative to ν, or otherwise stated, if and only if the equalityν = dν dν (·)ν holds true. Under such an absolute continuity assumption, a mapping u(·) : When ν andν are each one absolutely continuous with respect to the other, we will say that they are absolutely continuously equivalent. Now suppose that the mapping u(·) : I → H has bounded variation and denote by du the differential measure associated with u (see [23,44]); if in addition, u(·) is right continuous, then Conversely, if there exists some mappingû(·) ∈ L 1 ν (I, H ) such that u(t) = u(T 0 ) + ]T 0 ,t]û dν for all t ∈ I , then u(·) is of bounded variation and right continuous and du =û dν; soû(·) is a density of the vector measure du relative to ν. Then, putting , according to Moreau and Valadier [49], for ν-almost every t ∈ I , the following limits exist in H and In particular, the last equality ensures that and dλ dν (t) = 0, whenever ν({t}) > 0. (4.4) Above and in the rest of the paper λ denotes the Lebesgue measure.
Definition 1 A mapping u : I → H is a solution of the measure differential inclusion (4.1) if: (i) u(·) is of bounded variation, right continuous, and satisfies u(T 0 ) = u 0 and u(t) ∈ C(t) for all t ∈ I ; (ii) there exists a positive Radon measure ν absolutely continuously equivalent to μ + λ and with respect to which the differential measure du of u(·) is absolutely continuous with du dν (·) as an L 1 ν (I, H )-density and The following proposition concerning a particular chain rule for differential measures will be needed. Its statement is a consequence of a more general result from Moreau [44].

Proposition 1 Let H be a Hilbert space, ν be a positive Radon measure on the closed bounded interval I , and u(·)
: I → H be a right continuous with bounded variation mapping such that the differential measure du has a density du dν relative to ν. Then, the function Φ : I → R with Φ(t) := u(t) 2 is a right continuous with bounded variation function whose differential measure dΦ satisfies, in the sense of ordering of real measures, The next result is a substitute of Grownwall's lemma relative to Radon measures. We refer, for example, to [37] for its statement (see also [38]).

Lemma 1
Let ν be a positive Radon measure on [T 0 , T ] and let g(·) ∈ L 1 ν ([T 0 , T ], R + ). Assume that, for a fixed real number θ ≥ 0, one has, for all t ∈]T 0 , T ], Then, for all t ∈ [T 0 , T ], We establish now a stability property of the subdifferential of the distance function from a continuous moving set.

Proposition 2
Let E be a metric space, C : E ⇒ H be a set-valued mapping with nonempty closed convex sets of a normed space X , and let t 0 ∈ cl Q with Q ⊂ E. Assume that there exists a non-negative real-valued function η : Let (t n ) n be a sequence in Q tending to t 0 and let (x n ) n be a sequence in H converging to some x ∈ C(t 0 ) with x n ∈ C(t n ) for all n. Then, for all z ∈ X, Proof Let (t n ) n and (x n ) n be as in the statement. Fix any z ∈ X . Then, for each real τ > 0, we have, for all n, This justifies the desired inequality We can now prove, using some ideas from [26], the theorem concerning the above measure differential inclusion. The case of prox-regular sets C(t) will be treated elsewhere.
(ii) for each real r > 0, the functions ( f (·, x)) x∈r B H are equicontinuous and there exists some non-negative function L r (·) ∈ L 1 λ (I, R) such that Then, for each u 0 ∈ C(T 0 ), the following perturbed sweeping process has one and only one right continuous with bounded variation solution.
Proof I-First, let us suppose that and set The function v(·) is increasing and right continuous with v(T 0 ) = 0. Let (ε n ) n be a sequence of positive real numbers with ε n ↓ 0. For each n ∈ N, let 0 = V n 0 < V n 1 < · · · < V n q n = V be a partition such that Put V n 1+q n := V + ε n . For each n ∈ N, consider the partition of I associated with the subsets and note that (J m j ) q m j=0 is a refinement of (J n j ) q n j=0 whenever m ≥ n.
Since v(·) is increasing and right continuous, it is easy to see that, for each j = 0, 1, . . . , q n , the set J n j is either empty or an interval of the form [r, s[ with r < s. Furthermore, we have J n q n = {T }. This produces an integer p(n) ∈ N and a finite sequence and observe that η n i → 0 as n → ∞. Fix any i ∈ {0, . . . , p(n) − 1}. From (4.14) we have, by the variation assumption of C(·), On the other hand, from assumption (i), and this latter inequality combined with (4.16) yields Noting by (4.12) that The latter inequality, combined with (4.12), yields Consequently, by (4.15), and hence by (4.6) and (4.11) Step 1. Construction of the sequence (u n (·)). Following [16,45], define the mapping u n (·) : I → H by u n (T ) := u n p(n) and (4.22) We observe that u n (·) is well defined on I and it is right continuous with bounded variation on each interval [t n i , t n i+1 ], so it is right continuous with bounded variation on the whole interval I . Furthermore, the definition of u n (·) can be rewritten, for any t ∈ I , as and δ n (s) := t n i if t ∈ [t n i , t n i+1 [ and δ n (T ) := t n p(n)−1 . Since, by (4.6), the measure λ is absolutely continuous with respect to ν, it has dλ dν (·) as a density in L ∞ ν (I, R + ) relative to ν and then by (4. This tells us that the vector measure du n has the latter integrand as a density in L ∞ ν (I, H ) relative to ν, so by the first equality in (4.3) du n dν (·) is a density of du n with respect to ν, (4.23) and, for ν-almost every t ∈ I , (4.24) Taking (4.21) into account, it results that On the other hand, by (4.6) again, the measure (l + 1)(β(·) + 1)λ is absolutely continuous with respect to ν, thus it has d((l+1)(β(·)+1)λ) dν as a density relative to ν with, for ν-almost every t ∈ I , Note also, by (4.12) and (4.18), that f ρ n (t), u n (δ n (t)) ≤ (l + 1)(β(t) + 1), for all t ∈ I, which ensures, for ν-almost every t ∈ I , we also see by (2.5) that, for ν-almost every t ∈ I , du n dν (t) + f (ρ n (t), u n (δ n (t))) dλ dν (t) ∈ −N C(θ n (t)); u n (θ n (t)) , and hence according to (2.6) and (4.25) du n dν (t) + f (ρ n (t), u n (δ n (t))) dλ dν (t) ∈ −∂d C(θ n (t)) u n (θ n (t)) . (4.30) Step 2. Cauchy property of (u n (·)) n . Consider any integers n, m ∈ N. Since u n 0 = u 0 ∈ C(t n 0 ) and u n i+1 = proj C(t n i+1 ) (u n i − η n i y n i ), we note by (4.22) and (4.29) that u n (θ n (t)) ∈ C(θ n (t)) for all t ∈ I. (4.31) This allows us to write, for every t ∈ I , and hence, according to the variation assumption on C(·) and to the fact that one of the partitions (J m j ) q m j=0 and (J n j ) q n j=0 is a refinement of the other (depending on either n ≤ m or m < n),

(4.37)
On the other hand, for every t ∈ I , write and observe by the equicontinuity assumption in (ii) that the first expression {·} dλ dν (t) in the right-hand side tends to 0 as n → ∞ since there is some real r > 0 such that u n (t) ≤ r , for all t ∈ I and n ∈ N, according to (4.18), (4.19) and (4.22). By the inequality [due to (4.28)] and by (4.4), we also see that u n (δ n (t)) − u n (t) → 0 and hence the second expression {·} dλ dν (t) tends to 0 as n → ∞ according to the Lipschitz property of f (t, ·) on r B H in the assumption (ii). Consequently, and, for we have ]T 0 ,T ] ϕ n,m (t) dν(t) → 0 as n, m → ∞ by the Lebesgue dominated convergence theorem (because u n (t) ≤ r as seen above). From this and (4.36), for any n, m, it ensures that, for ν-almost every t ∈ I , thus, putting ψ n,m (t) := u n (t) − u m (t) 2 and noting that u n (T 0 ) = u m (T 0 ), we deduce that, for all t ∈ I , Noting that L r (s) dλ dν (s)ν({s}) = 0 for all s (since dλ dν (s) = 0 if ν(s) > 0 according to (4.4)) we can apply Lemma 1 and this yields This ensures that the sequence (u n (·)) n satisfies the Cauchy property with respect to the norm of uniform convergence on the space of all bounded mappings from I into H . Consequently, this sequence (u n (·)) n converges uniformly on I to some mapping u(·). This also tells us that the mapping u(·) does not depend on the partition V n 0 < · · · < V n q n of [0, V ] satisfying (4.7). Furthermore, by (4.28), extracting a subsequence if necessary, we may suppose that du n dν (·) n converges weakly in L 2 ν (I, H ) to some mapping h(·) ∈ L 2 ν (I, H ), so, for every t ∈ I ,

h(s) dν(s) weakly in H.
Since du n dν (·) is, by (4.23), a density of du n relative to ν, we also have u n (t) = u 0 + ]T 0 ,t] du n dν (s) dν(s), thus it ensues that u(t) = u 0 + ]T 0 ,t] h(s) dν(s), and this tells us that u(·) is right continuous with bounded variation on I , and the vector measure du has h(·) ∈ L 2 ν (I, H ) as a density relative to ν and du dν (·) = h(·) ν-a.e. We also deduce that Step 3. Let us prove that u(·) is a solution. First, for each t ∈ I , noting by (4.10) that 0 ≤ θ n (t) − t ≤ ε n , and writing by (4.28) we see that, as n → ∞, θ n (t) ↓ t and u n (θ n (t)) → u(t).
The latter inequality means, for each t ∈ I 0 , that −ζ(t) ∈ ∂d C(t) (u(t)) hence which finishes the proof of existence of a solution in the case where

II. Case where
First, from (4.40), we note that the mapping u(·) in the above case is also a solution with the measure μ + λ in place of ν therein, since the measure μ + λ is absolutely continuous with respect to ν and vice versa. Let T 0 , T 1 , . . . , T p be a subdivision of [T 0 , T ] such that, for each i = 1, . . . , p, For each i = 1, . . . , p, denote by μ i the Radon measure induced on [T i−1 , T i ] by μ and set ν i := μ i + λ. Then, the part I provides a right continuous with bounded variation mapping u 1 : , du 1 has du 1 dν 1 as a density in L 1 ν 1 ([T 0 , T 1 ], H ) relative to ν 1 , and Similarly, there is a right continuous with bounded variation mapping u 2 : [T 1 , , H ) relative to ν 2 , and So, by induction, we obtain a finite sequence of right continuous with bounded variation mappings u i : Then, the mapping u : 1, . . . , p) is well defined and right continuous with bounded variation, and the inclusions u(t) ∈ C(t), for all t ∈ [T 0 , T ], along with the equality u(T 0 ) = u 0 are obviously fulfilled. On the other hand, putting and considering the Radon measure ν 0 := μ + λ on [T 0 , T ], we easily see that With respect to the positive Radon measure ν := ν 1 + ν 2 absolutely continuously equivalent to μ+λ hence to ν i , the measures du i and λ have densities in L 1 ν ([T 0 , T ], H ) and L 1 ν ([T 0 , T ], R + ) respectively, and Noting that u 1 (·) and u 2 (·) are bounded on thus Proposition 1 says that, for all t ∈ [T 0 , T ], and Lemma 1 entails, for all t ∈ [T 0 , T ], that u 1 (t) − u 2 (t) 2 ≤ 0, which confirms the uniqueness of solution and finishes the proof of the theorem.

A variant with velocity in the moving set
In this section we are interested in the following variant of the sweeping process: By a solution, we mean an absolutely continuous mapping u(·) : [0, T ] → H with u(0) = u 0 such that the above inclusion is fulfilled for almost every t ∈ [0, T ]. We have to be careful with such a variant. Indeed, even in the simple case where A 0 and A 1 are the null operators, that is, the system is reduced to 3) tells us that the set-valued operator N (C(t); ·) is surjective and hence (5.2) has at least one solution u(·). Such boundedness condition of C(t) will be assumed in our analysis below.
Remark 1 Writing the inclusion in (5.1) as we see that it is equivalent tȯ By setting it appears that (5.1) is equivalent to the differential evolution inclusion We emphasize that, in the latter differential inclusion, the convex function g(t, ·) depends on the time t. Instead of continuing in this direction, our aim here is to show how an adaptation of Moreau's catching up algorithm leads to a constructive proof of existence of a solution to (5.1).

Assume that C(0) ⊂ R B H and the nonempty closed sets C(t) of H have a continuous variation in the sense that there is some nondecreasing continuous function v(·)
and note by (2.1) that is also bounded from below on ρB H (containing its effective domain), since the linear operators A 0 and A 1 are bounded. By (2.3) the set-valued operator η n A 0 + A 1 + N (C(t n 1 ); ·) is surjective, so we can choose some z n 1 ∈ H such that and clearly z n 1 ∈ C(t n 1 ) ⊂ ρB H . Put u n 1 = u n 0 + η n z n 1 . Now suppose that u n 0 , u n 1 , . . . , u n i , z n 1 , z n 2 , . . . , z n i are constructed. As above, the set-valued operator η n A 0 + A 1 + N (C(t n i+1 ); ·) is surjective, so we can find z n i+1 ∈ C(t n i+1 ) such that and we set u n i+1 := u n i + η n z n i+1 . We then obtain by induction finite sequences (u n i ) n i=0 and (z n i ) n i=1 such that, for all i = 0, . . . , n − 1, Clearly, the mapping u n (·) is Lipschitz continuous on [0, T ], and ρ is a Lipschitz Furthermore, for every t ∈ [0, T ], one has u n (t) = u 0 + t 0u n (s) ds, hence u n (t) ≤ u 0 + ρT.
Using the linearity of A 0 and the definition of u n i+1 , we see that So, defining the function θ n from [0, T ] to [0, T ] by θ n (0) = t n 1 and θ n (t) = t n i+1 for any t ∈]t n i , t n i+1 ], the latter inclusion becomes and we also note that sup t∈[0,T ] |θ n (t) − t| → 0 as n → ∞.
Now, let us prove the convergence of the sequences (u n (·)) n , (u n (·)) n and ( f n (·)) n . We have, for all n, which entails, according to (5.7), that u n (θ n (t)) → u(t) weakly in H as n → ∞. On the other hand, and using (5.9) and taking the limit as n → ∞ give The latter equality being true for all z ∈ H , we deduce that u(t) = u 0 + t 0 ζ(s) ds, and this guarantees thatu(·) = ζ(·) almost everywhere hencė according to (5.9) again. Furthermore, since f n (t) = f (θ n (t)) and f (·) is continuous, we have, for every t ∈ [0, T ], that f n (t) → f (t) strongly in H as n → ∞. Let us prove thatu(t) ∈ C(t), for almost every t ∈ [0, T ]. First, using the assumption on the variation of C(·), we note thaṫ u n (t) = z n i+1 ∈ C(θ n (t)) ⊂ C(t) + |v(θ n (t)) − v(t)|B H , for a.e. t ∈]t n i , t n i+1 ], Then D ε is closed and convex in L 2 ([0, T ], H ), hence weakly closed, andu n ∈ D ε for large n, by (5.11) since v(·) is uniformly continuous on [0, T ]. The weak convergence ofu n tou in L 2 ([0, T ], H ) implies thatu ∈ D ε for all ε > 0. Since every C(t) is closed, the claim follows. Now let us prove the inclusion in (5.1). Put We then note that the inclusion (5.7) is equivalent by (2.4) to the inequality σ C(θ n (t)), ζ n (t) + −ζ n (t),u n (t) ≤ 0, a.e. sinceu n (t) ∈ C(θ n (t)), and integrating on [0, T ] we get Furthermore, using the strong convergence of f n (t) to f (t) for all t ∈ [0, T ] along with the inequality f n (t) ≤ β, we see that ( f n (·)) n converges strongly in L 2 ([0, T ], H ) as n → ∞. This combined with the weak convergence of (u n (·)) n tou(·) in On the other hand, we have A 0 = ∇ϕ 0 , for the continuous convex function ϕ 0 (x) = 1 2 A 0 x, x . Therefore, the absolute continuity of ϕ 0 • u and ϕ 0 • u n gives where the inequality is due to the weak lower semicontinuity of ϕ 0 on H and to the fact that u n (T ) → u(T ) weakly in H as n → ∞. We have also lim inf From the properties of A 1 , it is easy to verify that the function x(·) →  Since the sequences (u n (·)) n and (u n (·)) n converge weakly in L 2 ([0, T ], H ) to u(·) andu(·) respectively, it results that (recall ζ n (t) := −A 0 u n (θ n (t)) − A 1 (u n (t)) + f n (t)), Since C(t) ⊂ C(θ n (t)) + |v(θ n (t)) − v(t)|B H (according to the assumption on C(·)), we also observe that It is easily seen that T 0 |v(t) − v(θ n (t))| ds → 0 as n → ∞, thus lim inf We then deduce that lim inf (5.17) Using all the inequalities (5.12)-(5.17) together, it follows that by taking lim inf n→∞ on both sides of (5.12). On the other hand, for almost every t ∈ [0, T ], the inclusionu(t) ∈ C(t) yields Taking the latter inequality into account, it results from (5.18) that, for almost every t ∈ [0, T ], which means, according to (2.4), This translates the desired inclusion (5.1) and completes the proof of the theorem.
Next is a uniqueness result related to Theorem 5.1 when the linear operator A 0 is coercive.
Theorem 5.2 Assume in addition to the hypotheses in Theorem 5.1, that A 0 is coercive, that is, for all x ∈ H, for some real constant α 0 > 0. Then, for any initial point u 0 ∈ H, there exists one and only one Lipschitz continuous solution of (5.1).

Application
As a direct application of Theorem (5.2) we obtain an existence and uniqueness result for the evolution variational inequality given in (1.6). Proof For i = 0, 1 we note by A i the linear, bounded and symmetric operators associated respectively with a i (·, ·), that is, a i (u, v) = A i u, v for all u, v ∈ H . Since C has convex values, the evolution variational inequality of type (1.6) can be rewritten in the form

Numerical experiments
In this section, we will give some numerical simulation to illustrate the theoretical results discussed in the last sections. In order to solve numerically problem (5.1), we will use the following algorithm discussed in the proof of Theorem 5.1. Let us suppose that the dimension of H is finite, i.e., dim R (H ) < +∞. For n ∈ N, let 0 = t n 0 < t n 1 < · · · < t n i < · · · < t n n = T, be a finite partition of the interval [0, T ]. We denote by η n i = t n i+1 − t n i the length of the time step.
For simplicity, we will suppose that η n i = η n = T n , i = 0, 1, . . . , n which means that t n i = i T n . The approximation of f (t n i ) will be denoted by f n i .
Algorithm 1 Fix n ≥ 2 and set η n = T n , u n 0 = u 0 and f n 0 = f (t n 0 ). For i = 0, 1, . . . , n − 1 -Compute f n i+1 = f (t n i+1 ) -Solve for z n i+1 the following variational inequalities (see Remark 2) f n i+1 − A 0 u n i ∈ η n A 0 + A 1 + N C(t n i+1 ) z n i+1 (6.1) -Update u n i+1 = u n i + η n z n i+1 . end Remark 2 The discretized variational inclusion (6.1) is equivalent to z n i+1 ∈ η n A 0 + A 1 + N C(t n i+1 ) Since C(t n i+1 ) is bounded, convex and closed in a finite dimensional space, by the classical result of Stampacchia, the variational inequality (6.1) has a solution. If one of the matrices A 0 or A 1 is positive definite, then this solution is unique and the operator [η n A 0 + A 1 + N C(t n i+1 ) ] −1 is single valued and non-expansive. Since A 0 and A 1 are symmetric, (6.2) is equivalent to solve the following optimization problem min z∈C(t n i+1 ) We note that this optimization problem is convex since A 1 , A 0 are positive semidefinite, η n > 0 and the set C(t n k+1 ) is (closed and) convex. The choice of the adequate solver for solving the optimization problem (6.3) depends on the structure of the set C(t). If the set C(t) is polyhedral, i.e., described by linear inequalities and equalities of the form Ax ≤ b and C x = d with A ∈ R m×n , b ∈ R m , C ∈ R p×n and d ∈ R p , then it is possible to use any quadratic programming solver (e.g., quadprog in Matlab). If the set C(t) is described by finitely many nonlinear inequalities and linear equalities C(t)= x ∈ H : g j (t, x) ≤ 0, j=1, 2, . . . , m and h k (t, x)=0, k = 1, 2, . . . , p , then we can use any nonlinear programming solver (e.g., Sequential Quadratic Programming, interior point method or fmincon in Matlab). , t ∈ [0, 1], it is easy to check that the unique solution of (5.1) is given by

By the way of conclusion
In this paper, using tools from convex analysis, we studied the well-posedness of some variants of the sweeping process within the framework of measure differential inclusions and evolution variational inequalities. We proved that the perturbed measure differential inclusion (4.1) has a unique right continuous solution with bounded variation. Under the assumption that the moving set C(t) has a continuous variation for every t ∈ [0, T ] with C(0) bounded and the coercivity of the linear operator A 0 , we proved that the sweeping process (1.5) with velocity in the moving set has a unique Lipschitz continuous solution. There remain many issues that need answers and further investigation. For example, as a consequence of the preceding assumption on C, it results that the set C(t) is bounded for every t ∈ [0, T ]. This assumption is essential in the proof of Theorem 5.1. It would be interesting to extend this result to the case of unbounded convex moving sets. As shown in the counter-example 1, the sweeping process problem (1.5) generated by an unbounded moving set can fail to have a solution for some f . We think that some compatibility conditions on f are needed to prove the existence of at least one solution. In some applications, the assumption of the convexity of C(t) is not satisfied, it will be also interesting to investigate the case of prox-regular sets C(t). This is out of the scope of the current manuscript and will be the aim of a future work.