Existence of solutions for a Lipschitzian vibroimpact problem with time-dependent constraints

We study a mechanical system with a finite number of degrees of freedom, subjected to perfect time-dependent frictionless unilateral (possibly nonconvex) constraints with inelastic collisions on active constraints. The dynamic is described in the form of a second-order measure differential inclusion. Under some regularity assumptions on the data, we establish several properties of the set of admissible positions, which is not necessarily convex but assumed to be uniformly prox-regular. Our approach does not require any second-order information or boundedness of the Hessians of the constraints involved in the problem and are specific to moving sets represented by inequalities constraints. On that basis, we are able to discretize our problem by the time-stepping algorithm and construct a sequence of approximate solutions. It is shown that this sequence possesses a subsequence converging to a solution of the initial problem. This methodology is not only used to prove an existence result but could be also used to solve numerically the vibroimpact problem with time-dependent nonconvex constraints.


Introduction
Vibroimpact systems are dynamical multibody systems subjected to perfect nonpenetration conditions that generate vibrations and impacts. Because of the impact laws, the systems involve discontinuities in the velocity and the acceleration may contain Dirac measures. Hence, vibroimpact systems cannot be modeled by ordinary differential equations, and one uses measure differential inclusions (see, e.g., [3, 20-24, 30, 35]).
In this paper, we consider a mechanical system with a finite number of degrees of freedom, subjected to perfect time-dependent unilateral constraints. More precisely, let I = [0, T], T > 0, be a bounded time real interval and d ∈ N * := {1, 2, . . . }. Let g : I × R d → R d and f i : I × R d → R, i ∈ {1, . . . , m} be some functions and m ∈ N * . We denote by q ∈ R d the representative point of the system in generalized coordinates and define the set of ( 1 ) where N C(t) (q(t)) is the Clarke normal cone [13, p. 51] to C(t) at q(t), t ∈ I.
Denote by ∇f i (t, ·)(q) the derivative of f i (t, q) with respect to the second variable q and by ∂f i (·, q) the derivative of f i with respect to the first variable t. In what follows, given a set ⊂ R d , we denote its interior and boundary, respectively, by int( ) and ∂ .
Since N C(t) (q) = ∅ if q(t) / ∈ C(t), if q is a solution of (1), then q(t) must belong to C(t) for all t ∈ I. If q(t) ∈ int(C(t)) for all t ∈ I, then N C(t) (q(t)) = {0} for all t ∈ I, so (1) becomes q = g(t, q), which is an ordinary differential equation.
Observe that the set T (t, q) is a polyhedral convex closed set for each pair (t, q). The inclusion (2) will be proved in Sect. 4.2. Note that the functionq may be discontinuous at some t ∈ I if J(t, q(t)) is nonempty. Therefore, in general, we cannot find a solution q of (1) for which there exists a differentiable derivativeq. Hence, we look for a solution q of (1) whose derivativeq is of bounded variation. The latter implies thatq is differentiable almost everywhere on I. Then,q can be understood as a Stieltjes measure. Therefore, (1) can be extended in the distributional sense: dqg(·, q(·)) dt ∈ -N C(·) (q(·)) dt, g ·, q(·) dt, ϕ = I g t, q(t) , ϕ(t) dt, ξ (·) dt : C I, R d → R; Since the relation (2) does not uniquely defineq(t + ), we will follow [21] to impose the following inealstic impact laẇ q t + = P T (t,q(t)) q t -, where P T (t,q(t)) (q(t -)) is the nearest point ofq(t -) in T (t, q(t)). In fact, J.-J. Moreau introduced the notion of inelastic shocks in 1983 in the paper [21] (see also [21,22]).
There are many existence results for the vibroimpact problems with time-independent constraints (i.e., when the set of admissible positions does not depend on time: C(t) = C for t ∈ [0, T]). In the single-constraint case, the results have been established by using the position-based algorithm in [32][33][34] and by using the velocity-based algorithm in [15,16,[18][19][20]. In the multiconstraint case, several results have been obtained in [6,25,26,28].
For vibroimpact problems with time-dependent constraints (i.e., when the set of admissible positions C(t) depends on time), there are few solution existence theorems. Let us list some important results related to this case that are known in the literature: Schatzman [35] established an existence result by considering a generalization of the Yosida-type approximation proposed in [31].
Assuming that the set of admissible positions at any instant is defined as a finite intersection of complements of convex sets, Bernicot and Lefebvre-Lepot [7] obtained an existence theorem.
Paoli [27,29] proposed a time-stepping approximation scheme for the problem and proved its convergence, which gives as a byproduct a global existence result when the set of admissible positions at any instant is defined by a finite family of C 2 functions.
Attouch, Cabot and Redont [3] studied the dynamics of elastic shocks via epigraphical regularization of the nonsmooth convex potential and established an asymptotic analysis of the solutions when time t → +∞.
Cabot and Paoli [12] studied the convergence of trajectories and the exponential decay of the energy function associated to a vibroimpact problem with a linear dissipation term.
Attouch, Manigé and Redont [4] studied a nonsmooth second-order differential inclusion involving a Hessian-driven damping with applications to nonelastic shock laws. The existence of solutions for these second-order differential problems has been studied by Bernicot and Venel [9] in a general and abstract framework. More precisely, the set C(t) of admissible positions is assumed in [9] to be Lipschitz continuous in the Hausdorff distance sense and satisfies an "admissibility" property (see Sect. 2.3 [9]). The authors also considered a particular case, where the constraints are C 2 functions and have bounded second-order derivatives (see Sect. 4 in [9]). The assumptions used in this paper require less regularity on the data of the problem and could be seen as a complementary result of Theorem 3.2 and an improvement of Theorem 4.6 in [9] (see Remark 4.2 for more details).
In this paper, we give explicit conditions for the constraints without requiring any second-order differentiability information on the data involved in the constraints. We will follow the time-stepping scheme of [27] to prove the convergence of the approximate solutions. An illustrative example is given to clarify the applicability of the obtained result.
Our main result is an analog of the Peano solution existence theorem [17, Theorem 2.1, p. 10] for ordinary differential equations. Among other things, the proof relies on the Ascoli-Arzelà theorem, and the Banach-Alaoglu theorem applied to the Radon measure space M(0, T; R d ), which is the dual space of the space of all continuous functions from [0, T] to R d . Note that, as shown by Bounkhel [10], one can obtain existence theorems for first-and second-order nonconvex sweeping processes with perturbations by applying a fixed-point theorem.
The paper is organized as follows. In Sect. 2, we recall some preliminaries. In Sect. 3, we formulate our regularity assumptions and deduce several properties of the set of admissible positions and its Clarke's normal cone. Section 4 presents the time-discretization scheme to construct a sequence of approximate solutions and establishes the main result of the paper. The convergence of the sequence of approximate solutions is investigated in Sect. 4.1. In Sects. 4.2 and 4.3, we prove that the limit trajectory is a solution of problem (P). To check the applicability of our result and to compare them with the existing ones, an example is presented in Sect. 5. Some concluding remarks are given in the final section.

Preliminaries
First, we recall some basic concepts and facts from nonsmooth analysis, which are widely used in what follows. We mainly follow the references [5,13,14] and [20]. Our notation is standard in variational analysis; see, e.g., [13].
Let the Euclidean space R d be equipped with a standard scalar product ·, · and the Euclidean norm · . The open ball (resp., closed ball) in R d with center x and radius r is denoted by B(x, r) (resp.,B(x, r)). The open unit ball and closed unit ball are denoted, respectively, by B andB.
The distance function d C (·) :  is defined by Let C be a closed subset of R d and x ∈ C.
is called the Clarke tangent cone to C at x. The Clarke normal cone to C at x is defined by polarity with T C (x): for all x ∈ U.
The set of such vectors, which is denoted by N P C (x), is said to be the proximal normal cone of C at x.

Definition 2.5
The set C is said to be r-prox-regular (or uniformly prox-regular with constant r > 0) whenever, for all x ∈ C, for all ξ ∈ N P C (x) ∩ B, and for all t ∈ (0, r), one has x ∈ P C (x + tξ ).
The following proposition provides a representation for the Clarke normal cone to a set, given by inequalities constraints, under some suitable assumptions.
where the supremum is taken over all finite partitions a = x 0 < x 1 < · · · < x n = b of [a, b].
If Var(f , [a, b]) < +∞, then one says that f is a function of bounded variation on [a, b] and The next proposition is a consequence of the Ascoli-Arzelà Theorem and the Banach-Alaoglu Theorem, which gives sufficient conditions for the existence of a convergence subsequence of a sequence of absolutely continuous functions. (ii) There exists a positive function c(·) ∈ L 1 (I, R) such that ẋ k (t) ≤ c(·) for almost all t ∈ I. Then, there exists a subsequence, still denoted by {x k (·)}, converging to an absolutely continuous function x(·) from I to X in the sense that (a) x k (·) converges uniformly to x(·) over compact subsets of I; (b)ẋ k (·) converges weakly toẋ(·) in L 1 (I, X).

Assumption A2
There is μ > 0 with the property that for all t ∈ [0, T] and q ∈ C(t) there Remark 3.1 From Assumption A1(i), it follows that We are going to present some characterizations of the set of admissible positions C(t) and the Clarke's normal cone N C(t) (q). Thanks to Assumptions A1 and A2, the following proposition is valid.
Following the technique used in [1], we obtain the following proposition, which gives sufficient conditions to obtain Lipschitz continuity of the moving constraint set with respect to the Hausdorff distance.

Proposition 3.2 Under Assumptions
Proof Fix a real number ϑ such that ϑ ≥ μ -1 L. Choose a subdivision , by the mean-value theorem there exists λ ∈ (0, 1) such that

By (3) and the inclusion x ∈ C(t) we obtain
where the inequality is valid due to the choice of ϑ. Since i ∈ {1, . . . , m} can be chosen arbitrarily, this implies that the vector

An existence result for the vibroimpact problem
The approximate solutions will be constructed by the following time-discretization scheme. Let N be a positive natural number and h = T/N , we define t n = nh for all 0 ≤ n ≤ N and Here, argmin x∈C(t n+1 ) V nx denotes the solution set of the minimization problem min x∈C(t n+1 ) V nx .
In this scheme, we use the approximation Clearly, this leads to (4). We define the discrete velocities as The sequence of approximate solutions q N is given by For the existence of a solution to our problem we will need the following assumptions: Assumption A3 For all q ∈ R d , g(·, q) is measurable on [0, T] and for all t ∈ [0, T], g(t, ·) is continuous on R d . Moreover, there exist L g > 0 and F ∈ L 1 (0, T; R) such that for almost every t ∈ [0, T] one has , and for all j, k ∈ J(t, q) and j = k, one has Proposition 4.1 Under Assumptions A1(i) and A2, for any t ∈ I and q ∈ C(t), the Clarke normal cone to C(t) at q can be computed by the formula Proof If q ∈ int(C(t)), then the Clarke tangent cone is equal to the whole space Hence, by Proposition 2.1 we obtain the desired formula for N C(t) (q).
From Proposition 4.1 we can deduce the next formula for computing the corresponding Clarke tangent cone: Lemma 4.1 Let t ∈ [0, T], q ∈ C(t) and v = v(t, q) be the vector that exists by Assumption A2. There exist ρ > 0, τ ∈ (0, ρ ] and θ ∈ (0, ρ ] such that for all t ∈ I, |t -t| ≤ τ , and for all q from the open ball B(q, θ ) centered at q with radius θ , Hence, Our main result is the next theorem. To make the proof of this theorem easier to understand, we present it in the forthcoming three subsections.

Convergence of the approximate solutions
In this subsection, we shall prove that the discrete sequence {q N } constructed in the latter section converges to a limit, which will later be verified to be a solution of problem (P). More precisely, we will prove that {p N } is uniformly bounded and it has bounded variation in Propositions 4.2 and 4.3.

Properties of the limit trajectory
In this subsection, we will prove that the limit trajectory q satisfies the properties (P1)-(P3). The definitions of q N and p N imply that Passing to the limit as N → +∞, by the dominated convergence theorem [11,Theorem 4.2,p. 90] we obtain Hence,q = p ∈ BV ([0, T]; R d ), which implies that q is Lipschitz continuous with rank κ on [0, T].

Proposition 4.4 For all t ∈ [0, T], q(t) ∈ C(t).
Proof Indeed, for all t ∈ [0, T] and for all N > N 1 , there exists n ∈ {0, . . . N -1} such that t ∈ [t n , t n+1 ]. Then, for all i ∈ {1, . . . , m}, Since q is Lipschitz continuous with modulus κ, we have Since {q N } converges uniformly to q on [0, T], f i (t n , q N (t n )) = f i (t n , Q n ) ≤ 0, and (11) holds for all N > N 1 , we can conclude that f i (t, q(t)) ≤ 0. The proof is complete. We are now going to show that the limit trajectory satisfies property (P3). By the definition of p N , the Stieltjes measure dq N = dp N is a sum of Dirac's measures dp N and where the constants λ n i are given in Remark 4.1. Then, (8) can be rewritten as λ n i ∇f i (t n+1 , ·)(Q n+1 ) ≤ P n -P n-1 + h G n .
By Assumption A1(ii), for fixed n, there exists v such that ∇f i (t n+1 , ·)(Q n+1 ), v ≤ -μ. Hence, For every fixed i, we have The proof is complete.
. By the above lemma, N i is uniformly bounded, then there exists a subsequence of { N i } converging weakly * to nonnegative measure i in M(0, T; R). Therefore, U N has a subsequence that converges weakly * to U in M(0, . Since ∇f i (t, ·)(q(t)) ∈ N C(t) (q(t)), we obtain U ∈ N C(·) (q(·)) dt.
Moreover, for all n ∈ {0, . . . , N -1}, we have (t n , q(t n )) ∈ C and Let ε n := Q n+1q(t n ) . From Remark 3.1 and Lemma 4.5 it follows that In addition, We also have where ω ϕ denotes the modulus of continuity of ϕ. Therefore, letting N go to ∞ in (13) we obtain The proof is complete.

Checking the impact law and the initial data
In this subsection, we will prove that the limit trajectory satisfies the impact law (P4) and the initial data (P5).
Proof Let t ∈ I be chosen arbitrarily. Consider an index i such that f i (t, q(t)) = 0. We have Dividing both sides by ε and letting ε → 0, we obtain ∂f i ·, q(t) (t) + ∇f i (t, ·) q(t) ,q t + ≤ 0.
From this it follows that 0 ≤ x i,n ≤ i∈J 2 (t,q(t)) Since {x n } is a convergent sequence, there exists l > 0 such that for each i ∈ J 2 (t, q(t)) we have 0 ≤ x i,n < l for all n. Hence, there exists a subsequence of {x i,n }, denoted by {x i,n } and a nonnegative number x * i such that for all i ∈ J 2 (t, q(t)) Since the sequence {x n } converges to x * , the sequence {x n } also converges to x * . We have From this we obtain the limit We have shown that k∈J(t,q) R + ∇f i (t, ·)(q) is closed. Hence, by (16) we obtain the desired result.
As T (t, q(t)) is a closed convex subset of R d , the above is equivalent tȯ q t + = P T (t,q(t)) q t -.
Remark 4.2 A similar existence result was proved in [9,Theorem 4.6]. Let us mention that our proof does not require any second-order information or boundedness on the constraints f i such as (A3) and (A4) used in [9]. In fact, the boundedness conditions on |∇ 2 f i (t, ·)(q)| and |∂ 2 f i (·, q)(t)| + |∂(∇f i (·, ·)(q))(t)| used in [9] are not necessary in our analysis. Moreover, the condition (R q ) used in [9] is replaced here by the weak uniform Slater condition A2. Our existence result is more specific to constraints inequalities, uses less regularity assumptions on the constraints f i and could be seen as complementary to [9,Theorem 3.2]. In fact, Theorem 3.2 in [9] gives a global existence result for second-order differential inclusions involving a general abstract prox-regular and Lipschitz continuous set C(t). When applying this result to the particular case of finite inequality constraints two main questions arise: under which conditions on the data f i the set C(t) is Lipschitz continuous? and is prox-regular? It is well known that the sublevel of prox-regular functions may fail to be prox-regular and also the prox-regularity of sets is not stable under intersection (see [2] for more details). Our aim here is to give some verifiable and practical conditions on the data f i to satisfy both the prox-regularity and Lipschitz continuity properties of the set C(t) in (18). Another way to obtain Theorem 4.1 is to assume A1-A3 to prove via Propositions 3.1 and 3.2 the prox-regularity and the Lipschitz continuity of the set C(t) given in (18) and then apply the general Theorem 3.2 in [9]. For the convenience of the reader, we prefer to give a direct and self-contained proof specific to constraints inequalities based on the time-stepping algorithm. We mention that this technique for proving the existence result for nonsmooth second-order differential inclusion problems was also used in [7,8,27]. The following example shows that the Assumptions (A3) and (A4) in [9] could not be satisfied.