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Existence of solutions for a Lipschitzian vibroimpact problem with time-dependent constraints
Fixed Point Theory and Algorithms for Sciences and Engineering volume 2022, Article number: 3 (2022)
Abstract
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.
1 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 \(d\in {\mathbb{N}}^{*}:=\{1,2,\dots \}\). Let \(g: I\times \mathbb{R}^{d} \rightarrow \mathbb{R}^{d}\) and \(f_{i}: I\times \mathbb{R}^{d} \rightarrow \mathbb{R}\), \(i \in \{1,\ldots, m\}\) be some functions and \(m \in {\mathbb{N}}^{*}\). We denote by \(q\in \mathbb{R}^{d}\) the representative point of the system in generalized coordinates and define the set of admissible positions at each instant \(t\in I\) by
and the set of active constraints by \(J(t, q)= \{i \in \{1,\ldots, m\} \mid f_{i }(t, q)=0 \}\). The vibroimpact system given by g and the functions \(f_{i}\) is formally described by the following second-order differential inclusion in \(\mathbb{R}^{d}\):
where \({\mathcal{N}}_{C(t)}(q(t))\) is the Clarke normal cone [13, p. 51] to \(C(t)\) at \(q(t)\), \(t\in I\).
Denote by \(\nabla f_{i}(t, \cdot )(q)\) the derivative of \(f_{i}(t, q)\) with respect to the second variable q and by \(\partial f_{i}(\cdot,q)\) the derivative of \(f_{i}\) with respect to the first variable t. In what follows, given a set \(\Omega \subset \mathbb{R}^{d}\), we denote its interior and boundary, respectively, by \(\operatorname{int}(\Omega )\) and ∂Ω.
Since \({\mathcal{N}}_{C(t)}(q)=\emptyset \) if \(q(t)\notin C(t)\), if q is a solution of (1), then \(q(t)\) must belong to \(C(t)\) for all \(t\in I\). If \(q(t) \in \operatorname{int}(C(t))\) for all \(t\in I\), then \({\mathcal{N}}_{C(t)}(q(t))=\{0\}\) for all \(t\in I\), so (1) becomes \(\ddot{q}=g(t, q)\), which is an ordinary differential equation.
If \(q(t)\in \operatorname{int} (C(t))\) for all \(t\in (t_{0}, t_{1})\cup (t_{1}, t_{2})\), \(q(t_{1})\in \partial C(t_{1})\), then
where
Observe that the set \({\mathcal{T}}(t,q)\) is a polyhedral convex closed set for each pair \((t,q)\). The inclusion (2) will be proved in Sect. 4.2.
Note that the function q̇ may be discontinuous at some \(t\in I\) if \(J(t, q(t))\) is nonempty. Therefore, in general, we cannot find a solution q of (1) for which there exists a differentiable derivative q̇. Hence, we look for a solution q of (1) whose derivative q̇ is of bounded variation. The latter implies that q̇ is differentiable almost everywhere on I. Then, q̈ can be understood as a Stieltjes measure. Therefore, (1) can be extended in the distributional sense:
where \(\mathit{BV}([0, T]; \mathbb{R}^{d})\) stands for the space of all functions of bounded variation from \([0,T]\) to \(\mathbb{R}^{d}\). More precisely, the second inclusion is taken in the Radon measure space \({\mathcal{M}}(0,T;\mathbb{R}^{d})\), which is the dual space of the space of all continuous functions from \([0,T]\) to \(\mathbb{R}^{d}\), denoted by \(C([0,T],\mathbb{R}^{d})\). For \(\varphi \in C(I,\mathbb{R}^{d})\) and for \(\xi (\cdot ) \in -{\mathcal{N}}_{C(\cdot )}(q(\cdot ))\),
Since the relation (2) does not uniquely define \(\dot{q}(t^{+})\), we will follow [21] to impose the following inealstic impact law
where \(\mathbb{P}_{\mathcal{T}(t,q(t))}(\dot{q}(t^{-}))\) is the nearest point of \(\dot{q}(t^{-})\) in \(\mathcal{T}(t,q(t))\). In fact, J.-J. Moreau introduced the notion of inelastic shocks in 1983 in the paper [21] (see also [21, 22]).
To sum up, we are interested in investigating the following problem.
Problem \(({\mathcal{P}})\). Let \((q_{0},p_{0}) \in C(0) \times \mathcal{T}(0,q_{0})\). Find \(q:[0,T]\to \mathbb{R}^{d}\), with \(T >0\), such that
(P1) q is absolutely continuous on \([0,T]\), \(\dot{q} \in {\mathit{BV}}(0,T;\mathbb{R}^{d})\);
(P2) \(q(t)\in C(t)\) for all \(t\in [0,T]\);
(P3) \(d\dot{q}-g(\cdot, q(\cdot ))\,dt \in -{\mathcal{N}}_{C(\cdot )}(q( \cdot )) \,dt\);
(P4) \(\dot{q}(t^{+}) = \mathbb{P}_{\mathcal{T}(t,q(t))}(\dot{q}(t^{-}))\) for all \(t\in [0,T]\);
(P5) \(q(0)=q_{0}\) and \(\dot{q}(0)=p_{0}\).
Under some appropriate regularity assumptions on the data, we will prove the existence of at least one solution to problem \(({\mathcal{P}})\). Namely, by using a time-discretization scheme, we will construct a sequence of approximate solutions that has a subsequence converging to a solution of \(({\mathcal{P}})\).
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 \(t\in [0,T]\)). In the single-constraint case, the results have been established by using the position-based algorithm in [32–34] and by using the velocity-based algorithm in [15, 16, 18–20]. 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 \(t\to +\infty \).
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 \({\mathcal{M}}(0,T;\mathbb{R}^{d})\), which is the dual space of the space of all continuous functions from \([0,T]\) to \(\mathbb{R}^{d}\). 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 \(({\mathcal{P}})\). 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.
2 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 \(\mathbb{R}^{d}\) be equipped with a standard scalar product \(\langle \cdot, \cdot \rangle \) and the Euclidean norm \(\Vert \cdot \Vert \). The open ball (resp., closed ball) in \(\mathbb{R}^{d} \) with center x and radius r is denoted by \(\mathbb{B} (x, r) \) (resp., \(\bar{\mathbb{B}} (x, r) \)). The open unit ball and closed unit ball are denoted, respectively, by \(\mathbb{B}\) and \(\bar{\mathbb{B}}\).
The distance function \(d_{C}(\cdot ): \mathbb{R}^{d}\rightarrow \mathbb{R}\), where C is a nonempty subset of \(\mathbb{R}^{d}\), is defined by setting \(d_{C}(x)=\inf \{\Vert x-y\Vert \mid y\in C\}\). For \(\rho >0\), the set \(U_{\rho }(C) = \{x \in \mathbb{R}^{d}\mid d_{C}(x) { < } \rho \}\) is called the ρ-enlargement \(U_{\rho }(C)\) of C. For x in \(\mathbb{R}^{d}\), the set of the nearest points of x in C is called the projection of x onto C and is defined by \(\mathbb{P}_{C}(x)= \{ y\in C \mid \Vert y-x\Vert = d_{C}(x) \}\).
A function \(f: Y\to \mathbb{R}\) defined on \(Y \subset \mathbb{R}^{d} \) is said to be Lipschitz continuous with modulus \(L>0\) on Y if \(\vert f(y)-f(y^{\prime })\vert \leq L\Vert y-y^{\prime }\Vert \) for all \(y, y^{\prime }\in Y\).
Definition 2.1
Let f be Lipschitz continuous near x in \(\mathbb{R}^{d}\) and let v be any vector in \(\mathbb{R}^{d}\). Clarke’s generalized directional derivative of f at x in the direction v, denoted by \(f^{0}(x; v)\), is defined by
Let C be a closed subset of \(\mathbb{R}^{d}\) and \(x\in C\).
Definition 2.2
The set \({\mathcal{T}}_{C}(x):=\{ v\in \mathbb{R}^{d} \mid d^{0}_{C}(x; v)=0\}\) is called the Clarke tangent cone to C at x. The Clarke normal cone to C at x is defined by polarity with \({\mathcal{T}}_{C}(x)\):
Definition 2.3
A vector \(v\in \mathbb{R}^{d}\) is a proximal subgradient of a function \(f:\mathbb{R}^{d}\to \mathbb{R}\) at x if there exist a real number \(\sigma \geq 0\) and a neighborhood U of x such that
for all \(x'\in U\).
Definition 2.4
A vector \(v\in \mathbb{R}^{d}\) is a proximal normal vector to C at \(x \in C\) when it is a proximal subgradient of the indicator function of C, that is, when there exist a constant \(\sigma \geq 0\) and a neighborhood U of x such that \(\langle v, x^{\prime }-x\rangle \leq \sigma \Vert x^{\prime }-x \Vert ^{2}\) for all \(x^{\prime }\in U\cap C\). The set of such vectors, which is denoted by \({\mathcal{N}}^{P}_{C} (x)\), 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 \(x \in C\), for all \(\xi \in {\mathcal{N}}^{P}_{C}(x) \cap \mathbb{B}\), and for all \(t\in (0,r)\), one has \(x \in \mathbb{P}_{C}(x + t\xi )\).
Remark 2.1
If C is uniformly prox-regular, then \({\mathcal{N}}^{P}_{C}(x) = {\mathcal{N}}_{C}(x) \).
The following proposition provides a representation for the Clarke normal cone to a set, given by inequalities constraints, under some suitable assumptions.
Proposition 2.1
(See [13] Corollary 2 of Theorem 2.4.7)
Let C be given as follows:
and let x be such that \(f_{i}(x) = 0\) for \(i = 1,\dots,m\). Then, if each \(f_{i}\) is differentiable at x and if the gradients \(\nabla f_{i}(x), i=1,\dots,m\), are positively linearly independent, we have
Lemma 2.1
(See [2, Lemma 3.2])
Let \(C\subset \mathbb{R}^{d}\) and \(x, y \in C\) with \(\Vert x-y\Vert < 2\rho \), where \(\rho \in (0,+\infty ]\). Then, for any \(\tau \in [0, 1]\) one has \(x + \tau (y - x) \in U_{\rho }(C)\).
Definition 2.6
Let \(f:[a,b] \to \mathbb{R}^{d}\) be a function. The total variation of f on \([a,b]\) is the nonnegative extended real number
where the supremum is taken over all finite partitions \(a=x_{0}< x_{1}<\cdots <x_{n}=b\) of \([a,b]\). If \(\operatorname{Var}(f,[a,b]) < +\infty \), then one says that f is a function of bounded variation on \([a,b]\) and writes \(f\in \mathit{BV}([a,b],\mathbb{R}^{d})\).
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.
Proposition 2.2
(See [5, Theorem 4, p. 13])
Let \(\{x_{k}(\cdot )\}\) be a sequence of absolutely continuous functions from an interval \(I\subset {\mathbb{R}}\) to a Banach space X satisfying
-
(i)
For all \(t\in I\), \(\{x_{k}(t)\}_{k}\) is a relatively compact subset of X;
-
(ii)
There exists a positive function \(c(\cdot )\in L^{1}(I,{\mathbb{R}})\) such that \(\Vert \dot{x}_{k}(t)\Vert \leq c(\cdot )\) for almost all \(t\in I\).
Then, there exists a subsequence, still denoted by \(\{x_{k}(\cdot )\}\), converging to an absolutely continuous function \(x(\cdot )\) from I to X in the sense that
-
(a)
\(x_{k}(\cdot )\) converges uniformly to \(x(\cdot )\) over compact subsets of I;
-
(b)
\(\dot{x}_{k}(\cdot )\) converges weakly to \(\dot{x}(\cdot )\) in \(L^{1}(I,X)\).
3 The framework
We now propose some regularity assumptions. In the notation of Sect. 1, let
Assumption A1
There exists an extended real \(\rho \in (0,+\infty ]\) such that
-
(i)
for all \(i\in \{1,\ldots, m\}\), \(f_{i}\) is differentiable on \(U_{\rho }(C)\) and its derivative \(\nabla f_{i}(\cdot,\cdot ):U_{\rho }(C)\to \mathbb{R}\) is Lipschitz continuous with rank L;
-
(ii)
there is \(\gamma >0\) such that for all \(t\in [0,T]\) and \(i\in \{1,\ldots, m\}\), for all \(q_{1}, q_{2} \in U_{\rho }(C(t))\),
$$\begin{aligned} \bigl\langle \nabla f_{i}(t,\cdot ) (q_{1})-\nabla f_{i}(t,\cdot ) (q_{2}), q_{1}-q_{2} \bigr\rangle \geq -\gamma \Vert q_{1}-q_{2} \Vert ^{2}; \end{aligned}$$ -
(iii)
for all \(t\in [0,T]\) and for all \(i\in \{1,\dots,m\}\), one has \(\Vert \nabla f_{i}(t,\cdot )(q) \Vert \leq L\) for all \(q\in U_{\rho }(C(t))\).
Assumption A2
There is \(\mu >0\) with the property that for all \(t\in [0,T]\) and \(q\in C(t)\) there exists \(v=v(t,q)\in \mathbb{R}^{d}\) with \(\Vert v \Vert =1\) such that for all \(i\in \{1,\ldots, m\}\), one has
Remark 3.1
From Assumption A1(i), it follows that
-
(i)
For each \(i \in \{1,\dots, m \}\), for all \(t,t^{\prime }\in [0,T]\) and \(q, q^{\prime }\in \mathbb{R}^{d}\),
$$\begin{aligned} \bigl\vert \partial f_{i}(\cdot,q) (t)- \partial f_{i} \bigl(\cdot,q^{\prime }\bigr) \bigl(t^{\prime }\bigr) \bigr\vert \leq L \bigl( \bigl\vert t-t^{\prime } \bigr\vert + \bigl\Vert q-q^{\prime } \bigr\Vert \bigr); \end{aligned}$$ -
(ii)
for each \(i \in \{1,\dots, m \}\), for all \(t,t^{\prime }\in [0,T]\), \(q, q^{\prime }\in U_{\rho }(C(t))\),
$$\begin{aligned} \bigl\Vert \nabla f_{i}(t,\cdot ) (q)-\nabla f_{i} \bigl(t^{\prime }, \cdot \bigr) \bigl(q^{\prime }\bigr) \bigr\Vert \leq L\bigl( \bigl\vert t-t^{\prime } \bigr\vert + \bigl\Vert q-q^{\prime } \bigr\Vert \bigr). \end{aligned}$$
Remark 3.2
From Assumptions A1 and A2, it follows that for all \(i\in \{1,\dots,m\}\), \(\mu \leq \Vert \nabla f_{i}(t,\cdot )(q)\Vert \leq L\) for all \(t\in [0,T]\) and \(\vert \partial f_{i}(\cdot,q)(t)\vert \leq L\) for all \(q\in U_{\rho }(C(t))\). In particular, \(\nabla f_{i}(t,\cdot )(q)\neq 0\) for all \(i\in \{1,\dots,m\}\).
We are going to present some characterizations of the set of admissible positions \(C(t)\) and the Clarke’s normal cone \({\mathcal{N}}_{C(t)}(q)\). Thanks to Assumptions A1 and A2, the following proposition is valid.
Proposition 3.1
(See [2, Theorem 3.1])
Suppose that Assumptions A1(i)–(ii) and A2hold, then, for all \(t\in [0,T]\), the set \(C(t)\) is r-prox-regular with \(r=\min \{\rho, \frac{\mu }{\gamma }\}\).
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 A1(i) and A2, \(C(\cdot )\) is ϑ-Lipschitzian on \([0, T]\), with \(\vartheta \geq \frac{L}{\mu }\).
Proof
Fix a real number ϑ such that \(\vartheta \geq \mu ^{-1}L\). Choose a subdivision
of \([0,T]\) such that \(T_{k}-T_{k-1}<\frac{1}{\vartheta } \rho \). Fix any k and select \(s,t\in I_{k}:=[T_{k-1}, T_{k}]\). Then, take any \(i\in \{1,\ldots, m\}\). Put \(u(s,t)=\vartheta \vert s-t\vert \). For any \(x\in C(t)\), define \(y:=x+u(s,t)v\). Since \(t,s\in I_{k}\), we have \(\Vert y-x\Vert =\vartheta \vert s-t\vert < \rho \). This proves that \(y\in \operatorname{int}(U_{\rho }(C(t)))\). By Lemma 2.1, for all \(\lambda \in [0,1]\) we have
Now, we consider the expression \(f_{i}(t,x+u(s,t)v)-f_{i}(t,x)\). Since \(f_{i}(s,\cdot )\) is differentiable on \(U_{\rho }(C(t))\), by the mean-value theorem there exists \(\lambda \in (0,1)\) such that
with \(x_{\lambda }=\lambda x+(1-\lambda )(x+u(s,t)v)\). Hence, by Remark 3.1, we have
By (3) and the inclusion \(x\in C(t)\) we obtain
where the inequality is valid due to the choice of ϑ. Since \(i\in \{1,\ldots, m\}\) can be chosen arbitrarily, this implies that the vector \(x+u(s,t)v=x+\vartheta \vert s-t\vert v\) belongs to \(C(s)\). Hence, \(x\in C(s)+\vartheta \vert s-t\vert (-v)\). It follows that
Thus, \(C(\cdot )\) is ϑ-Lipschitzian on \([T_{k-1}, T_{k}]\). Hence, we can infer that \(C(\cdot )\) is ϑ-Lipschitzian on \([0,T]\). □
4 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 \(0 \leq n\leq N\) and
-
1.
\(Q_{-1}=q_{0}-h p_{0}\), \(Q_{0}=q_{0}\),
-
2.
for all \(n\in \{0,\dots, N\}\),
$$\begin{aligned} G_{n}= \int _{t_{n}}^{t_{n+1}}g(s,Q_{n}) \,ds \end{aligned}$$and
$$\begin{aligned} V_{n}=2Q_{n}-Q_{n-1}+h^{2}G_{n},\qquad Q_{n+1}\in \mathop {\operatorname {argmin}}_{x\in C(t_{n+1})} \Vert V_{n} -x \Vert . \end{aligned}$$(4)
Here, \(\mathop {\operatorname {argmin}}_{x\in C(t_{n+1})} \Vert V_{n} -x \Vert \) denotes the solution set of the minimization problem \(\min_{x\in C(t_{n+1})} \Vert V_{n} -x \Vert \).
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
and
For the existence of a solution to our problem we will need the following assumptions:
Assumption A3
For all \(q\in \mathbb{R}^{d}\), \(g(\cdot,q)\) is measurable on \([0,T]\) and for all \(t\in [0,T]\), \(g(t,\cdot )\) is continuous on \(\mathbb{R}^{d}\). Moreover, there exist \(L_{g}>0\) and \(F\in L^{1}(0,T;\mathbb{R})\) such that for almost every \(t\in [0,T]\) one has
Assumption A4
For all \(t\in [0,T]\), \(q\in U_{\rho }(C(t))\), and for all \(j, k \in J(t, q)\) and \(j\neq k\), one has
Proposition 4.1
Under Assumptions A1(i) and A2, for any \(t\in I\) and \(q\in C(t)\), the Clarke normal cone to \(C(t)\) at q can be computed by the formula
Proof
If \(q\in \operatorname{int}(C(t))\), then the Clarke tangent cone is equal to the whole space \(\mathbb{R}^{d}\). Therefore, \({\mathcal{N}}_{C(t)}(q) =\{0\}\). Now, we consider the case when q is on the boundary \(\partial C(t)\) of \(C(t)\). Then, \(J(t,q)\neq \emptyset \). From Assumption A2 it follows that \(\{\nabla f_{i}(t,\cdot )(q) \mid i\in J(t,q) \}\) is positively linearly independent. Hence, by Proposition 2.1 we obtain the desired formula for \({\mathcal{N}}_{C(t)}(q)\). □
From Proposition 4.1 we can deduce the next formula for computing the corresponding Clarke tangent cone:
Lemma 4.1
Let \(t\in [0,T]\), \(q\in C(t)\) and \(v=v(t,q)\) be the vector that exists by Assumption A2. There exist \(\rho ^{\prime }>0\), \(\tau \in (0, \rho ^{\prime }]\) and \(\theta \in (0, \rho ^{\prime }]\) such that for all \(t^{\prime }\in I\), \(\vert t^{\prime }- t \vert \leq \tau \), and for all \(q^{\prime }\) from the open ball \(\mathbb{B}(q, \theta )\) centered at q with radius θ,
Proof
Let \(q\in C(t)\), v be defined in A2. For all \(t^{\prime }\in I\), \(q^{\prime }\in \mathbb{R}^{d}\) such that \(\Vert q^{\prime }- q\Vert \leq \rho \), and for any \(i\in \{1,\dots,m\}\), by Remark 3.1(ii) we have
Hence,
Choose \(\tau = \theta = \min \{\mu /3L, \rho \}\). Then, we have \(\langle \nabla f_{i}(t^{\prime }, \cdot )(q^{\prime }), v \rangle \leq - \frac{\mu }{3}\). □
Our main result is the next theorem.
Theorem 4.1
Suppose that Assumptions A1–A3hold. Let \((q_{0}, p_{0})\in C(0)\times {\mathcal{T}}(0, q_{0})\). Then, there is a subsequence of \(\{q_{N}\}\), still denoted by \(\{q_{N}\}\), of the approximate solutions that converges uniformly on \([0, T]\) to a limit q satisfying (P1)–(P3). Furthermore, if Assumption A4holds, then q also satisfies (P4) and (P5), and it is a solution of problem \(({\mathcal{P}})\) on \([0, T]\).
To make the proof of this theorem easier to understand, we present it in the forthcoming three subsections.
4.1 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 \(({\mathcal{P}})\). More precisely, we will prove that \(\{p_{N}\}\) is uniformly bounded and it has bounded variation in Propositions 4.2 and 4.3.
Lemma 4.2
For all \(n\in \{0,\dots, N-1 \}\), one has
Proof
By definition of the scheme, for all \(x \in C(t_{n+1})\), we have
Hence,
By definition, \(V_{n} - Q_{n+1} = h(P_{n-1}-P_{n}+hG_{n})\), hence
If \(Q_{n+1}\in \operatorname{int}(C(t_{n+1}))\), we can choose \(\varepsilon >0\) sufficiently small such that \(x_{1}= Q_{n+1}+\varepsilon E\) and \(x_{2}=Q_{n+1}-\varepsilon E\) belong to \(C(t_{n+1})\), where \(E = (1,\dots,1) \in \mathbb{R}^{d}\). Then, we have
Otherwise \(J(t_{n+1},Q_{n+1}) \neq \emptyset \). We know that by (5), the Clarke’s tangent cone of \(C(t_{n+1})\) at \(Q_{n+1}\) is
Hence, we need to show that
Indeed, by Assumption A2, \(\operatorname{int}({\mathcal{T}}_{C(t_{n+1})}(Q_{n+1}))\ne \emptyset \). Note that
Take any \(\bar{w} \in \operatorname{int}({\mathcal{T}}_{C(t_{n+1})}(Q_{n+1}))\). We will prove that \(Q_{n+1}+s\bar{w}\in C(t_{n+1})\) for \(s>0\) sufficiently small. For any \(s\geq 0\), there exists \(q_{\lambda }:=Q_{n+1}+\lambda s\bar{w}\) with \(\lambda \in (0,1)\), such that
For s small enough such that \(\Vert s\bar{w} \Vert \leq \rho \), we have \(Q_{n+1}+s\bar{w} \in U_{\rho }(C(t_{n+1}))\). By Remark 3.1(ii),
Then, \(\langle \nabla f_{i}(t_{n+1},\cdot )(Q_{n+1}+\lambda s\bar{w}) - \nabla f_{i}(t_{n+1},\cdot )(Q_{n+1}), s\bar{w}\rangle \leq \lambda Ls^{2} \Vert \bar{w} \Vert ^{2}\). Hence,
Since \(\langle \nabla f_{i}(t_{n+1},\cdot )(Q_{n+1}), \bar{w} \rangle <0\), we can choose s small enough such that
This implies that \(Q_{n+1}+s\bar{w} \in C(t_{n+1})\). Now, we choose \(x=Q_{n+1}+s\bar{w}\) satisfying \(x\in C(t_{n+1})\), by (7) we obtain
Letting \(s\to 0\), one has
By Assumption A2, there exits a unit vector \(v(t_{n+1},Q_{n+1})\in \operatorname{int}({\mathcal{T}}_{C(t_{n+1})}(Q_{n+1}))\). Therefore, for all \(v\in {\mathcal{T}}_{C(t_{n+1})}(Q_{n+1})\), the sequence \(\{v_{k}\}_{k\in \mathbb{N}^{*}}\), which is defined by
for all \(k\geq 1\), converges to v. We also see that \(v_{k}\in \operatorname{int}({\mathcal{T}}_{C(t_{n+1})}(Q_{n+1}))\) for all \(k\geq 1\). Hence, \(\operatorname{int}({\mathcal{T}}_{C(t_{n+1})}(Q_{n+1}))\) is dense in \({\mathcal{T}}_{C(t_{n+1})}(Q_{n+1})\). This leads to
which implies that \(P_{n-1}-P_{n} +h G_{n} \in {\mathcal{N}}_{C(t_{n+1})}(Q_{n+1})\). □
Remark 4.1
One can reformulate (6) as follows: For all \(n\in \{ 0, \ldots, N-1\}\), there exist nonnegative real numbers \(\lambda ^{n}_{i}\), \(i = 1,\ldots, m\) such that \(\lambda _{i}^{n}=0\) for all \(i \notin J(t_{n+1}, Q_{n+1})\), and
Lemma 4.3
For each \(i\in J(t_{n+1}, Q_{n+1})\) and \(\Vert P_{n}\Vert \leq \frac{\rho N}{2T}\), one has
Proof
For all \(i \in J(t_{n+1}, Q_{n+1})\), we have \(f_{i}(t_{n+1},Q_{n+1})=0 \geq f_{i}(t_{n},Q_{n})\). Thus,
where \(q^{n}_{\alpha _{i}} = \alpha _{i} Q_{n}+(1-\alpha _{i})Q_{n+1}\) for some \(\alpha _{i}\in (0,1)\). It follows that
Since \(\Vert P_{n}\Vert \leq \frac{\rho N}{2T}\), by Lemma 2.1 we know that \(q^{n}_{\alpha _{i}}\in U_{\rho }(C(t_{n+1}))\). By Assumption A1(ii), we obtain (9). □
Lemma 4.4
Let \(N > N^{0}\), where \(N^{0}= \max \{ \frac{T}{2}, \frac{6TL}{\mu \theta } \} \). Then, for all \(n\in \{0,\dots,N-1\}\), we have
Proof
Let \(w=\frac{6L}{\mu } v(t_{n},Q_{n})\), where \(v(t_{n},Q_{n})\) is the unit vector defined in Assumption A2 for \((t,x)=(t_{n},Q_{n})\), i.e., for all \(i \in \{1,\dots, m\}\), one has \(\langle \nabla f_{i}(t_{n},\cdot )(Q_{n}), v(t_{n},Q_{n}) \rangle \leq -\mu \). Then,
Indeed, by Remark 3.2 and the mean-value theorem, we have
By the mean-value theorem, there exists \(q^{n}_{\alpha }= \alpha Q_{n}+(1-\alpha ) (Q_{n} +hw)\) with \(\alpha \in (0,1)\), such that
Since \(N\geq \frac{6TL}{\mu \theta }\), \(q^{n}_{\alpha }\in B(Q_{n},\theta )\). By Lemma 4.1, we have
Therefore, for all \(i\in \{1,\dots,m\}\),
We have proved that \(Q_{n}+hw \in C(t_{n+1})\). As \(Q_{n+1} \in \mathop {\operatorname {argmin}}_{x\in C(t_{n+1})} \Vert V_{n} -x \Vert \), it follows that
Thus, \(\Vert P_{n-1}-P_{n} + hG_{n} \Vert \leq \Vert P_{n-1}-w+hG_{n}\Vert \). Hence, we obtain \(\Vert P_{n} \Vert \leq 2\Vert P_{n-1}\Vert +2h\Vert G_{n}\Vert + \Vert w\Vert \), which yields the conclusion. □
Proposition 4.2
There exist \(N^{1}> N^{0}\) and \(\kappa >0\) such that
Proof
We now define two real sequences \(\{\kappa _{k}\}_{k\in \mathbb{N}}\) and \(\{\tau _{k}\}_{k\in \mathbb{N}^{*}}\) by setting \(\kappa _{0}=\Vert p_{0}\Vert +1\),
and
It is easy to see that the series \(\sum_{k=1}^{\infty }\tau _{k}\) is a divergent sum, hence, there exists \(k_{0}\geq 1\) such that \(\sum_{k=1}^{k_{0}} \tau _{k}>T\). Let \(\kappa =\kappa _{k_{0}}\). Define
and
We now prove that for all \(N>N^{1}\) and we can construct a finite family of real numbers \((\tau _{k}^{N})_{1\leq k \leq k_{0}}\) such that \(\tau _{0}^{N}=0<\tau _{1}^{N}<\cdots <\tau _{k_{0}^{N}}^{N}=T\) with \(1\leq k_{0}^{N}\leq k_{0}\) and for all \(k\in \{1,\dots,k_{0}^{N}\}\), in each interval \([\tau _{k-1}^{N},\tau _{k}^{N})\), one has
Consider the interval \([0, \tau _{1}]\) instead of \([0,T]\). From Assumption A2, we can define a vector \(w_{0}=\frac{6L}{\mu }v(t_{0},Q_{0})\). Note that \(\Vert P_{-1}\Vert =\Vert p_{0}\Vert \leq \kappa _{0}\leq \kappa \), by Lemma 4.4 we have \(\Vert P_{0}\Vert \leq \bar{\kappa }\). Since \(0< h=\frac{T}{N}\leq \frac{\theta }{2\bar{\kappa }}\),
Moreover, \(\vert t_{1}-t_{0}\vert \leq h \leq \tau /2 < \tau \), we have \((t_{1}, Q_{1})\in \mathbb{B}(t_{0}, \tau )\times \mathbb{B}(Q_{0}, \theta )\). We will prove that \(w_{0}-P_{0}\in {\mathcal{T}}_{C(t_{1})}(Q_{1})\). Indeed, for all \(i \in J(t_{1},Q_{1})\), by Lemma 4.3 one has
From the latter inequality, it follows that \(w_{0}-P_{0}\in {\mathcal{T}}_{C(t_{1})}(Q_{1})\). Since \(P_{-1}-P_{0}+hG_{0} \in {\mathcal{N}}_{C(t_{0})}(Q(0))\), we obtain
This yields \(\langle P_{-1}-w_{0}+hG_{0}, w_{0}-P_{0}\rangle \leq -\Vert P_{0} -w_{0} \Vert ^{2}\), which implies that
Hence,
Next, we will prove by induction that
Indeed, let \(n\in \{0,\ldots, N-1\}\). Suppose that
Then,
and by Lemma 4.4 we infer that \(\Vert P_{n}\Vert \leq \bar{\kappa }\). Since \(0< h\leq \frac{\theta }{2\bar{\kappa }}\),
Moreover, as \(\vert t_{n+1}-t_{n}\vert \leq h < \tau \), we have \((t_{n+1}, Q_{n+1})\in B(t_{n},\tau )\times B(Q_{n}, \theta )\). For all \(i \in J(t_{n+1},Q_{n+1})\), by Lemma 4.3 one has
It follows that \(w_{0}-P_{n} \in {\mathcal{T}}_{C(t_{n+1})}(Q_{n+1})\). Therefore,
This yields
Hence,
We have shown that \(\Vert P_{n}\Vert \leq \kappa _{1}\) for all \(n\in \{0,\dots, N\}\) on the interval \([0,\tau _{1}]\). Putting \(\tau _{0}^{N}=0\), we define \(\tau _{1}^{N}=\min \{\tau ^{N}_{0}+\tau _{1}, T \}\). If \(\tau ^{N}_{0}+\tau _{1}< T\), we have \(\tau _{1}^{N}-\tau ^{N}_{0}=\tau _{1}\). If \(T>\tau _{1}^{N}\), then \(k_{0}>1\), \((t_{N+1}, Q_{N+1})\in C\) and \(\Vert P_{N+1}\Vert \leq \kappa _{1}\leq \kappa \).
Assume now that \(\tau ^{N}_{0}+\tau _{1}< T\). By Lemma 4.1 and Assumption A2, we can define a vector \(w_{1}=\frac{6L}{\mu }v(t_{N+1}, Q_{N+1})\). For the sake of simplicity, we will recount the index from 0 instead of \(N+1\). By the same argument, we can prove that \(\Vert P_{n}\Vert \leq \kappa _{2}\) for all \(n\in \{0,\dots, N-1\}\) on the interval \([\tau ^{N}_{1},\tau ^{N}_{1}+\tau _{2}]\). We now can divide the interval \([0,T]\) into subintervals \([\tau ^{N}_{i}, \tau ^{N}_{i}+\tau _{i+1}]\) for \(i\in \{1,\dots,k_{0} \}\). Repeating the same argument for finitely many steps, we obtain the desired result. □
Proposition 4.3
There exists \(\kappa ^{\prime }>0\) such that, for all \(N>N^{1}\), we have
Proof
We decompose \([0, T]\) into the subintervals \([\tau ^{N}_{k}, \tau ^{N}_{k+1}]\), \(k\in \{0,\ldots, k_{0}^{h}-1\}\), which were defined in the proof of Proposition 4.2. Consider the interval \([\tau ^{N}_{0}, \tau ^{N}_{1}]\). We have shown that
for all \(n\in \{0,\dots,N-1\}\). We now prove that the closed ball \(\bar{\mathbb{B}} (w_{0}-P_{n}, \frac{1}{2} )\subset { \mathcal{T}}_{C(t_{n+1})}(Q_{n+1})\). Indeed, let \(a \in \bar{\mathbb{B}} (w_{0}-P_{n}, \frac{1}{2} )\). Then, \(\Vert a -(w_{0}-P_{n})\Vert \leq \frac{1}{2}\). As in the proof of Proposition 4.2, one has \(\langle \nabla f_{i}(t_{n+1},\cdot )(Q_{n+1}),w_{0}-P_{n}\rangle \leq -\frac{L}{2}\) for all \(n\in \{0,\dots, N-1\}\). Then,
This proves that \(a\in {\mathcal{T}}_{C(t_{n+1})}(Q_{n+1})\). Since the tangent cone \({\mathcal{T}}_{C(t_{n+1})}(Q_{n+1})\) is closed and convex [13, p. 51], for every \(x\in \mathbb{R}^{d}\), by [20, Lemma 4.3, p. 22] we have
Applying this with \(x=P_{n-1}-P_{n}+hG_{n}\), we obtain
where \(\bar{P} = \mathbb{P}_{{\mathcal{T}}_{C(t_{n+1})}(Q_{n+1})}(P_{n-1}-P_{n}+hG_{n})\). It follows that
Recall that \(P_{n-1}-P_{n}+hG_{n}\in {\mathcal{N}}_{C(t_{n+1})}( Q_{n+1})\) (see Lemma 4.2). Since \({\mathcal{N}}_{C(t_{n+1})}(Q_{n+1})\) is the dual cone of \({\mathcal{T}}_{C(t_{n+1})}(Q_{n+1})\), \(\bar{P}=0\). We obtain
It follows that \(\Vert P_{n-1}-P_{n}\Vert \leq h (1+\Vert F\Vert _{L^{1}(0,T; \mathbb{R}^{d}) }+ 2\kappa +\frac{12L}{\mu } ) \Vert G_{n} \Vert +\Vert P_{n-1}-w_{0}\Vert ^{2}-\Vert P_{n}-w_{0}\Vert ^{2}\) for \(n=0,\ldots,N-1\). Adding these inequalities, we obtain
Similarly, we can obtain the same result for all the subintervals \([\tau ^{N}_{i}, \tau ^{N}_{i+1}]\), where \(i\in \{1,\dots,k_{0} \}\). Since the number of the subintervals \([\tau ^{N}_{i}, \tau ^{N}_{i+1}]\) is finite, the proof is complete. □
From Propositions 4.2 and 4.3 we can infer that the sequence \(\{q_{N}\}\) is uniformly Lipschitz continuous and that the sequence \(\{p_{N}\}\) is uniformly bounded in \(L^{\infty }(0, T; \mathbb{R}^{d})\) and in \(\mathit{BV}([0, T]; \mathbb{R}^{d})\). For any \(t\in [0,T]\), it is clear that \(q_{N}(t)\) is bounded for all N. Moreover, since \(p_{N}\) is the derivative of \(q_{N}\), by Proposition 2.2, there exists a subsequence of \(\{q_{N}\}\), still denoted by \(\{q_{N}\}\), converging uniformly to an absolutely continuous function q over \([0,T]\). In addition, by [20, Theorem 2.1], we can extract subsequences of \(\{p_{N}\}\), still denoted by \(\{p_{N}\}\) and find \(p\in \mathit{BV}([0, T]; \mathbb{R}^{d})\) such that
4.2 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 \(N\to +\infty \), by the dominated convergence theorem [11, Theorem 4.2, p. 90] we obtain
Hence, \(\dot{q}= p\in \mathit{BV}([0, T]; \mathbb{R}^{d})\), which implies that q is Lipschitz continuous with rank κ on \([0, T]\).
Proposition 4.4
For all \(t\in [0, T]\), \(q(t)\in C(t)\).
Proof
Indeed, for all \(t\in [0, T]\) and for all \(N>N^{1}\), there exists \(n\in \{0, \ldots N-1\}\) such that \(t\in [t_{n}, t_{n+1}]\). Then, for all \(i\in \{1,\ldots, m\}\),
Since q is Lipschitz continuous with modulus κ, we have
Since \(\{q_{N}\}\) converges uniformly to q on \([0, T]\), \(f_{i}(t_{n}, q_{N}(t_{n}))=f_{i}(t_{n}, Q_{n})\leq 0\), and (11) holds for all \(N>N^{1}\), we can conclude that \(f_{i}(t, q(t))\leq 0\).
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 \(d\dot{q}_{N}=d{p}_{N}\) is a sum of Dirac’s measures
Define
and
where the constants \(\lambda _{i}^{n}\) are given in Remark 4.1. Then, (8) can be rewritten as
Lemma 4.5
For all \(i \in \{1,\ldots, m\}\) and for all \(N>N^{1}\) we have
Proof
Let \(i \in \{1,\ldots, m\}\), \(n\in \{ 0, \ldots, N-1\}\). By (8) we have
By Assumption A1(ii), for fixed n, there exists v such that \(\langle \nabla f_{i}(t_{n+1}, \cdot )(Q_{n+1}), v\rangle \leq -\mu \). Hence,
For every fixed i, we have
Hence,
The proof is complete. □
Let \(\Lambda ^{N}_{i}(t) = \sum_{n=0}^{N-1} \lambda _{i}^{n} \delta (t-t_{n})\). By the above lemma, \(\Lambda ^{N}_{i}\) is uniformly bounded, then there exists a subsequence of \(\{\Lambda ^{N}_{i}\}\) converging weakly∗ to nonnegative measure \(\Lambda _{i}\) in \({\mathcal{M}}(0, T; \mathbb{R})\). Therefore, \(U_{N}\) has a subsequence that converges weakly∗ to U in \({\mathcal{M}}(0, T; \mathbb{R}^{d})\) with \(U(t) = \sum_{i=1}^{m} \Lambda _{i}(t)\nabla f_{i}(t,\cdot )(q(t))\). Since \(\nabla f_{i}(t,\cdot )(q(t)) \in {\mathcal{N}}_{C(t)}(q(t))\), we obtain \(U \in {\mathcal{N}}_{C(\cdot )}(q(\cdot ))\,dt\).
Lemma 4.6
The sequence \(\{g_{N}\}\) converges weakly∗ to \(g(\cdot, q)\,dt \) in \({\mathcal{M}}(0,T;\mathbb{R}^{d}) \), where \(g(\cdot,q)\,dt\) is the measure of density \(g(\cdot, q)\) with respect to Lebesgue’s measure on \([0, T]\).
Proof
Let \(\varphi \in C([0, T]; \mathbb{R}^{d})\). By the definition of \(g_{N}\), we have
Moreover, for all \(n\in \{0, \ldots, N-1\}\), we have \((t_{n}, q(t_{n}))\in C\) and
Let \(\varepsilon _{n}:= \Vert Q_{n+1}-q(t_{n})\Vert \). From Remark 3.1 and Lemma 4.5 it follows that
In addition,
We also have
where \(\omega _{\varphi }\) denotes the modulus of continuity of φ. Therefore, letting N go to ∞ in (13) we obtain
The proof is complete. □
Passing (12) to the limit yields \(dp-g(\cdot, q) \,dt\in -{\mathcal{N}}_{C(\cdot )}(q(\cdot )) \,dt\).
4.3 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).
Lemma 4.7
If \(J(t,q) \neq \emptyset \), then \(\dot{q}(t^{+})\in {\mathcal{T}}(t, q(t))\).
Proof
Let \(t\in I\) be chosen arbitrarily. Consider an index i such that \(f_{i}(t, q(t))=0\). We have
Dividing both sides by ε and letting \(\varepsilon \to 0\), we obtain
We have shown that \(\dot{q}(t^{+})\in {\mathcal{T}}(t, q(t))\).
Similarly, we can prove that \(\dot{q}(t^{-}) \in -{\mathcal{T}}(t, q(t))\). □
Lemma 4.8
For each \(i\in J(t_{n+1}, Q_{n+1})\) and \(\Vert P_{n}\Vert \leq \frac{\rho N}{2T}\), one has
Proof
For all \(i \in J(t_{n+1}, Q_{n+1})\), \(f_{i}(t_{n+1},Q_{n+1})=0 \geq f_{i}(t_{n},Q_{n})\). Thus,
where \(t^{n}_{\alpha } = \alpha t_{n}+(1-\alpha )t_{n+1}\) and \(q^{n}_{\beta } = \beta Q_{n}+(1-\beta )Q_{n+1}\) for some \(\alpha,\beta \in (0,1)\), satisfying
and
Hence,
Since \(h\Vert P_{n}\Vert \leq \frac{\rho }{2}\), by Lemma 2.1 we know that \(q^{n}_{\beta }\in U_{\rho }(C(t_{n+1}))\). Therefore, by Remark 3.1(i),
Then, by Assumption A1(ii), one has
Hence,
The proof is complete. □
Proposition 4.5
For all \(t\in (0,T)\), one has \(\dot{q}(t^{+})=\mathbb{P}_{{\mathcal{T}}(t, q)}( \dot{q}(t^{-}))\).
Proof
Step 1: We consider the case that \(J(t, q(t))=\emptyset \). Since \(f_{i}\) are continuous for all \(i\in \{1,\ldots, m\}\), we may define \(\rho _{t}\in (0, \min (\rho, t, T-t))\) such that, for all \(i \in \{1,\ldots, m\}\) we have
and we define \(N_{t} > \max \{ N^{1}, \frac{4T(\kappa +1)}{\rho _{t}} \} \) such that \(\Vert q-q_{N}\Vert _{C([0, T]; \mathbb{R}^{d})}\leq \frac{\rho _{t}}{4}\) for all \(N>N_{t}\). Then, for all \(\tilde{\rho }\in (0,\rho _{t}]\) and for all \(N>N_{t}\), we define
It follows that
and
By relation (8) we have
Moreover, for all \(n\in \{n_{-},\ldots, n_{+}\}\) we have \(t_{n}=nh\in [t-\frac{\tilde{\rho }}{4(\kappa +1)}, t+ \frac{\tilde{\rho }}{4(\kappa +1)} ]\) and
It follows that \(f_{i}(t_{n+1}, Q_{n+1})< 0\) and \(\lambda ^{n}_{i}=0\) for all \(i\in \{1,\ldots, m\}\) and for all \(n\in \{n_{-}, \dots,n_{+}\}\). Thus,
Letting N go to infinity, we obtain that \(\Vert p(t^{+})-p(t^{-})\Vert = 0\). This means that
Step 2: Now, let \(t\in (0,T)\) be such that \(J(t,q(t))\neq\emptyset \). Consider the case if \(J(t,q(t))=\{1,\ldots,m\}\), we let \(\rho _{t}=\frac{1}{2}\min (\rho, t, T-t)\). Otherwise, using the continuity of the mappings \(f_{i}, i\in \{1,\ldots, m\}\) we may define \(\rho _{t}\) in \((0, \min (\rho, t, T-t) )\) such that, for all \(i\in \{1,\ldots, m\}\setminus J(t,q(t))\) we have
Then, by the uniform convergence of \((q_{N})\) to q on \([0,T]\), we can define
such that \(\Vert q-q_{N}\Vert _{C([0, T]; \mathbb{R}^{d})}\leq \frac{\rho _{t}}{4}\) for all \(N>N_{t}\). We will show that for all \(N>N_{t}\) and for all \(t_{n}\in [t-\frac{\rho _{t}}{4(\kappa +1)}, t+ \frac{\rho _{t}}{4(\kappa +1)} ]\), \(J(t_{n+1}, Q_{n+1})\subset J(t,q(t))\). Indeed, let \(N>N_{t}\) and \(t_{n}\in [t-\frac{\rho _{t}}{4(\kappa +1)}, t+ \frac{\rho _{t}}{4(\kappa +1)} ]\). We have
In addition, we have
Therefore, \(J(t_{n+1}, Q_{n+1})\subset J(t,q(t))\). Represent \(J(t,q(t))\) as \(J(t,q(t))=J_{1}(t,q(t))\cup J_{2}(t,q(t))\) with
and
Since \(J_{1}(t,q(t))\) is a finite set, we may define
Now, let \(\tilde{\rho }\in (0, \tilde{\rho }_{t}]\) and \(N>\tilde{N}_{t}\). As before, we define
which implies that
and
Thus,
Since \(J(t_{n+1}, Q_{n+1}) \subset J(t,q(t))\), \(i \notin J(t_{n+1}, Q_{n+1})\) implies that \(i \in J_{1}(t,q(t))\). Thus,
If \(J_{2}(t,q(t))=\emptyset \) using the same arguments as in Step 1, we can obtain that \(\dot{q}(t^{+})=\dot{q}(t^{-})\). Moreover, since \(q(s)\in C(s)\) for all \(s\in [0,T]\), \(\dot{q}(t^{+})\in {\mathcal{T}}(t, q(t))\). It follows that \(\dot{q}(t^{+})=\dot{q}(t^{-})\in {\mathcal{T}}(t, q(t))\) and therefore we have \(\dot{q}(t^{-})=\dot{q}(t^{+})=\mathbb{P}_{{\mathcal{T}}(t,q(t))}( \dot{q}(t^{-}))\). For the case where \(J_{2}(t,q(t))\neq\emptyset \), we rewrite (14) as follows
Before continuing the proof, we prove the following two technical lemmas.
Lemma 4.9
We have
Proof
We can estimate the last two terms of (15) as follows
and, let \(\Delta ^{n}_{i}(t) = \lambda ^{n}_{i} (\nabla f_{i}(t_{n+1}, \cdot )(Q_{n+1})- \nabla f_{i}(t, \cdot )(q(t)))\), using Lemma 4.5 and Remark 3.1(ii) we have
From (15), it follows that
We now will prove that the set \(S:= \sum_{i \in J_{2}(t,q(t))}\mathbb{R}_{+}\nabla f_{i}(t, \cdot )(q(t))\) is a closed subset of \(\mathbb{R}\). Indeed, let \(\{x_{n}\}\), with \(x_{n}=\sum_{i\in J_{2}(t,q(t))} x_{i,n} \nabla f_{i}(t, \cdot )(q(t))\), be a sequence in S converging to some \(x^{*}\). By Assumption A2, there exists \(v=v(t,q(t))\) such that \(\Vert v\Vert =1\) and
From this it follows that
Since \(\{x_{n}\}\) is a convergent sequence, there exists \(l>0\) such that for each \(i\in J_{2}(t,q(t))\) we have \(0\leq x_{i,n}< l\) for all n. Hence, there exists a subsequence of \(\{x_{i,n}\}\), denoted by \(\{x_{i,n^{\prime }}\}\) and a nonnegative number \(x^{*}_{i}\) such that for all \(i\in J_{2}(t,q(t))\)
Since the sequence \(\{x_{n}\}\) converges to \(x^{*}\), the sequence \(\{x_{n^{\prime }}\}\) also converges to \(x^{*}\). We have
From this we obtain the limit
We have shown that \(\sum_{k\in J(t,q) }\mathbb{R}_{+}\nabla f_{i}(t,\cdot )(q)\) is closed. Hence, by (16) we obtain the desired result. □
Lemma 4.10
For all \(i \in J_{2}(t, q(t))\), one has
Proof
By Lemma 4.7, \(\dot{q}(t^{+})\in {\mathcal{T}}(t, q(t))\). Hence, for each \(i \in J_{2}(t,q(t))\),
We only need to prove that
Let \(i\in J_{2}(t, q(t))\) and \(\tilde{\rho }\in (0, \tilde{\rho }_{t}]\). By the definition of \(J_{2}(t, q(t))\), there exists a subsequence \(\{N_{\alpha }\}_{\alpha \in \mathbb{N}}\) strictly increasing to infinity such that, for all \(\alpha \in \mathbb{N}\) we have \(N_{\alpha }>\tilde{N}_{t}\). Let \(h_{\alpha }=T/N_{\alpha }\), then there exists \(nh_{\alpha }\in [ t-\frac{\tilde{\rho }}{4(\kappa +1)},t+ \frac{\tilde{\rho }}{4(\kappa +1)} ]\) such that \(f_{i}(t_{n+1}, Q_{n+1})=0\), i.e., \(i \in J(t_{n+1}, Q_{n+1})\). We define
By Lemma 4.8 we have
It follows that
We can estimate the second and fourth terms of the right-hand side of (17) as follows
and
If \(n_{\alpha }=n_{+}\), the third term of the right-hand side of (17) vanishes. Otherwise, we rewrite it as follows
where
Since \(i \notin J(t_{n+1}, Q_{n+1})\) for all \(n\in \{n_{\alpha }+1, \ldots, n_{+}\}\) by the definition of \(n_{\alpha }\) and the inclusion \(J(t_{n+1}, Q_{n+1})\subset J(t, q(t))\), Assumption A4 implies that the second term of the right-hand side of this last inequality is nonnegative. Furthermore, the last term can be estimated as
Then, passing the right-hand side of (17) to the limit and recalling that \(P_{n_{+}}=p_{N} (t+\frac{\tilde{\rho }}{4(\kappa +1)} )\), we obtain
This means that \(\partial f_{i}(\cdot,q(t))(t)+\langle \nabla f_{i}(t, \cdot )(q(t)), \dot{q}(t^{+})\rangle \geq 0\). □
We now continue the proof of Proposition 4.5. We have \(\dot{q}(t^{+})\in {\mathcal{T}}(t,q(t))\) and
Hence, there exist nonnegative real numbers \(\bar{\lambda }_{i}\), for \(i\in J_{2}(t, q(t))\), such that
for all \(w\in {\mathcal{T}}(t, q(t))\)
However, using the previous proposition, for all \(w\in {\mathcal{T}}(t, q(t))\) and for all \(i\in J_{2}(t, q(t))\), we have
Hence,
As \({\mathcal{T}}(t, q(t))\) is a closed convex subset of \(\mathbb{R}^{d}\), the above is equivalent to
The proof is complete. □
Finally, we observe that the limit trajectory satisfies the initial data. Indeed, with (10) we have immediately \(q(0)=q_{0}\). Moreover, \(p_{0}\in {\mathcal{T}}(0, q_{0})\) we can prove that \(\dot{q}(0^{+})=p_{0}=\mathbb{P}_{{\mathcal{T}}(0, q_{0})} (p_{0})\) by the same kind of computations. Indeed, if \(t=t_{0}=0\), we may define \(\rho _{t_{0}}\in (0, \min (\rho, T))\) such that
and we define \(N_{t_{0}}\) (respectively, \(\tilde{\rho }_{t_{0}}\) and \(\tilde{N}_{t_{0}}\) if \(J(t_{0}, q(t_{0})) \neq\emptyset \)). in the same way as previously. Then, for all \(\tilde{\rho }\in (0, \rho _{t_{0}}]\) and for all \(N >h_{t_{0}}\) (respectively, for all \(\tilde{\rho }\in (0, \tilde{\rho }_{t_{0}}]\) and for all \(N> \tilde{N}_{t_{0}}\) if \(J(t_{0},q(t_{0}))\neq\emptyset \)) we define
We obtain
Using the same computation as above, we obtain \(\dot{q}(0^{+})=p_{0}\).
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 \(\vert \nabla ^{2} f_{i}(t,\cdot )(q)\vert \) and \(\vert \partial ^{2} f_{i}(\cdot,q)(t)\vert +\vert \partial ( \nabla f_{i}(\cdot,\cdot ) (q) )(t)\vert \) 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.
5 Example
Let \(t\in [0,1]\) and for \(i\in \{1,2\}\), \(f_{i}:[0,T]\times \mathbb{R}^{2}\to \mathbb{R}\) be defined by
and
Consider the problem \(\mathcal{P}\) with the set \(C(t) = \{q =(x,y)\in \mathbb{R}^{2}\mid f_{i}(t,q)\leq 0, i\in \{1,2 \}\}\) and \(g(t,q) = 0\).
Observe that \(f_{i}(\cdot,\cdot )\), \(i\in \{1,2\}\) are differentiable and their derivatives are Lipschitz continuous with rank \(L=\frac{\sqrt{5}}{2}\). This shows that the Assumption A1(i) holds. Note that \(\partial f_{1}(\cdot,q)(t) = \partial f_{2}(\cdot,q)(t) = -1\) and
and
Assumption A1(ii) is always true for \(v=(0,1)\) and \(\mu = 1\). We also have \(\Vert f_{i}(t,\cdot )(x,y)\Vert \leq L\) and therefore, Assumption A1(iii) holds. Assumption A2 is satisfied with the choice of \(\gamma = \frac{1}{2}\). If \(J(t,q) =\{1,2\}\) we have
Hence, Assumption A4 holds. We have shown that Assumptions A1–A4 are satisfied for the above problem. By Theorem 4.1, the problem has a solution.
Note that the second-order derivative with respect to the second variable q of \(f_{1}\) (of \(f_{2}\)) does not exist at \(q=(0,y)\) (at \(q=(4,y)\), respectively) for any \(y\in \mathbb{R}\). Hence, \(f_{1},f_{2} \notin C^{2}([0,1]\times \mathbb{R}^{2};\mathbb{R})\). This shows that the assumptions proposed in [7, 9, 27] cannot be applied to ensure the existence solution for this example.
6 Conclusions
In this paper, we have presented some regularity conditions for the data to ensure the existence of solutions for a class of vibroimpact problems. These conditions require neither the second-order differentiability nor convexity of the constraint functions. Some assumptions relate to the uniformly prox-regularity of the set of admissible positions. We also give an example to illustrate the applicability of the provided assumptions.
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Adly, S., Thieu, N.N. Existence of solutions for a Lipschitzian vibroimpact problem with time-dependent constraints. Fixed Point Theory Algorithms Sci Eng 2022, 3 (2022). https://doi.org/10.1186/s13663-022-00713-y
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DOI: https://doi.org/10.1186/s13663-022-00713-y