State-dependent Sweeping Process Research Articles

Well-Posedness of History/State-Dependent Implicit Sweeping Processes

This paper is devoted to the study of a new class of implicit state-dependent sweeping processes with history-dependent operators. Based on the methods of convex analysis, we prove the equivalence of the history/state dependent implicit sweeping process and a nonlinear differential equation, which, through a fixed point argument for history-dependent operators, enables us to prove the existence, uniqueness, and continuous dependence of the solution in a very general framework. Moreover, we present some new convergence results with respect to perturbations in the data, including perturbations of the associated moving sets. Finally, the theoretical results are applied to prove the well-posedness of a history-dependent quasi-static contact problem.

A Continuation Principle for Periodic BV-Continuous State-Dependent Sweeping Processes

We consider a Caratheodory differential equation with a state-dependent convex constraint that changes BV-continuously in time (a perturbed BV-continuous state-dependent sweeping process). By setti...

Well-posedness and nonsmooth Lyapunov pairs for state-dependent maximal monotone differential inclusions

ABSTRACTIn this paper, we introduce a class of state-dependent maximal monotone differential inclusions. The existence and uniqueness of solutions are obtained by using an implicit discretization scheme and a kind of hypo-monotonicity assumption. In addition, a characterization for nonsmooth Lyapunov pairs associated with such systems is provided. Our result can be applied to study state-dependent sweeping processes and Lur'e dynamical systems. It is new even the involved maximal monotone operators depend only on the time.

On evolution quasi-variational inequalities and implicit state-dependent sweeping processes

In this paper, we study a variant of the state-dependent sweeping process with velocity constraint. The constraint \begin{document}$ {C(\cdot, u)} $\end{document} depends upon the unknown state \begin{document}$ u $\end{document} , which causes one of the main difficulties in the mathematical treatment of quasi-variational inequalities. Our aim is to show how a fixed point approach can lead to an existence theorem for this implicit differential inclusion. By using Schauder's fixed point theorem combined with a recent existence and uniqueness theorem in the case where the moving set \begin{document}$ C $\end{document} does not depend explicitly on the state \begin{document}$ u $\end{document} (i.e. \begin{document}$ C: = C(t) $\end{document} ) given in [ 4 ], we prove a new existence result of solutions of the quasi-variational sweeping process in the infinite dimensional Hilbert spaces with a velocity constraint. Contrary to the classical state-dependent sweeping process, no conditions on the size of the Lipschitz constant of the moving set, with respect to the state, is required.

Reduction approach to second order perturbed state-dependent sweeping process

In this paper, we present a new approach to solving second order nonconvex perturbed sweeping process in finite dimensional setting. It consists in a reduction of the problem to a first order one without use of the standard methods of fixed point theory. The perturbation, that is the external force applied on the system is not necessary with bounded values.

State-Dependent Implicit Sweeping Process in the Framework of Quasistatic Evolution Quasi-Variational Inequalities

This paper deals with the existence and uniqueness of solutions for a class of state-dependent sweeping processes with constrained velocity in Hilbert spaces without using any compactness assumption, which is known to be an open problem. To overcome the difficulty, we introduce a new notion called hypomonotonicity-like of the normal cone to the moving set, which is satisfied by many important cases. Combining this latter notion with an adapted Moreau’s catching-up algorithm and a Cauchy technique, we obtain the strong convergence of approximate solutions to the unique solution, which is a fundamental property. Using standard tools from convex analysis, we show the equivalence between this implicit state-dependent sweeping processes and quasistatic evolution quasi-variational inequalities. As an application, we study the state-dependent quasistatic frictional contact problem involving viscoelastic materials with short memory in contact mechanics.

Truncated nonconvex state-dependent sweeping process: implicit and semi-implicit adapted Moreau’s catching-up algorithms

In this paper, we deal with the existence of solutions for perturbed state-dependent Moreau’s sweeping processes. Two ways are investigated to realize such a study, depending on the nature of the used scheme, namely implicit or semi-implicit. In both cases, our evolution problem is described in a general Hilbert space by a prox-regular moving set controlled through the truncated Hausdorff–Pompeiu distance. The normal cone involved is perturbed by a sum of a single-valued mapping and a multimapping.

On state-dependent sweeping process in Banach spaces

In this paper we prove, in a separable reflexive uniformly smooth Banach space, the existence of solutions of a perturbed first order differential inclusion governed by the proximal normal cone to a moving set depending on the time and on the state. The perturbation is assumed to be separately upper semicontinuous.

Unbounded state-dependent sweeping processes with perturbations in uniformly convex and q-uniformly smooth Banach spaces

In this paper, the existence of solutions for a class of first and second order unbounded state-dependent sweeping processes with perturbation in uniformly convex and $q$-uniformly smooth Banach spaces are analyzed by using a discretization method. The sweeping process is a particular differential inclusion with a normal cone to a moving set and is of a great interest in many concrete applications. The boundedness of the moving set, which plays a crucial role for the existence of solutions in many works in the literature, is not necessary in the present paper. The compactness assumption on the moving set is also improved.

Moreau-Yosida Regularization of State-Dependent Sweeping Processes with Nonregular Sets

The existence and the convergence (up to a subsequence) of the Moreau-Yosida regularization for the state-dependent sweeping process with nonregular (subsmooth and positively alpha-far) sets are established. Then, by a reparametrization technique, the existence of solutions for bounded variation continuous state-dependent sweeping processes with nonregular (subsmooth and positively alpha-far) sets is proved. An application to vector hysteresis is discussed, where it is shown that the Play operator with positively alpha-far sets is well defined for bounded variation continuous inputs.

Galerkin-like Method and Generalized Perturbed Sweeping Process with Nonregular Sets

In this paper we present a new method to solve differential inclusions in Hilbert spaces. This method is a Galerkin-like method where we approach the original problem by projecting the state into a $n$-dimensional Hilbert space but not the velocity. We prove that the approached problem always has a solution and that, under some compactness conditions, the approached problems have a subsequence which converges strongly pointwisely to a solution of the original differential inclusion. We apply this method to the generalized perturbed sweeping process governed by nonregular sets (equi-uniformly subsmooth or positively $\alpha$-far). This differential inclusion includes Moreau's sweeping process, the state-dependent sweeping process, and second-order sweeping process for which we give very general existence results.

Reduction of state dependent sweeping process to unconstrained differential inclusion

In this article we discuss the differential inclusion known as state dependent sweeping process for a class of prox-regular non-convex sets. We associate with any state dependent sweeping process with such sets an unconstraint differential inclusion whose any solution is a solution of the state sweeping process too. We prove a theorem on the existence of a global solution of nonconvex state dependent sweeping process with unbounded perturbations. The perturbations are not required to be convex valued.

Delay perturbed state-dependent sweeping process

In this paper, we prove a result on the existence of solutions of a differential inclusion governed by a class of nonconvex state-dependent sweeping process with perturbation depending on all the variables and with delay. The moving set involved in the process is prox-regular and depends both on the time and on the state.

Existence of solutions to the state dependent sweeping process with delay

Existence of solutions to the state dependent sweeping process with delay

Nonconvex Differential Variational Inequality and State-Dependent Sweeping Process

We prove the existence of solutions of a differential variational inequality involving a prox-regular set in an infinite dimensional Hilbert space via a new existence result of a non-convex state-dependent sweeping process.