In this paper, we study a new variant of Moreau's sweeping process with velocity constraint. Base... more In this paper, we study a new variant of Moreau's sweeping process with velocity constraint. Based on an adapted version of Moreau's catching-up algorithm, we show the well-posedness (in the sense existence and uniqueness) of this problem in a general framework. We show the equivalence between this implicit sweeping process and a quasistatic evolution variational inequality. It is well known that the variational formulations of many mechanical problems with unilateral contact and friction lead to an evolution variational inequality. As an application, we reformulate the quasistatic antiplane frictional contact problem for linear elastic materials with short memory as an implicit sweeping process with velocity constraint. The link between the implicit sweeping process and the quasistatic evolution variational inequality is possible thanks to some standard tools from convex analysis and is new in the literature.
The main concern of this paper is the study of degenerate sweeping process involving uniform prox... more The main concern of this paper is the study of degenerate sweeping process involving uniform prox-regular sets via an unconstrained differential inclusion by showing that the sets of solutions of the two problems coincide. This principle of reduction to unconstrained evolution problem can be seen as a penalization of the subdifferential of the distance function. Using this reduction technique, an existence and uniqueness result of a Lipschitz perturbed version of the degenerate sweeping process is proved in the finite dimensional setting. An application is given to quasistatic unilateral dynamics in nonsmooth mechanics where the moving set is described by a finite number of inequalities. We provide sufficient verifiable conditions ensuring both the prox-regularity and the Lipschitz continuity with respect to the Hausdorff distance of the moving set.
HAL (Le Centre pour la Communication Scientifique Directe), 2016
The class of subsmooth sets strictly contains the class of closed convex sets and the class of pr... more The class of subsmooth sets strictly contains the class of closed convex sets and the class of prox-regular sets. The present paper is concerned with the study of perturbed sweeping process differential inclusions where the moving set is nonconvex and non prox-regular and depends both on the time and on the state. We prove the existence of solution, in particular, under the subsmoothness of the moving set.
We establish a result asserting the existence of solutions for a perturbed state dependent sweepi... more We establish a result asserting the existence of solutions for a perturbed state dependent sweeping process in an infinite dimensional Hilbert space.
State-Dependent Sweeping Process with Perturbation
Springer eBooks, 2013
ABSTRACT We prove, via a new projection algorithm, the existence of solutions for differential in... more ABSTRACT We prove, via a new projection algorithm, the existence of solutions for differential inclusion generated by sweeping process with closed convex sets depending on state.
Discrete and Continuous Dynamical Systems - Series S, 2018
In this paper, we study a variant of the state-dependent sweeping process with velocity constrain... more In this paper, we study a variant of the state-dependent sweeping process with velocity constraint. The constraint C(•, u) depends upon the unknown state u, 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 C does not depend explicitly on the state u (i.e. C := C(t)) given in [3], 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.
Siam Journal on Control and Optimization, Oct 1, 2022
In this paper, we analyze and discuss the well-posedness of a new variant of the so-called sweepi... more In this paper, we analyze and discuss the well-posedness of a new variant of the so-called sweeping process, introduced by J.J. Moreau in the early 70's [18] with motivation in plasticity theory. In this variant, the normal cone to the (mildly nonconvex) prox-regular moving set C(t), supposed to have an absolutely continuous variation, is perturbed by a sum of a Carathéodory mapping and an integral forcing term. The integrand of the forcing term depends on two time-variables, that is, we study a general integro-differential sweeping process of Volterra type. By setting up an appropriate semi-discretization method combined with a new Gronwall-like inequality (differential inequality), we show that the integro-differential sweeping process has one and only one absolutely continuous solution. We also establish the continuity of the solution with respect to the initial value. The results of the paper are applied to the study of nonlinear integro-differential complementarity systems which are combination of Volterra integro-differential equations with nonlinear complementarity constraints. Another application is concerned with non-regular electrical circuits containing timevarying capacitors and nonsmooth electronic device like diodes. Both applications represent an additional novelty of our paper.
The primary goal of this paper is to study a nonlinear complementarity system (NCS, for short) wi... more The primary goal of this paper is to study a nonlinear complementarity system (NCS, for short) with a nonlinear and nonhomogeneous partial differential operator and mixed boundary conditions, and a simultaneous distributed-boundary optimal control problem governed by (NCS), respectively. First, we formulate the weak formulation of (NCS) to a mixed variational inequality with double obstacle constraints (MVI, for short), and prove the existence and uniqueness of solution to (MVI). Then, a power penalty method is applied to (NCS) for introducing an approximating mixed variational inequality without constraints (AMVI, for short). After that, a convergence result that the unique solution of (MVI) can be approached by the unique solution of (AMVI) when a penalty parameter tends to infinity, is established. Moreover, we explore the solvability of the simultaneous distributed-boundary optimal control problem described by (MVI), and consider a family of approximating optimal control problems driven by (AMVI). Finally, we provide a result on asymptotic behavior of optimal controls, system states and minimal values to approximating optimal control problems.
This paper is devoted to the study, for the first time in the literature, of optimal control prob... more This paper is devoted to the study, for the first time in the literature, of optimal control problems for sweeping processes governed by integro-differential inclusions of the Volterra type with different classes of control functions acting in nonconvex moving sets, external dynamic perturbations, and integral parts of the sweeping dynamics. We establish the existence of optimal solutions and then obtain necessary optimality conditions for a broad class of local minimizers in such problems. Our approach to deriving necessary optimality conditions is based on the method of discrete approximations married to basic constructions and calculus rules of first-order and second-order variational analysis and generalized differentiation. The obtained necessary optimality conditions are expressed entirely in terms of the problem data and are illustrated by nontrivial examples that include applications to optimal control models of non-regular electrical circuits.
This paper is devoted to the study of a perturbed differential inclusion governed by a nonconvex ... more This paper is devoted to the study of a perturbed differential inclusion governed by a nonconvex sweeping process in a Hilbert space. The sweeping process is perturbed by a sum of an integral forcing term which the integrand depends on two time-variables and a maximal monotone operator. By using a semi-regularization method combined with a Gronwall-like inequality we prove solvability of the initial value problem.
Discussiones Mathematicae. Differential Inclusions, Control and Optimization, 2008
In this paper, some fixed point principle is applied to prove the existence of solutions for dela... more In this paper, some fixed point principle is applied to prove the existence of solutions for delay second order differential inclusions with three-point boundary conditions in the context of a separable Banach space. A topological property of the solutions set is also established.
ftp ejde.math.txstate.edu (login: ftp) EXISTENCE OF VIABLE SOLUTIONS FOR NONCONVEX DIFFERENTIAL INCLUSIONS
Abstract. We show the existence result of viable solutions to the differential inclusion ˙x(t) ∈ ... more Abstract. We show the existence result of viable solutions to the differential inclusion ˙x(t) ∈ G(x(t)) + F (t, x(t)) x(t) ∈ S on [0, T], where F: [0, T] × H → H (T> 0) is a continuous set-valued mapping, G: H → H is a Hausdorff upper semi-continuous set-valued mapping such that G(x) ⊂ ∂g(x), where g: H → R is a regular and locally Lipschitz function and S is a ball, compact subset in a separable Hilbert space H. 1.
Abstract. We prove the existence of solutions to the differential inclusion ẍ(t) ∈ F (x(t), ẋ(t... more Abstract. We prove the existence of solutions to the differential inclusion ẍ(t) ∈ F (x(t), ẋ(t)) + f(t, x(t), ẋ(t)), x(0) = x0, ẋ(0) = y0, where f is a Carathéodory function and F with nonconvex values in a Hilbert space such that F (x, y) ⊂ γ(∂g(y)), with g a regular locally Lipschitz function and γ a linear operator. 1.
In this paper, we study a new variant of Moreau's sweeping process with velocity constraint. Base... more In this paper, we study a new variant of Moreau's sweeping process with velocity constraint. Based on an adapted version of Moreau's catching-up algorithm, we show the well-posedness (in the sense existence and uniqueness) of this problem in a general framework. We show the equivalence between this implicit sweeping process and a quasistatic evolution variational inequality. It is well known that the variational formulations of many mechanical problems with unilateral contact and friction lead to an evolution variational inequality. As an application, we reformulate the quasistatic antiplane frictional contact problem for linear elastic materials with short memory as an implicit sweeping process with velocity constraint. The link between the implicit sweeping process and the quasistatic evolution variational inequality is possible thanks to some standard tools from convex analysis and is new in the literature.
The main concern of this paper is the study of degenerate sweeping process involving uniform prox... more The main concern of this paper is the study of degenerate sweeping process involving uniform prox-regular sets via an unconstrained differential inclusion by showing that the sets of solutions of the two problems coincide. This principle of reduction to unconstrained evolution problem can be seen as a penalization of the subdifferential of the distance function. Using this reduction technique, an existence and uniqueness result of a Lipschitz perturbed version of the degenerate sweeping process is proved in the finite dimensional setting. An application is given to quasistatic unilateral dynamics in nonsmooth mechanics where the moving set is described by a finite number of inequalities. We provide sufficient verifiable conditions ensuring both the prox-regularity and the Lipschitz continuity with respect to the Hausdorff distance of the moving set.
HAL (Le Centre pour la Communication Scientifique Directe), 2016
The class of subsmooth sets strictly contains the class of closed convex sets and the class of pr... more The class of subsmooth sets strictly contains the class of closed convex sets and the class of prox-regular sets. The present paper is concerned with the study of perturbed sweeping process differential inclusions where the moving set is nonconvex and non prox-regular and depends both on the time and on the state. We prove the existence of solution, in particular, under the subsmoothness of the moving set.
We establish a result asserting the existence of solutions for a perturbed state dependent sweepi... more We establish a result asserting the existence of solutions for a perturbed state dependent sweeping process in an infinite dimensional Hilbert space.
State-Dependent Sweeping Process with Perturbation
Springer eBooks, 2013
ABSTRACT We prove, via a new projection algorithm, the existence of solutions for differential in... more ABSTRACT We prove, via a new projection algorithm, the existence of solutions for differential inclusion generated by sweeping process with closed convex sets depending on state.
Discrete and Continuous Dynamical Systems - Series S, 2018
In this paper, we study a variant of the state-dependent sweeping process with velocity constrain... more In this paper, we study a variant of the state-dependent sweeping process with velocity constraint. The constraint C(•, u) depends upon the unknown state u, 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 C does not depend explicitly on the state u (i.e. C := C(t)) given in [3], 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.
Siam Journal on Control and Optimization, Oct 1, 2022
In this paper, we analyze and discuss the well-posedness of a new variant of the so-called sweepi... more In this paper, we analyze and discuss the well-posedness of a new variant of the so-called sweeping process, introduced by J.J. Moreau in the early 70's [18] with motivation in plasticity theory. In this variant, the normal cone to the (mildly nonconvex) prox-regular moving set C(t), supposed to have an absolutely continuous variation, is perturbed by a sum of a Carathéodory mapping and an integral forcing term. The integrand of the forcing term depends on two time-variables, that is, we study a general integro-differential sweeping process of Volterra type. By setting up an appropriate semi-discretization method combined with a new Gronwall-like inequality (differential inequality), we show that the integro-differential sweeping process has one and only one absolutely continuous solution. We also establish the continuity of the solution with respect to the initial value. The results of the paper are applied to the study of nonlinear integro-differential complementarity systems which are combination of Volterra integro-differential equations with nonlinear complementarity constraints. Another application is concerned with non-regular electrical circuits containing timevarying capacitors and nonsmooth electronic device like diodes. Both applications represent an additional novelty of our paper.
The primary goal of this paper is to study a nonlinear complementarity system (NCS, for short) wi... more The primary goal of this paper is to study a nonlinear complementarity system (NCS, for short) with a nonlinear and nonhomogeneous partial differential operator and mixed boundary conditions, and a simultaneous distributed-boundary optimal control problem governed by (NCS), respectively. First, we formulate the weak formulation of (NCS) to a mixed variational inequality with double obstacle constraints (MVI, for short), and prove the existence and uniqueness of solution to (MVI). Then, a power penalty method is applied to (NCS) for introducing an approximating mixed variational inequality without constraints (AMVI, for short). After that, a convergence result that the unique solution of (MVI) can be approached by the unique solution of (AMVI) when a penalty parameter tends to infinity, is established. Moreover, we explore the solvability of the simultaneous distributed-boundary optimal control problem described by (MVI), and consider a family of approximating optimal control problems driven by (AMVI). Finally, we provide a result on asymptotic behavior of optimal controls, system states and minimal values to approximating optimal control problems.
This paper is devoted to the study, for the first time in the literature, of optimal control prob... more This paper is devoted to the study, for the first time in the literature, of optimal control problems for sweeping processes governed by integro-differential inclusions of the Volterra type with different classes of control functions acting in nonconvex moving sets, external dynamic perturbations, and integral parts of the sweeping dynamics. We establish the existence of optimal solutions and then obtain necessary optimality conditions for a broad class of local minimizers in such problems. Our approach to deriving necessary optimality conditions is based on the method of discrete approximations married to basic constructions and calculus rules of first-order and second-order variational analysis and generalized differentiation. The obtained necessary optimality conditions are expressed entirely in terms of the problem data and are illustrated by nontrivial examples that include applications to optimal control models of non-regular electrical circuits.
This paper is devoted to the study of a perturbed differential inclusion governed by a nonconvex ... more This paper is devoted to the study of a perturbed differential inclusion governed by a nonconvex sweeping process in a Hilbert space. The sweeping process is perturbed by a sum of an integral forcing term which the integrand depends on two time-variables and a maximal monotone operator. By using a semi-regularization method combined with a Gronwall-like inequality we prove solvability of the initial value problem.
Discussiones Mathematicae. Differential Inclusions, Control and Optimization, 2008
In this paper, some fixed point principle is applied to prove the existence of solutions for dela... more In this paper, some fixed point principle is applied to prove the existence of solutions for delay second order differential inclusions with three-point boundary conditions in the context of a separable Banach space. A topological property of the solutions set is also established.
ftp ejde.math.txstate.edu (login: ftp) EXISTENCE OF VIABLE SOLUTIONS FOR NONCONVEX DIFFERENTIAL INCLUSIONS
Abstract. We show the existence result of viable solutions to the differential inclusion ˙x(t) ∈ ... more Abstract. We show the existence result of viable solutions to the differential inclusion ˙x(t) ∈ G(x(t)) + F (t, x(t)) x(t) ∈ S on [0, T], where F: [0, T] × H → H (T> 0) is a continuous set-valued mapping, G: H → H is a Hausdorff upper semi-continuous set-valued mapping such that G(x) ⊂ ∂g(x), where g: H → R is a regular and locally Lipschitz function and S is a ball, compact subset in a separable Hilbert space H. 1.
Abstract. We prove the existence of solutions to the differential inclusion ẍ(t) ∈ F (x(t), ẋ(t... more Abstract. We prove the existence of solutions to the differential inclusion ẍ(t) ∈ F (x(t), ẋ(t)) + f(t, x(t), ẋ(t)), x(0) = x0, ẋ(0) = y0, where f is a Carathéodory function and F with nonconvex values in a Hilbert space such that F (x, y) ⊂ γ(∂g(y)), with g a regular locally Lipschitz function and γ a linear operator. 1.
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