Time-dependent Maximal Monotone Research Articles

Perturbed evolutionary differential hemivariational inequalities involving time-dependent maximal monotone operators

Perturbed evolutionary differential hemivariational inequalities involving time-dependent maximal monotone operators

On a Non-Convex Lagrange Optimal Control Problem

Abstract In this paper, we are concerned with an iterative differential inclusion governed by the time-dependent maximal monotone operator with perturbation. The approach to solve our problem is based on the Yosida approximation technique. The theoretical result is applied to prove an existence result for a Lagrange optimal control problem without assumptions concerning convexity.

Periodic systems with time dependent maximal monotone operators

Periodic systems with time dependent maximal monotone operators

Invariant Closed Sets with Respect to Differential Inclusions with Time-Dependent Maximal Monotone Operators

Invariant Closed Sets with Respect to Differential Inclusions with Time-Dependent Maximal Monotone Operators

EXISTENCE AND RELAXATION OF SOLUTIONS FOR A DIFFERENTIAL INCLUSION WITH MAXIMAL MONOTONE OPERATORS AND PERTURBATIONS

A differential inclusion with a time-dependent maximal monotone operator and a perturbation is studied in a separable Hilbert space. The perturbation is the sum of a time-dependent single-valued operator and a multivalued mapping with closed nonconvex values. A particular feature of the single-valued operator is that its sum its with the identity operator multiplied by a positive square-integrable function is a monotone operator. The multivalued mapping is Lipschitz continuous with respect to the phase variable. We prove the existence of a solution and the density in the corresponding topology of the solution set of the initial inclusion in the solution set of the inclusion with the convexified multivalued mapping. For these purposes, new distances between maximal monotone operators are introduced.

Upper semicontinuous convex-valued selectors of a Nemytskii operator with nonconvex values and evolution inclusions with maximal monotone operators

On the space of continuous functions, we consider a multivalued superposition operator, a Nemytskii operator valued in the space of p-integrable functions with 1≤p<∞. The Nemytskii operator is generated by a multivalued mapping defined on the direct product of a segment of the real line and a separable reflexive Banach space and having closed values in this space. At each point of the phase variable, the mapping either has a closed graph and is convex-valued or it is lower semicontinuous in some neighborhood of this point. In recent years, such mappings are called mappings with mixed semicontinuity properties. We prove that the restriction of the Nemytskii operator onto any compact subset of the space of continuous functions with the sup-norm has a multivalued selector with convex closed values that is upper semicontinuous in the weak topology of the space of p-integrable functions. This result generalizes some well-known results of the author valid in the case when the multivalued mapping generating the Nemytskii operator has compact values. A multivalued selector theorem is used to study an evolution inclusion in a separable Hilbert space with a time-dependent maximal monotone operator and a composite perturbation that is the sum of two multivalued mappings. The values of the first multivalued mapping are closed, bounded, not necessarily convex sets. The second multivalued mapping has closed values and possesses mixed semicontinuity properties. The existence of an absolutely continuous solution is proved. The proof is based on the theorem on an upper semicontinuous convex-valued selector and the author's theorem on selectors continuous in parameter and passing through fixed points of a parameter-dependent multivalued mapping with closed, nonconvex, decomposable values in the space of p-integrable functions as well as on the classical Ky Fan's fixed point theorem. Our approach is fairly simple, concise and clear as compared to other known approaches based on discretization.

Functional differential inclusions with maximal monotone operators and nonconvex perturbations

In this paper, we study the existence of solutions for functional differential inclusions governed by time-dependent maximal monotone operators with nonconvex perturbations depending on all the variables.

A measure differential inclusion with time-dependent maximal monotone operators

A measure differential inclusion with time-dependent maximal monotone operators

Lipschitz perturbation to evolution inclusion driven by time-dependent maximal monotone operators

Lipschitz perturbation to evolution inclusion driven by time-dependent maximal monotone operators

Convergence of proximal solutions for evolution inclusions with time-dependent maximal monotone operators

This article studies the solutions of time-dependent differential inclusions which is motivated by their utility in optimization algorithms and the modeling of physical systems. The differential inclusion is described by a time-dependent set-valued mapping having the property that, for a given time instant, the set-valued mapping describes a maximal monotone operator. By successive application of a proximal operator, we construct a sequence of functions parameterized by the sampling time that corresponds to the discretization of the continuous-time system. Under certain mild assumptions on the regularity with respect to the time argument, and using appropriate tools from functional and variational analysis, this sequence is then shown to converge to the unique solution of the original differential inclusion. The result is applied to develop conditions for well-posedness of differential equations interconnected with nonsmooth time-dependent complementarity relations, using passivity of underlying dynamics (equivalently expressed in terms of linear matrix inequalities).

Existence and relaxation of solutions for evolution differential inclusions with maximal monotone operators

In this paper, we consider evolution problems involving time-dependent maximal monotone operators in Hilbert spaces. Existence and relaxation theorems are proved.

BV Continuous Solutions of an Evolution Inclusion with Maximal Monotone Operator and Nonconvex-Valued Perturbation. Existence Theorem

An evolution inclusion with the right-hand side containing a time-dependent maximal monotone operator and a multivalued mapping with closed nonconvex values is studied in a separable Hilbert space. The dependence of the maximal monotone operator on time is described with the help of the distance between maximal monotone operators in the sense of Vladimirov. This distance as a function of time has bounded variation with an upper bound given by a nonatomic positive Radon measure. By a solution of the inclusion one means a continuous function of bounded variation whose differential measure (Stieltjes measure) is absolutely continuous with respect to the positive Radon measure above, and the values of the density of this differential measure with respect to the Radon measure belong to the right-hand side of the inclusion almost everywhere. Under the traditional assumptions on the perturbation (measurability, Lipschitzianity in the phase variable in the Hausdorff metric, linear growth condition), the existence of solutions is proven and some properties of the solution set are established.

Evolution Inclusions Governed by Time-Dependent Maximal Monotone Operators with a Full Domain

In this paper, we study the existence of solutions for evolution inclusions governed by time-dependent maximal monotone operators with a full domain. Without assumptions concerning time-regularity on the time-dependent maximal monotone operators, and by using the Moreau-Yosida regularization technique, we establish the existence of solutions in Hilbert spaces. The theoretical result is applied to prove the existence of solutions for semicoercive sweeping processes with velocity constraint.

Multi-valued perturbation to evolution problems involving time dependent maximal monotone operators

In this paper, we study the existence of solutions for evolution problems of the form \begin{document}$ -\frac{du}{dr}(t) \in A(t)u(t) + F(t, u(t))+f(t, u(t)) $\end{document} , where, for each \begin{document}$ t $\end{document} , \begin{document}$ A(t) : D(A(t)) \to 2 ^H $\end{document} is a maximal monotone operator in a Hilbert space \begin{document}$ H $\end{document} with continuous, Lipschitz or absolutely continuous variation in time. The perturbation \begin{document}$ f $\end{document} is separately integrable on \begin{document}$ [0, T] $\end{document} and separately Lipschitz on \begin{document}$ H $\end{document} , while \begin{document}$ F $\end{document} is scalarly measurable and separately scalarly upper semicontinuous on \begin{document}$ H $\end{document} , with convex and weakly compact values. Several new applications are provided.

Nonlinear Multivalued Periodic Systems

We consider a first-order periodic system involving a time-dependent maximal monotone map, a subdifferential term, and a multivalued perturbation F(t, x). We prove existence theorems for the “convex” problem (that is, F is convex valued and for the “nonconvex” problem (that is, F is nonconvex valued). Also, we establish the existence of extremal trajectories (that is, solutions when the multivalued perturbation F(t, x) is replaced by ext F(t, x), the extreme points of F(t, x)). Also, we show that every solution of the convex problem can be approximated uniformly by certain extremal trajectories (“strong relaxation” theorem). Finally, we illustrate our result by examining a nonlinear periodic feedback control system.

Well-Posedness and Output Regulation for Implicit Time-Varying Evolution Variational Inequalities

A class of evolution variational inequalities (EVIs), which comprises ordinary differential equations (ODEs) coupled with variational inequalities (VIs) associated with time-varying set-valued mappings, is proposed in this paper. We first study the conditions for existence and uniqueness of solutions. The central idea behind the proof is to rewrite the system dynamics as a differential inclusion which can be decomposed into a single-valued Lipschitz map, and a time-dependent maximal monotone operator. Regularity assumptions on the set-valued mapping determine the regularity of the resulting solutions. Complementarity systems with time-dependence are studied as a particular case. We then use this result to study the problem of designing state feedback control laws for output regulation in systems described by EVIs. The derivation of control laws for output regulation is based on the use of internal model principle, and two cases are treated: First, a static feedback control law is derived when full state feedback is available, In the second case, only the error to be regulated is assumed to be available for measurement and a dynamic compensator is designed. As applications, we demonstrate how control input resulting from the solution of a variational inequality results in regulating the output of the system while maintaining polyhedral state constraints. Another application is seen in designing control inputs for regulation in power converters.

Viscosity Results for a Class of Evolution Inclusions with Maximal Monotone Operators

ABSTRACTIn the finite dimensional setting, we investigate the existence of viscosity subsolutions of Hamilton–Jacobi–Bellman equations related to control problems subject to evolution inclusions governed by time-dependent maximal monotone operators with Young measures.

Control problems governed by time-dependent maximal monotone operators

The paper concerns on an infinite dimensional Hilbert space, the existence and uniqueness of absolutely continuous solutions, for Lipschitz single-valued perturbations of evolution problems involving maximal-monotone operators. This result allows us to extend to optimal control problems associated with such equations, the relaxation theorems with Young measures proved recently in [S. Saidi, L. Thibault and M.F. Yarou, Numer. Funct. Anal. Optim. 34 (2013) 1156–1186].

Functional evolution equations involving time dependent maximal monotone operators in Banach spaces

Functional evolution equations involving time dependent maximal monotone operators in Banach spaces