Differential Variational Inequalities Research Articles

A differential variational inequality in the study of contact problems with wear

We start with a mathematical model which describes the sliding contact of a viscoelastic body with a moving foundation. The contact is frictional and the wear of the contact surfaces is taken into account. We prove that this model leads to a differential variational inequality in which the unknowns are the displacement field and the wear function. Then, inspired by this model, we consider a general differential variational inequality in reflexive Banach spaces, governed by four parameters. We prove the unique solvability of the inequality as well as the continuous dependence of its solution with respect to the parameters. The proofs are based on arguments of monotonicity, compactness, convex analysis and lower semicontinuity. Then, we apply these abstract results to the mathematical model of contact for which we deduce the existence of a unique solution as well as the existence of optimal control for an associate optimal control problem. We also present the corresponding mechanical interpretations.

The Convergence Results of Differential Variational Inequality Problems

In this work, we suggest a differential variational inequality in reflexive Banach spaces and construct a sequence with a set of constraints and a penalty parameter. We use the penalty method to prove a unique solution to the problem and make suitable assumptions to prove the convergence of the sequence. The proof is based on arguments for compactness, symmetry, pseudomonotonicity, Mosco convergence, inverse strong monotonicity and Lipschitz continuity. Finally, we discuss the boundary value problem for the differential variational inequality problem as an application.

Challenges in Optimization with Complex PDE-Systems

The workshop concentrated on various aspects of optimization problems with systems of nonlinear partial differential equations (PDEs) or variational inequalities (VIs) as constraints. In particular, discussions around several keynote presentations in the areas of optimal control of nonlinear or non-smooth systems, optimization problems with functional and discrete or switching variables leading to mixed integer nonlinear PDE constrained optimization, shape and topology optimization, feedback control and stabilization, multi-criteria problems and multiple optimization problems with equilibrium constraints as well as versions of these problems under uncertainty or stochastic influences, and the respectively associated numerical analysis as well as design and analysis of solution algorithms were promoted. Moreover, aspects of optimal control of data-driven PDE constraints (e.g. related to machine learning) were addressed.

Feedback control for nonlinear evolutionary equations with applications

In the paper we are concerned with the feedback control system governed by nonlinear evolutionary equations involving weakly continuous operators. By using the Rothe method and a surjectivity result for weakly continuous operators, we first present the solvability for the evolutionary equation. Then we show the existence of solutions to the feedback control system. We also consider an existence result for an optimal control problem. Moreover, we apply the main results to a class of differential variational inequalities, evolutionary hemivariational inequalities and the non-stationary Navier–Stokes–Voigt equation with a subgradient inclusion condition.

On periodic solutions to a class of delay differential variational inequalities

<p style='text-indent:20px;'>In this paper, we introduce and study a class of delay differential variational inequalities comprising delay differential equations and variational inequalities. We establish a sufficient condition for the existence of periodic solutions to delay differential variational inequalities. Based on some fixed point arguments, in both single-valued and multivalued cases, the solvability of initial value and periodic problems are proved. Furthermore, we study the conditional stability of periodic solutions to this systems.</p>

On the Differential Variational Inequalities of Parabolic-Parabolic Type

On the Differential Variational Inequalities of Parabolic-Parabolic Type

On the delay differential variational inequalities of parabolic–elliptic type

In this work, we consider a model of functional differential variational inequalities formulated by a delay differential inclusion and an elliptic variational inequality in Banach spaces. We prove the global solvability of the problem and we have given some sufficient conditions to ensure the existence of decay solutions. Moreover, the existence of a global attractor for the semiflow governed by our system is presented thanks to the framework of Melnik and Valero [On attractors of multivalued semi-flows and differential inclusions. Set-Valued Anal. 1998;6:83–111]. The application to partial differential inclusions driven by obstacle constraints is given to illustrate our abstract results.

The Nonemptiness and Compactness of Mild Solution Sets for Riemann-Liouville Fractional Delay Differential Variational Inequalities

The Nonemptiness and Compactness of Mild Solution Sets for Riemann-Liouville Fractional Delay Differential Variational Inequalities

On a new system of fractional delay differential equations coupled with fuzzy variational inequalities

Fuzzy variational inequalities (FVIs) are modeling tools used to characterize a variety of fuzzy decision making problems arising in mathematical optimization, control theory, operations research, and game theory. Fractional order delay differential equations are also finding applications in all disciplines including chemistry, physics, and finance. The aim of this paper is to introduce and study a new dynamical fuzzy system, named the fractional differential fuzzy variational inequality (FDFVI) with delay, which bridges these two areas of research and broadens the class of problems amenable to be studied under the fuzzy environments. By using the KKM theorem and monotonicity arguments, we show that the solution set of the FVI is nonempty, convex and compact. We also establish the upper semicontinuity of the solution mapping U of the FVI involved in the FDFVI with delay. Moreover, we obtain an existence of the mild solution for the FDFVI with delay by employing Bohnenblust-Karlin's fixed point theorem. In addition, we provide an approximating algorithm to find a solution of the FDFVI with delay. Finally, we give two numerical examples to illustrate our main results.

Existence and Relaxation Results for Second Order Multivalued Systems

We consider nonlinear systems driven by a general nonhomogeneous differential operator with various types of boundary conditions and with a reaction in which we have the combined effects of a maximal monotone term A(x) and of a multivalued perturbation F(t,x,y) which can be convex or nonconvex valued. We consider the cases where D(A)neq mathbb{R}^{N} and D(A)= mathbb{R}^{N} and prove existence and relaxation theorems. Applications to differential variational inequalities and control systems are discussed.

On a class of controlled differential variational inequalities

By using various analytical and topological tools, weak sharp solutions are investigated for a class of controlled differential variational inequality involving convex functionals defined by path-independent curvilinear integrals. In order to illustrate the main results, an application is also provided.

Uniqueness and Hyers-Ulam Stability of Random Differential Variational Inequalities with Nonlocal Boundary Conditions

In this paper, we consider a new class of random differential variational inequalities (RDVIs) with nonlocal boundary conditions in Hilbert spaces. We apply the projection operator, Gronwall’s lemma and a result on the existence of a random differential inclusion to establish uniqueness and Hyers–Ulam stability results of the abstract inequality. As an illustrative application, the linear random differential complementarity systems and a price control model are investigated.

On a Class of Differential Variational Inequalities in Infinite-Dimensional Spaces

A new class of differential variational inequalities (DVIs), governed by a variational inequality and an evolution equation formulated in infinite-dimensional spaces, is investigated in this paper. More precisely, based on Browder’s result, optimal control theory, measurability of set-valued mappings and the theory of semigroups, we establish that the solution set of DVI is nonempty and compact. In addition, the theoretical developments are accompanied by an application to differential Nash games.

Generalized fractional evolution equations driven by fuzzy variational inequalities

The aim of this paper is to study a comprehensive dynamics system called fractional fuzzy differential variational inequality, which is composed of a nonlinear fractional differential equation with Atangana-Baleanu fractional derivative and a time-dependent fuzzy variational inequality. We explore an existence and uniqueness theorem for the dynamics system under consideration. Our proof is based on F-KKM theorem, Banach fixed point principle, and the theory of monotone operators.

Geometric optimal techniques to control the muscular force response to functional electrical stimulation using a non-isometric force-fatigue model

<p style='text-indent:20px;'>A recent force-fatigue parameterized mathematical model, based on the seminal contributions of V. Hill to describe muscular activity, allows to predict the muscular force response to external electrical stimulation (FES) and it opens the road to optimize the FES-input to maximize the force response to a pulse train, to track a reference force while minimizing the fatigue for a sequence of pulse trains or to follow a reference joint angle trajectory to produce motion in the non-isometric case. In this article, we introduce the geometric frame to analyze the dynamics and we present Pontryagin types necessary optimality conditions adapted to digital controls, used in the experiments, vs permanent control and which fits in the optimal sampled-data control frame. This leads to Hamiltonian differential variational inequalities, which can be numerically implemented vs direct optimization schemes.

An identification problem involving fractional differential variational inequalities

Abstract We study the solvability and stability for the problem of identifying parameter in a class of fractional differential variational inequalities. Our approach is based on a regularity analysis for fractional diffusion equations and fixed point techniques.

A new class of fuzzy fractional differential inclusions driven by variational inequalities

The goal of this paper is to consider a new fuzzy differential system consisting of a fuzzy fractional differential inclusion combined with a variational inequality, which is called fuzzy fractional differential variational inequality (FFDVI). Such a model captures the desired features of both fuzzy fractional differential inclusions and fractional differential variational inequalities within the same framework. By employing the set-valued version of the Krasnoselskii fixed point theorem, an existence of solutions is obtained for the FFDVI under some mild conditions. Moreover, an approximating algorithm is provided to find a solution of the FFDVI. Finally, two numerical examples are given to illustrate our main results.

Existence results for a class of semilinear differential variational inequalities with nonlocal boundary conditions

In the paper we study a class of semilinear differential variational systems with nonlocal boundary conditions, which are obtained by mixing evolution equations and generalized variational inequalities. Firstly, we show the properties of the solution set for generalized variational inequalities. Then, the existence results are established and proved mainly by the topological degree theory and the Yosida approximations of the generator of $C_0$-semigroup.

Dynamic variational inequality in fuzzy environments

This paper introduces a class of differential fuzzy variational inequalities, the model provides an efficient approach for solving many dynamic multi-objective optimization problems in fuzzy environments. An existence theorem of the Caratheodory weak solution for the model is established under some suitable assumptions. An algorithm, supported by the convergence analysis, is developed to find the solution. In addition, some examples are given to illustrate the application and the algorithm.

Optimal Retirement in a General Market Environment

We study an optimal retirement, consumption/portfolio selection problem of an economic agent in a non-Markovian environment. We show that under a suitable condition the optimal retirement decision is to retire when the individual’s wealth reaches a threshold level. We express the value and the optimal strategy by using the strong solution of the backward stochastic partial differential variational inequality (BSPDVI) associated with the dual problem. We derive properties of the value function and the optimal strategy by analyzing the strong solution and the free boundary of the BSPDVI. We also make a methodological contribution by proposing an approach to investigate properties of the strong solution and the stochastic free boundary of BSPDVI by combining a probabilistic method and the theory of backward stochastic partial differential equations (BSPDEs).