Differential Variational Inequalities Research Articles

New existence and stability results of mild solutions for fuzzy fractional differential variational inequalities

The aim of this work is to study the stability for a class of fuzzy fractional differential variational inequalities modeled by variational inequalities coupled with fuzzy fractional differential inclusions. By introducing the parametric problem with the help of two parameters in the involved mapping and constraint set, we first establish an existence result for the parametric fuzzy fractional differential variational inequality (PFFDVI), and we also establish the compactness and continuous dependence on the initial value for the PFFDVI. Furthermore, when the parameters are perturbed, we obtain the closedness of solution mapping for the PFFDVI. Then we further investigate the upper semicontinuity of solution mapping for the PFFDVI under some mild conditions. Finally, in order to illustrate the obtained results, we give a numerical example at the end of this paper.

Portfolio-consumption choice problem with voluntary retirement and consumption constraints

We study the investment–consumption choice problem with voluntary retirement and upside/downside consumption constraints in infinite horizon, which can be formulated as a two-phase mixed stochastic control problem including stopping time coupled with controls. By verification theorem, we show the strong solution of a fully nonlinear differential variational inequality, satisfying some special properties, is the value function of this problem. Then we eliminate the nonlinearity through Legendre transformation, and obtain the closed-form formulas of the optimal investment–consumption feedback strategy, the optimal retirement strategy and the value function by solving the dual equation. Since the introduction of consumption constraints, these formulas are both segmented and semi-explicit, and bring certain difficulties to the analysis of the properties of the optimal strategies and the value function. The regularity of the feedback strategy and the value function are obtained, as well as their monotonicity or non-monotonicity with respect to various parameters are discussed by some stochastic analysis methods and differential equation techniques.

A stochastic fractional differential variational inequality with Lévy jump and its application

This paper investigates a stochastic fractional differential variational inequality with Lévy jump (SFDVI with Lévy jump), which comprises a stochastic fractional differential equation with Lévy jump and a stochastic variational inequality. By employing the successive approximation method as well as the projection technique, we establish unique existence of the solution for the SFDVI with Lévy jump under some mild conditions. Moreover, the main results are applied to obtain the unique existence of the solution for the spatial price equilibrium problem in stochastic environments.

A class of delay differential variational inequalities with boundary conditions

In this paper, we introduce a new class of delay differential variational inequalities. Delay differential variational inequalities with initial conditions have been already introduced and studied in literatures. The novelty of this work is to consider such problems with boundary conditions. In the first step, we prove a result on the existence of solutions for delay differential variational inequalities with boundary conditions. In the second step, we study the numerical aspect of the problem, where a numerical scheme is considered and the convergence analysis for the approach is provided. As an application, we consider dynamic Nash equilibrium with time delay. A numerical illustration is provided by considering the problem of optimal allocation of distributed generations.

Existence and convergence for stochastic differential variational inequalities

In this paper, we consider a class of stochastic differential variational inequalities (for short, SDVIs) consisting of an ordinary differential equation and a stochastic variational inequality. The existence of solutions to SDVIs is established under the assumption that the leading operator in the stochastic variational inequality is $P$-function and $P_{0}$-function, respectively. Then, by using the sample average approximation and time stepping methods, two approximated problems corresponding to SDVIs are introduced and convergence results are obtained.

Fractional partial differential variational inequality

In this present paper, we introduce and study a dynamical systems involving fractional derivative operator and nonlocal condition, which is constituted of a fractional evolution equation and a time-dependent variational inequality, and is named as fractional partial differential variational inequality (FPDVI, for short). By employing the estimates involving the one-and two-parameter Mittag-Leffler functions, fixed-point theory for set-value mappings, and non-compactness measure theory, we develop a general framework to establish the existence of smooth solutions to (FPDVI).

Euler scheme for solving a class of stochastic differential variational inequalities with some applications

In this paper, we introduce and study Euler scheme for solving a class of stochastic differential variational inequalities (SDVIs). We show the convergence of the sequence construed by this Euler scheme for solving SDVIs under some mild conditions. Moreover, we apply the obtained results to simulate the electrical circuits with diodes and the collapse of the bridge problems in stochastic environment.

Asymptotically periodic solutions for fractional differential variational inequalities

Asymptotically periodic solutions for fractional differential variational inequalities

Time Optimal Feedback Control for 3D Navier–Stokes-Voigt Equations

In this article, we discuss a time optimal feedback control for asymmetrical 3D Navier–Stokes–Voigt equations. Firstly, we consider the existence of the admissible trajectories for the asymmetrical 3D Navier–Stokes–Voigt equations by using the well-known Cesari property and the Fillippove’s theorem. Secondly, we study the existence result of a time optimal control for the feedback control systems. Lastly, asymmetrical Clarke’s subdifferential inclusions and asymmetrical 3D Navier–Stokes–Voigt differential variational inequalities are given to explain our main results.

Penalty method for solving a class of stochastic differential variational inequalities with an application

The aim of this paper is to study the penalty method for solving a class of stochastic differential variational inequalities (SDVIs). The penalty problem for solving SDVIs is first constructed and the convergence of the sequences generated by the penalty problem is proved under some mild conditions. As an application, the convergence of the sequences generated by the penalty problem is obtained for solving a stochastic migration equilibrium problem with movement cost.

Complete Stability of Neural Networks With Extended Memristors.

The article considers a large class of delayed neural networks (NNs) with extended memristors obeying the Stanford model. This is a widely used and popular model that accurately describes the switching dynamics of real nonvolatile memristor devices implemented in nanotechnology. The article studies via the Lyapunov method complete stability (CS), i.e., convergence of trajectories in the presence of multiple equilibrium points (EPs), for delayed NNs with Stanford memristors. The obtained conditions for CS are robust with respect to variations of the interconnections and they hold for any value of the concentrated delay. Moreover, they can be checked either numerically, via a linear matrix inequality (LMI), or analytically, via the concept of Lyapunov diagonally stable (LDS) matrices. The conditions ensure that at the end of the transient capacitor voltages and NN power vanish. In turn, this leads to advantages in terms of power consumption. This notwithstanding, the nonvolatile memristors can retain the result of computation in accordance with the in-memory computing principle. The results are verified and illustrated via numerical simulations. From a methodological viewpoint, the article faces new challenges to prove CS since due to the presence of nonvolatile memristors the NNs possess a continuum of nonisolated EPs. Also, for physical reasons, the memristor state variables are constrained to lie in some given intervals so that the dynamics of the NNs need to be modeled via a class of differential inclusions named differential variational inequalities.

A computational approach to dynamic generalized Nash equilibrium problem with time delay

In this paper, a dynamic generalized Nash equilibrium problem with time delay is introduced and studied. An algorithm is developed for solving the model using differential variational inequality, and the convergence analysis for the algorithm is investigated. The highlights of this paper lie in the following aspects: (i) A static generalized Nash equilibrium problem is extended to a dynamic system with time delay. (ii) Both the utility functions and the constraint sets depend on the state variable and the rival players’ strategies. (iii) As a result of this new model, a numerical example involving commodity production in a continuous time system is given, and the numerical results benefit the decision makers to predict the change process of commodity price and its production volume.

Stability for a Class of Differential Set-Valued Inverse Variational Inequalities in Finite Dimensional Spaces

In this paper, we introduce and study a new class of differential set-valued inverse variational inequalities in finite dimensional spaces. By applying a result on differential inclusions involving an upper semicontinuous set-valued mapping with closed convex values, we first prove the existence of Carathéodory weak solutions for differential set-valued inverse variational inequalities. Then, by the existence result, we establish the stability for the differential set-valued inverse variational inequality problem when the constraint set and the mapping are perturbed by two different parameters. The closedness and continuity of Carathéodory weak solutions with respect to the two different parameters are obtained.

Convergence of a Class of Delayed Neural Networks with Real Memristor Devices

Neural networks with memristors are promising candidates to overcome the limitations of traditional von Neumann machines via the implementation of novel analog and parallel computation schemes based on the in-memory computing principle. Of special importance are neural networks with generic or extended memristor models that are suited to accurately describe real memristor devices. The manuscript considers a general class of delayed neural networks where the memristors obey the relevant and widely used generic memristor model, the voltage threshold adaptive memristor (VTEAM) model. Due to physical limitations, the memristor state variables evolve in a closed compact subset of the space; therefore, the network can be mathematically described by a special class of differential inclusions named differential variational inequalities (DVIs). By using the theory of DVI, and the Lyapunov approach, the paper proves some fundamental results on convergence of solutions toward equilibrium points, a dynamic property that is extremely useful in neural network applications to content addressable memories and signal-processing in real time. The conditions for convergence, which hold in the general nonsymmetric case and for any constant delay, are given in the form of a linear matrix inequality (LMI) and can be readily checked numerically. To the authors knowledge, the obtained results are the only ones available in the literature on the convergence of neural networks with real generic memristors.

Dissipative Dynamical Systems With Set-Valued Feedback Loops: Well-Posed Set-Valued Lur’e Dynamical Systems

Passivity and dissipativity, in Jan Willems’s sense, are known to be powerful tools for stability analysis and feedback control design. For instance, passive systems in negative feedback interconnection with slope-restricted, static, single-valued, smooth nonlinearities, known as Lur’e systems, have been thoroughly studied in automatic control, yielding the so-called absolute stability problem (with the famous Popov and circle criteria). On the other hand, large classes of nonsmooth systems (complementarity dynamical systems, relay systems, projected dynamical systems, evolution and differential variational inequalities, Moreau’s sweeping processes, and maximal monotone differential inclusions), with applications in circuits, mechanics, and economics, are interpreted as set-valued Lur’e systems, in which the feedback nonlinearity is a multivalued mapping. Therefore, the closed-loop system is a differential inclusion of a certain type, the well posedness of which must be analyzed as a prerequisite for stability and control. This introductory article focuses on the well posedness of such set-valued feedback systems. It is shown how the existence and uniqueness of solutions to these specific differential inclusions benefit a lot from the passivity of the system and the maximal monotonicity (which is a form of incremental passivity) of the feedback set-valued mapping. Available results are reviewed, many illustrative examples are given, and some open issues are highlighted.

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>