Differential Variational Inequalities Research Articles

Variational inequalities of multilayer elastic contact systems with interlayer friction: Existence and uniqueness of solution and convergence of numerical solution

Inspired by the layered structure models in pavement mechanics research, in this study, a class of multilayer elastic contact systems with interlayer frictional contact conditions and deformable supporting frictional contact conditions on the foundation has been constructed. Based on the nonlinear elastic constitutive equations, the corresponding system of partial differential equations and variational inequalities are respectively introduced. Under the framework of variational inequalities, the existence and uniqueness of solutions for such models, along with the approximation properties of finite element numerical solutions, are proven and analyzed. The aforementioned conclusions provide fundamental and broadly applicable theoretical support for addressing mechanical problems in multilayer elastic contact systems within the framework of variational inequalities. Finally, the numerical experimental results based on the mixed finite element method also substantiate our theoretical conclusions.

Dynamic spatial price equilibrium, dynamic user equilibrium, and freight transportation in continuous time: A differential variational inequality perspective

In this paper we provide a statement of dynamic spatial price equilibrium (DSPE) in continuous time as a basis for modeling freight flows in a network economy. The model presented describes a spatial price equilibrium due to its reliance on the notion that freight movements occur in response to differences between the local and distant prices of goods for which there is excess demand; moreover, local and distant delivered prices are equated at equilibrium. We propose and analyze a differential variational inequality (DVI) associated with dynamic spatial price equilibrium to study the Nash-like aggregate game at the heart of DSPE using the calculus of variations and optimal control theory. Our formulation explicitly considers inventory and the time lag between shipping and demand fulfillment. We stress that such a time lag cannot be readily accommodated in a discrete-time formulation. We provide an in-depth analysis of the DVI's necessary conditions that reveals the dynamic user equilibrium nature of freight flows obtained from the DVI, alongside the role played by freight transport in maintaining equilibrium commodity prices and the delivered-price-equals-local-price property of spatial price equilibrium. By intent, our contribution is wholly theoretical in nature, focusing on a mathematical statement of the defining equations and inequalities for dynamic spatial price equilibrium (DSPE), while also showing there is an associated differential variational inequality (DVI), any solution of which is a DSPE. The model of spatial price equilibrium we present integrates the theory of spatial price equilibrium in a dynamic setting with the path delay operator notion used in the theory of dynamic user equilibrium. It should be noted that the path delay operator used herein is based on LWR theory and fully vetted in the published dynamic user equilibrium literature. This integration is new and constitutes a significant addition to the spatial price equilibrium and freight network equilibrium modeling literatures. Among other things, it points the way for researchers interested in dynamic traffic assignment to become involved in dynamic freight modeling using the technical knowledge they already possess. In particular, it suggests that algorithms developed for dynamic user equilibrium may be adapted to the study of urban freight modelled as a dynamic spatial price equilibrium. As such, our work provides direction for future DSPE algorithmic research and application. However, no computational experiments are reported herein; instead, the computing of dynamic spatial price equilibria is the subject of a separate manuscript.

On stochastic fractional differential variational inequalities general system with Lévy jumps

This paper is focused on investigating a stochastic fractional differential variational inequalities general system possessing Lévy jumps (SFDVIGS possessing Lévy jumps), which consists of two systems, i.e., a stochastic variational inequalities general system (SVIGS) and a stochastic fractional differential equations general system (SFDEGS) possessing Lévy jumps. Applying Picard successively iterative technique and the projection method, we derive the unique existence of solutions to SFDVIGS possessing Lévy jumps under certain mild assumptions.

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.

An interior penalty method for a parabolic complementarity problem involving a fractional Black-Scholes operator

In this paper, an interior penalty method is proposed to solve a parabolic complementarity problem involving fractional Black–Scholes operator arising in pricing American options under a geometric Lévy process. The complementarity problem is first reformulated as a fractional partial differential variational inequality problem using the representations of fractional order operators and appropriate mathematical techniques. A penalty equation is then proposed to approximate the variational inequality problem by introducing a novel interior-point based penalty term. The existence and uniqueness of the solution to the penalized problem are proved, and an upper bound on the distance between the solutions to the penalty equation and the variational inequality problem is established. To test our method, we discretize the penalty equation by a finite difference method in space and the Crank–Nicolson method in time. We then present numerical experimental results to demonstrate the usefulness and effectiveness for the interior penalty method.

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.

Stability results for a new kind fractional partial differential variational inequalities

Stability results for a new kind fractional partial differential variational inequalities

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.

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.