Differential Variational Inequalities Research Articles

The mathematical foundations of dynamic user equilibrium

This paper is pedagogic in nature, meant to provide researchers a single reference for learning how to apply the emerging literature on differential variational inequalities to the study of dynamic traffic assignment problems that are Cournot-like noncooperative games. The paper is presented in a style that makes it accessible to the widest possible audience. In particular, we apply the theory of differential variational inequalities (DVIs) to the dynamic user equilibrium (DUE) problem. We first show that there is a variational inequality whose necessary conditions describe a DUE. We restate the flow conservation constraint associated with each origin-destination pair as a first-order two-point boundary value problem, thereby leading to a DVI representation of DUE; then we employ Pontryagin-type necessary conditions to show that any DVI solution is a DUE. We also show that the DVI formulation leads directly to a fixed-point algorithm. We explain the fixed-point algorithm by showing the calculations intrinsic to each of its steps when applied to simple examples.

Uniqueness and Hyers–Ulam Stability Results for Differential Variational Inequalities with Nonlocal Conditions

In this paper, we study the nonlocal problem for differential variational inequalities (DVIs) in Hilbert spaces. We proved that under some suitable conditions the considered problem is Hyers–Ulam stable and has a unique solution. It is shown how the abstract results can be applied to study the linear differential complementarity systems and a price control model. Some examples are given to illustrate the obtained results.

Collocation method for differential variational inequality problems

AbstractIn this paper, a Jacobi collocation method is presented for solving differential variational inequalities (DVIs). Differential variational inequalities consist of a differential equation and a variational inequality. A type of Jacobi‐Gauss collocation scheme with N knots is applied to the differential part of the problem whereas another type of Jacobi‐Gauss collocation scheme with N + 1 knots is applied to the variational part of it. So the DVI problem turns into a variational inequality problem. Electrical circuits with nonsmooth elements like ideal diodes are an important class of physical systems, which can be modeled as DVI problems. So in the numerical experiments, 1 example with smooth solutions and 4 illustrative examples of simple electrical circuits with ideal diodes are considered. Numerical results demonstrate the effectiveness of the proposed method but slow convergence for the proposed method for some examples. The reason for slow convergence in this method is that the solutions of these DVIs are nonsmooth.

Extending DualSPHysics with a Differential Variational Inequality: modeling fluid-mechanism interaction

This work details the coupling of a Smoothed Particle Hydrodynamics (SPH) fluid solver with a general-purpose Differential Variational Inequality (DVI) based non-smooth multibody dynamics solver, allowing for efficient and accurate modeling of fluid-mechanism interactions, an ubiquitous scenario in natural and industrial settings. The SPH fluid model (DualSPHysics) can deal with flow non-linearities, free-surface and intense topological changes, while the non-smooth dynamics model (Project Chrono) deals with discontinuous frictional contacts and kinematic restrictions. An open-source integrated framework to model fluid–structure–structure coupled systems is presented by implementing Project Chrono under DualSPHysics.The model is validated with fluid–structure–structure interaction cases. Both frictional and multi-restriction based behaviors are tested and simple convergence analysis are presented, showing that the model is capable of reproducing complex interactions. Several hypothetical cases are then presented, in order to demonstrate possible applications, showcasing a wide set of options useful for practitioners requiring the use of advanced fluid-mechanism models.

Existence result for differential variational inequality with relaxing the convexity condition

In this paper, a class of differential variational inequalities are studied, and a new approach is introduced to relax the convexity condition. Firstly, an existence theorem of Carathéodory weak solution for the differential variational inequalities is established. Secondly, an algorithm for solving the problem is developed and the convergence analysis for the algorithm is given. Finally, a numerical example is reported to illustrate the proposed algorithm.

Metrically Regular Differential Generalized Equations

In this paper we consider a control system coupled with a generalized equation, which we call a differential generalized equation (DGE). This model covers a large territory in control and optimization, such as differential variational inequalities, control systems with constraints, as well as necessary optimality conditions in optimal control. We study metric regularity and strong metric regularity of mappings associated with DGE by focusing in particular on the interplay between the pointwise versions of these properties and their infinite-dimensional counterparts. Metric regularity of a control system subject to inequality state-control constraints is characterized. A sufficient condition for local controllability of a nonlinear system is obtained via metric regularity. Sufficient conditions for strong metric regularity in function spaces are presented in terms of uniform pointwise strong metric regularity. A characterization of the Lipschitz continuity of the control part of the solution mapping as a function of time is established. Finally, a path-following procedure for a discretized DGE is proposed for which an error estimate is derived.

A CLASS OF DIFFERENTIAL INVERSE VARIATIONAL INEQUALITIES IN FINITE DIMENSIONAL SPACES

In this paper, we study a class of differential inverse variational inequality (for short, DIVI) in finite dimensional Euclidean spaces. Firstly, under some suitable assumptions, we obtain linear growth of the solution set for the inverse variational inequalities. Secondly, we prove existence theorems for weak solutions of the DIVI in the weak sense of Caratheodory by using measurable selection lemma. Thirdly, by employing the results from differential inclusions we establish a convergence result on Euler time dependent procedure for solving the DIVI. Finally, we give a numerical experiment to verify the validity of the algorithm.

Dynamic model of a supply chain network with sticky price

This article aims at studying the strategic behaviours of competing firms with sticky retail pricing for the product. We base our study in the context of a supply chain network comprising multiple manufacturers and retailers. The manufacturers are involved in the production of a homogeneous product while the retailers purchase the product and sell it to consumers in the end markets. The retail price of the product is sticky. A differential variational inequality model is proposed to handle the multiple agents and their independent behaviours. Furthermore, the existence and uniqueness of the solution to the dynamic supply chain network with sticky price are shown. A numerical example is provided to illustrate the model and the computational results of equilibrium behaviour are presented. This paper contributes to literature by introducing price stickiness into supply chain networks and developing a differential variational inequality model to analyze the dynamic strategies of firms in the decentralized supply chain network.

Extragradient methods for differential variational inequality problems and linear complementarity systems

In this paper, 2 extragradient methods for solving differential variational inequality (DVI) problems are presented, and the convergence conditions are derived. It is shown that the presented extragradient methods have weaker convergence conditions in comparison with the basic fixed‐point algorithm for solving DVIs. Then the linear complementarity systems, as an important and practical special case of DVIs, are considered, and the convergence conditions of the presented extragradient methods are adapted for them. In addition, an upper bound for the Lipschitz constant of linear complementarity systems is introduced. This upper bound can be used for adjusting the parameters of the extragradient methods, to accelerate the convergence speed. Finally, 4 illustrative examples are considered to support the theoretical results.

A New Numerical Approach to the Structural Analysis of Masonry Vaults

The present paper concentrates on the numerical modelling of masonry vaults, adopting a type of analysis first developed at the University of Parma for applied mechanics, based on the use of non-smooth dynamics software, through a Differential Variational Inequalities (DVI) formulation specifically developed for the 3D discrete elements method. It allows to follow large displacements and the opening and closure of cracks in dynamic field, typical of the masonry vaulted structures. Once the modelling instrument was calibrated, thanks to the comparison with the recurrent damage mechanisms previously analysed, it was also applied to foresee the behavior of the same structure with different actions and with different types of strengthening. The development of damage mechanisms, both in quasi-static cases (for insufficient lateral confinement or for possible soil settlements) and in the occurrence of seismic events, make this type of structures very difficult to be modelled precisely with other methods. Given the three-dimensional CAD model of a vault modeled with a great number of masonry units with specific positions and pattern, the method proved to be able to reproduce the behavior of the structure under both static and seismic loads, showing the mechanism of collapse, the network of contact forces, the displacements and other useful data. The aim is to inspect the possible influences in the structural behavior given by the discrete geometry and the changes in the mechanisms development given by different strengthening interventions. Once the modeling instrument will be calibrated, also through the comparison with real cases and with the results obtained through limit analysis, it will be possible to adopt it as a base also for the prevision of the future behavior of the vaults subjected to strengthening, avoiding uncalibrated and uncritical applications of materials based more on trends rather than on a thorough analysis for the specific case.

Partial differential variational inequalities involving nonlocal boundary conditions in Banach spaces

In this paper, we firstly introduce a complicated system obtained by mixing a nonlinear evolutionary partial differential equation and a mixed variational inequality in infinite dimensional Banach spaces in the case where the set of constraints is not necessarily bounded and the problem is driven by nonlocal boundary conditions, which is called partial differential variational inequality ((PDVI), for short). Then, we show that the solution set of the mixed variational inequality involved in problem (PDVI) is nonempty, bounded, closed and convex. Moreover, the upper semicontinuity and measurability properties for set-valued mapping U:[0,T]×E2→Cbv(E1) (see (3.7), below) are also established. Finally, several existence results for (PDVI) are obtained by using a fixed point theorem for condensing set-valued operators and theory of measure of noncompactness.

A regularized non-smooth contact dynamics approach for architectural masonry structures

A Non-Smooth Contact Dynamic (NSCD) formulation is used to analyze complex assemblies of rigid blocks, representative of real masonry structures. A model of associative friction sliding is proposed, expressed through a Differential Variational Inequality (DVI) formulation, relying upon the theory of Measure Differential Inclusion (MDI). A regularization is used in order to select a unique solution and to avoid problems of indeterminacy in redundant contacts. This approach, complemented with an optimized collision detection algorithm for convex contacts, results to be reliable for dynamic analyses of masonry structures under static and dynamic loads. The approach is comprehensive, since we implement a custom NSCD simulator based on the Project Chrono C++ framework, and we design custom tools for pre- and post-processing through a user-friendly parametric design software. Representative examples confirm that the method can handle 3-D complex structures, as typically are architectural masonry constructions, under both static and dynamic loading.

On the differential variational inequalities of parabolic-elliptic type

We consider a model of infinite dimensional differential variational inequalities formulated by a parabolic differential inclusion and an elliptic variational inequality. The existence of global solution and global attractor for the semiflow governed by our system is proved by using measure of noncompactness. Copyright © 2017 John Wiley & Sons, Ltd.

Differential variational inequalities in infinite banach spaces

In this paper, we consider a new differential variational inequality (DVI, for short) which is composed of an evolution equation and a variational inequality in infinite Banach spaces. This kind of problems may be regarded as a special feedback control problem. Based on the Browder's theorem and the optimal control theory, we show the existence of solutions to the mentioned problem.

Nonsmooth Contact Dynamics for the large-scale simulation of granular material

For the large-scale simulation of granular material, the Nonsmooth Contact Dynamics Method (NSCD) is examined. First, the equations of motion of nonsmooth mechanical systems are introduced and classified as a Differential Variational Inequality (DVI) that has a structure similar to Differential–Algebraic Equations (DAEs). Using a Galerkin projection in time, we derive nonsmooth extensions of the SHAKE and RATTLE schemes. A matrix-free Interior Point Method (IPM) is used for the complementarity problems that need to be solved in each time step. We demonstrate that in this way, the NSCD approach yields highly accurate results and is competitive compared to the Discrete Element Method (DEM).

A Class of Delay Differential Variational Inequalities

In the paper, we introduce a class of delay differential variational inequalities consisting of a system of delay differential equations and variational inequalities. The existence conclusion of Caratheodory’s weak solution for delay differential variational equalities is obtained. Furthermore, an algorithm for solving the delay differential variational inequality is shown, and the convergence analysis for the algorithm is given. Finally, a numerical example is given to verify the validity of the algorithm.

Fractional optimal control problem for variational inequalities with control constraints

In this article, the fractional optimal control problem for variational inequalities is considered. The fractional time derivative is considered in Riemann–Liouville sense. The existence and uniqueness of solutions to a class of fractional differential variational inequalities in a Sobolev space is studied. An application to a fractional variational inequality in a bounded domain with Dirichlet and Neumann boundary conditions is given. Constraints on controls are imposed. Necessary and sufficient optimality conditions for the fractional Cauchy problems with the quadratic performance functional are derived. Specifically, the Euler–Lagrange equations first order optimality condition with an adjoint problem are presented.

Multi-valued stochastic differential equations driven by G-Brownian motion and related stochastic control problems

ABSTRACTIn this paper, we prove the existence and uniqueness of a solution for a class of multi-valued stochastic differential equations driven by G-Brownian motion (MSDEG) by means of the Yosida approximation method. Moreover, we set up an optimality principle of stochastic control problem and prove the value function of the control problem is the unique viscosity solution of a class of nonlinear partial differential variational inequalities.

A class of differential fuzzy variational inequalities in finite-dimensional spaces

In this paper, a new class of differential fuzzy variational inequalities in finite dimensional spaces are introduced and studied. An existence theorem of the Caratheodory weak solution of differential fuzzy variational inequality is established under some suitable conditions. The convergence analysis for time-stepping method for finding the weak solution of the differential fuzzy variational inequality is given. Finally, an example is reported to verify the validity of the proposed algorithm.

A class of impulsive differential variational inequalities in finite dimensional spaces

In this paper, a new class of impulsive differential variational inequalities (IDVIs) are introduced and studied in finite dimensional Euclidean spaces. Some existence results on the solutions for the IDVIs are presented under suitable conditions and a convergence theorem of the discrete Euler time-dependent procedure for solving the IDVI is proved by using constructed discrete approximation methods for the impulsive differential inclusions (IDIs). The stability results concerned with the solutions of the IDVIs are also considered when the variation of initial data, impulsive perturbation and right-hand sides happens.