Three-Field Modelling of Nonlinear Nonsmooth Boundary Value Problems and Stability of Differential Mixed Variational Inequalities

The purpose of this paper is twofold. Firstly we consider nonlinear nonsmooth elliptic boundary value problems, and also related parabolic initial boundary value problems that model in a simplified way steady-state unilateral contact with Tresca friction in solid mechanics, respectively, stem from nonlinear transient heat conduction with unilateral boundary conditions. Here a recent duality approach, that augments the classical Babuška-Brezzi saddle point formulation for mixed variational problems to twofold saddle point formulations, is extended to the nonsmooth problems under consideration. This approach leads to variational inequalities of mixed form for three coupled fields as unknowns and to related differential mixed variational inequalities in the time-dependent case. Secondly we are concerned with the stability of the solution set of a general class of differential mixed variational inequalities. Here we present a novel upper set convergence result with respect to perturbations in the data, including perturbations of the associated nonlinear maps, the nonsmooth convex functionals, and the convex constraint set. We employ epiconvergence for the convergence of the functionals and Mosco convergence for set convergence. We impose weak convergence assumptions on the perturbed maps using the monotonicity method of Browder and Minty

A Note on Linear Differential Variational Inequalities in Hilbert Space

Part 2: Control of Distributed Parameter SystemsInternational audienceRecently a new class of differential variational inequalities has been introduced and investigated in finite dimensions as a new modeling paradigm of variational analysis to treat many applied problems in engineering, operations research, and physical sciences. This new subclass of general differential inclusions unifies ordinary differential equations with possibly discontinuous right-hand sides, differential algebraic systems with constraints, dynamic complementarity systems, and evolutionary variational systems. In this short note we lift this class of nonsmooth dynamical systems to the level of a Hilbert space, but focus to linear input/output systems. This covers in particular linear complementarity systems where the underlying convex constraint set in the variational inequality is specialized to an ordering cone.The purpose of this note is two-fold. Firstly, we provide an existence result based on maximal monotone operator theory. Secondly we are concerned with stability of the solution set of linear differential variational inequalities. Here we present a novel upper set convergence result with respect to perturbations in the data, including perturbations of the associated linear maps and the constraint set