A Note on Linear Differential Variational Inequalities in Hilbert Space

Recently 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.