Stability of Equilibrium Points of Projected Dynamical Systems

Abstract

We present a survey of the main results about asymptotic stability, exponential stability and monotone attractors of locally and globally projected dynamical systems, whose stationary points coincide with the solutions of a corresponding variational inequality. In particular, we show that the global monotone attractors of locally projected dynamical systems are characterized by the solutions of a corresponding Minty variational inequality. Finally, we discuss two special cases: when the domain is a polyhedron, the stability analysis for a locally projected dynamical system, at regular solutions to the associated variational inequality, is reduced to one of a standard dynamical system of lower dimension; when the vector field is linear, some global stability results, for locally and globally projected dynamical systems, are proved if the matrix is positive definite (or strictly copositive when the domain is a convex cone).