Non-linear damping for scattering-passive systems in the Maxwell class

Abstract

We start from a special class of scattering passive linear infinite-dimensional systems introduced in Staffans and Weiss (SIAM J. Control and Opt., 2012). This class is called the Maxwell class of systems, because it includes the scattering formulation of Maxwell’s equations, as well as various wave and beam equations. We generalize this class by allowing a nonlinear damping term. While the system may have unbounded linear damping (for instance, boundary damping), the nonlinear damping term N is “bounded” in the sense that it defined on the whole state space (but no actual continuity assumption is made on N). We show that this new class of nonlinear infinite dimensional systems is well-posed and scattering passive. Our approach uses the theory of maximal monotone operators and the Crandall-Pazy theorem about nonlinear contraction semigroups, which we apply to a Lax-Phillips type nonlinear semigroup that represents the whole system, with input and output signals.