Abstract nonlinear control systems

Abstract

We investigate abstract nonlinear infinite dimensional systems of the form: $\dot x(t) \in Ax(t) - {\mathcal{M}}(x(t)) + Bu(t)$ . These are obtained by subtracting a nonlinear maximal monotone (possibly multi-valued) operator ${\mathcal{M}}$ from the semigroup generator A of a linear system. While the linear system may have un-bounded linear damping (for instance, boundary damping), the operator ${\mathcal{M}}$ is "bounded" in the sense that it is defined on the whole state space. We show that under some assumptions, such nonlinear infinite dimensional systems have unique classical and generalized solutions. Moreover, these solutions are Lipschitz continuous on any finite time interval and right differentiable. 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 entire system, with states and input signals. We illustrate the theory with Maxwell’s equations in a bounded domain with a nonlinear conductor.