Projected incrementally scattering passive systems on closed convex sets

Abstract

In this article we show that the projected dynamical system obtained by restricting the state of an incrementally scattering passive system to a closed and convex subset $K$ of the state space (a real Hilbert space), is also an incrementally scattering passive system. First we show that the projection of a maximal dissipative operator to the tangent cones of $K$ is again maximal dissipative, hence, it determines a contraction semigroup. Using this result, we prove our earlier claim. Our results are based on the Crandall–Pazy theorem, Rockafellar’s theorem on sums of operators and Moreau’s decomposition theorem. We give an application of our results to Maxwell’s equations on a cylindrical domain, approximately describing a fault current limiter, restricting the average current through the cylinder (in the direction of its axis) so that its absolute value cannot exceed a given threshold.