Feedback theory extended for proving generation of contraction semigroups

Abstract

Recently, the following novel method for proving the existence of solutions for certain linear time-invariant PDEs was introduced: The operator associated to a given PDE is represented by a (larger) operator with an internal loop. If the larger operator (without the internal loop) generates a contraction semigroup, the internal loop is accretive, and some non-restrictive technical assumptions are fulfilled, then the original operator generates a contraction semigroup as well. Beginning with the undamped wave equation, this general idea can be applied to show that the heat equation and wave equations with damping are well-posed. In the present paper we show how this approach can benefit from feedback techniques and recent developments in well-posed systems theory, at the same time generalising the previously known results. Among others, we show how well-posedness of degenerate parabolic equations can be proved.