Feedback theory extended for proving generation of contraction semigroups

TL;DR

This paper extends feedback theory to prove the generation of contraction semigroups for a broader class of PDEs, including degenerate parabolic equations, by leveraging recent well-posed systems theory developments.

Contribution

It generalizes existing methods by integrating feedback techniques and modern system theory to establish contraction semigroup generation for complex PDEs.

Findings

Proves well-posedness of degenerate parabolic equations.

Extends feedback-based methods to broader PDE classes.

Shows the applicability to heat and wave equations with damping.

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.