On continuity of solutions for parabolic control systems and input-to-state stability

TL;DR

This paper investigates conditions ensuring the continuity of solutions in parabolic control systems and explores input-to-state stability, especially in relation to boundary control and operator-theoretic methods.

Contribution

It establishes minimal conditions for solution continuity and links input-to-state stability with integral-input-to-state stability in parabolic PDE control systems.

Findings

Continuity of mild solutions under minimal conditions

Input-to-state stability coincides with integral-input-to-state stability in certain cases

Operator-theoretic methods facilitate analysis of boundary controlled PDEs

Abstract

We study minimal conditions under which mild solutions of linear evolutionary control systems are continuous for arbitrary bounded input functions. This question naturally appears when working with boundary controlled, linear partial differential equations. Here, we focus on parabolic equations which allow for operator-theoretic methods such as the holomorphic functional calculus. Moreover, we investigate stronger conditions than continuity leading to input-to-state stability with respect to Orlicz spaces. This also implies that the notions of input-to-state stability and integral-input-to-state stability coincide if additionally the uncontrolled equation is dissipative and the input space is finite-dimensional.