Well-posedness of boundary control systems and application to ISS for coupled heat equations with boundary disturbances and delays

TL;DR

This paper establishes conditions ensuring the well-posedness of boundary control systems and applies these results to demonstrate exponential input-to-state stability for coupled heat equations with disturbances and delays.

Contribution

It provides explicit verifiable conditions for solution dependence on data and introduces a new boundedness estimate for input/output maps in infinite-dimensional systems.

Findings

Conditions for well-posedness of boundary control systems

Explicit criteria for exponential input-to-state stability

Application to coupled heat equations with disturbances and delays

Abstract

This paper studies the existence of solutions and, in particular, the well-posedness of a class of boundary control systems. Our main result provides explicit and verifiable conditions on the system data that guarantee continuous dependence of solutions on the initial data and $L^p$-inputs. The proof relies on a new boundedness estimate for the input/output maps of linear time-invariant infinite-dimensional systems with unbounded control and observation operators. The developed technique is applied to derive specific conditions for the exponential input-to-state stability of boundary-coupled heat equations with boundary disturbances and time-delays.