Controllability and Inverse Problems for Parabolic Systems with Dynamic Boundary Conditions

TL;DR

This paper reviews recent advances in controlling and solving inverse problems for parabolic systems with dynamic boundary conditions, highlighting the extension of classical methods like Carleman estimates to these complex scenarios.

Contribution

It demonstrates how classical control and inverse problem techniques can be adapted to parabolic systems with dynamic boundary conditions, addressing key difficulties and open problems.

Findings

Extension of Carleman estimates to dynamic boundary conditions

Lipschitz stability estimates for inverse problems

Identification of open problems in the field

Abstract

This review surveys previous and recent results on null controllability and inverse problems for parabolic systems with dynamic boundary conditions. We aim to demonstrate how classical methods such as Carleman estimates can be extended to prove null controllability for parabolic systems and Lipschitz stability estimates for inverse problems with dynamic boundary conditions of surface diffusion type. We mainly focus on the substantial difficulties compared to static boundary conditions. Finally, some conclusions and open problems will be mentioned.