Controllability and Inverse Problems for Hyperbolic and Dispersive Equations with Dynamic Boundary Conditions

TL;DR

This review discusses controllability and inverse problems for hyperbolic and dispersive equations with dynamic boundary conditions, emphasizing Carleman estimates and stability results, and highlighting challenges and future directions.

Contribution

It provides a comprehensive overview of recent advances in controllability and inverse problems for equations with dynamic boundary conditions, including new stability estimates and methodological insights.

Findings

Carleman estimates are effective for controllability and inverse problems.

Lipschitz stability estimates are established for source and coefficient identification.

Dynamic boundary conditions pose unique challenges compared to static ones.

Abstract

This review examines classical and recent results on controllability and inverse problems for hyperbolic and dispersive equations with dynamic boundary conditions. We aim to illustrate the applicability of Carleman estimates to establish exact controllability of such equations and derive Lipschitz stability estimates for inverse problems of source terms and coefficients with general dynamic boundary conditions. We highlight the challenges associated with dynamic boundary conditions compared to classical static ones. Finally, we conclude with a discussion of open problems and future research directions.