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

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.