Sliding mode control is a robust control strategy, especially suited for systems subject to parameter uncertainties and external disturbances. Motivated by the stabilization of the temperature distribution in a metallic beam by thermoelectric modules acting on the boundaries, we study a parabolic problem with adaptive sliding mode boundary control. The resulting closed-loop system leads to nonlinear, non-autonomous, and state-dependent boundary conditions.
While the control method is already employed in laboratory experiments, a rigorous mathematical analysis of the underlying control scheme has so far been missing. In this talk, we establish well-posedness of the closed-loop system and discuss the analytical challenges arising from the adaptive, discontinuous boundary feedback. We further comment on Lyapunov-based stability properties and the relation to the motivating application and numerical results.