Adaptive sliding mode boundary control of a perturbed diffusion process

This paper proposes a sliding-mode-based adaptive boundary control law for stabilizing a class of uncertain diffusion processes affected by a matched disturbance. The matched disturbance is assumed to be uniformly bounded along with its time derivative, whereas the corresponding upper bounding constants are not known. This motivates the use of adaptive control strategies. In addition, the spatially-varying diffusion coefficient is also uncertain. To achieve asymptotic stability of the plant origin in the L2$$ {L}_2 $$-sense in the presence of the disturbance, a discontinuous boundary feedback law is proposed where the gain of the discontinuous control term is adjusted according to a gradient-based adaptation law. A constructive Lyapunov analysis supports the stability properties of the considered closed-loop system, yielding sufficient convergence conditions in terms of suitable inequalities involving the controllers' tuning parameters. Simulation results are presented to corroborate the theoretical findings.