Adaptive Sliding Mode Boundary Control of a Diffusion Process Under Matched and Unmatched Boundary Disturbances

This paper investigates the robust boundary control of a diffusion process subject to matched and unmatched disturbances. The bounds of the matched disturbance are unknown, motivating the use of an adaptive sliding mode control law. It is applied at one boundary, where a matched disturbance acts alongside the control input. Additionally, an unmatched disturbance affects the opposite boundary. While the matched disturbance can be fully rejected, the unmatched disturbance fundamentally limits the achievable performance. It is shown that the adaptive gain remains bounded, and an ultimate bound on the L2-norm of the state is derived in terms of the unmatched disturbance magnitude. The bound is established via a transfer function analysis and an input-to-state stability argument. Numerical simulations illustrate the theoretical findings.