Adaptive Sliding Mode Boundary Control of a Perturbed Diffusion Process With Uncertain Input Gain

Robust boundary control of a diffusion process with spatially varying, unknown diffusivity, matched additive disturbance, and an uncertain, time-varying input gain is addressed. An adaptive sliding-mode boundary controller is proposed, combining a proportional and discontinuous term whose gain updates monotonically via a gradient-based adaptation law. No a priori bounds on the disturbances are required for tuning. Under standard boundedness and positivity assumptions for the uncertain drift disturbance and control gains, global asymptotic stability of the closed-loop origin in the L2 -sense is established via a tailored Lyapunov construction and Barbalat's lemma. Numerical simulations verify and illustrate the theoretical findings.