Bilateral Control for an Unstable 2-D Reaction-Diffusion Equation with Unequal Input Delays at the Boundaries

This paper develops a bilateral delay-compensation control strategy for an unstable 2-D reaction-diffusion system with unequal input delays. The main novelty compared to our previous work lies in the design of asymmetric backstepping transformations tailored to the unequal delays, which results in a more complex kernel equation associated with the larger delay and poses significant challenges in proving its well-posedness. To address this, we employ a combination of operator semigroup theory and Lyapunov function analysis. Based on this result, we prove the boundedness and invertibility of the backstepping transformations and establish the exponential stability of the closed-loop system under the proposed compensator.