Backstepping Boundary Control of a 1-D 2 × 2 Unstable Diffusion-Reaction PDE System with Distinct Input Delays

We consider the stabilization problem for a 1-D 2 × 2 unstable diffusion-reaction partial differential equation system with distinct input delays. The input delays are represented with a system of first-order hyperbolic PDEs, leading to a 2 + 2 × 2 mixed-type system of hyperbolic-parabolic type. We establish two integral transformations, which admit a target system with triangular trace terms. Exponential stability is shown for the target system in the H
¹
× H
¹
norm. The transformations admit a companion mixed-type gain kernel PDE system. We employ method of characteristics and Galerkin methods to show existence of solutions to the kernel PDE, guaranteeing the invertibility of the transformation and thus confirming the exponential stability results.