Output-Feedback Boundary Control of Reaction--Diffusion PDEs on Arbitrary Lipschitz Domains: A Target-Domain Approach

Abstract

We present a domain-extension framework for output-feedback boundary stabilization of reaction-diffusion equations on arbitrary bounded Lipschitz domains, including non-convex and multiply connected geometries. The plant is posed on an irregular domain whose boundary has actuated and uncontrolled portions. Just as backstepping transforms the plant dynamics into a stable target system, the method embeds the plant in a target domain, such as a ball or a rectangle, where a stabilizing design is already known. Every boundary portion through which the extension proceeds must carry actuation and the complementary collocated measurement. Uncontrolled portions are allowed when they are shared with the target boundary and have the same boundary-condition type. The gap between the two domains is filled with a virtual copy of the plant dynamics, coupled to the plant through interface conditions, and the concatenated state evolves exactly as the known closed loop on the target domain. Well-posedness and exponential stability of the physical state follow by restriction. The offline design data are inherited from the target design and are closed-form for constant-coefficient plants on balls and rectangles. Online simulation of the virtual PDE has the same computational character as a full-order PDE observer, a standard component of output-feedback designs. A new explicit Neumann-actuated backstepping law on n-balls enlarges the available target designs. Output feedback is obtained by lifting the target-domain observer, driven by a collocated interface measurement relayed through the virtual domain. Numerical experiments on star-shaped, horseshoe, and multiply connected domains, with a partitioned plant/controller implementation and a shared-wall cavity, test the designs.