Feedback Stabilization of a 1-D Linear Reaction-Diffusion Equation With Delay Boundary Control

• Published in: IEEE transactions on automatic control Vol. 64; no. 4; pp. 1415 - 1425
• Main Authors: Prieur, Christophe, Trelat, Emmanuel
• Format: Journal Article
• Language: English
• Published: New York IEEE 01.04.2019; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Boundary control; Computation; Computer simulation; Control systems; Controllers; Delay; delay control; Delays; Design methodology; Dimensional stability; Feedback; Feedback control; Liapunov functions; Lyapunov function; Lyapunov methods; Mathematical model; partial differential equation (PDE); Partial differential equations; Reaction-diffusion equations; Reaction–diffusion equation
• ISSN: 0018-9286; 1558-2523
• DOI: 10.1109/TAC.2018.2849560

The goal of this paper is to design a stabilizing feedback boundary control for a reaction-diffusion partial differential equation (PDE), where the boundary control is subject to a constant delay while the equation may be unstable without any control. For this system, which is equivalent to a parabolic equation coupled with a transport equation, a prediction-based control is explicitly computed by splitting the infinite-dimensional system into two parts: a finite-dimensional unstable part and a stable infinite-dimensional part. A finite-dimensional delayed controller is computed for the unstable part, and it is shown that this controller stabilizes the whole PDE. The proof is based on an explicit expression of the classical Artstein transformation combined with an adequately designed Lyapunov function. A numerical simulation illustrates the constructive feedback design method.