Boundary Control of Coupled Reaction-Advection-Diffusion Systems with Spatially-Varying Coefficients

TL;DR

This paper extends boundary stabilization techniques to coupled reaction-advection-diffusion systems with spatially-varying coefficients, establishing well-posedness and stability by linking to hyperbolic system kernels.

Contribution

It demonstrates the equivalence of control kernel equations for these complex systems to those of hyperbolic systems, enabling stability analysis.

Findings

Proves well-posedness of control kernel equations for systems with advection and spatially-varying coefficients.

Establishes H^1 stability for the closed-loop system.

Reveals a connection between backstepping kernels for parabolic and hyperbolic problems.

Abstract

Recently, the problem of boundary stabilization for unstable linear constant-coefficient coupled reaction-diffusion systems was solved by means of the backstepping method. The extension of this result to systems with advection terms and spatially-varying coefficients is challenging due to complex boundary conditions that appear in the equations verified by the control kernels. In this paper we address this issue by showing that these equations are essentially equivalent to those verified by the control kernels for first-order hyperbolic coupled systems, which were recently found to be well-posed. The result therefore applies in this case, allowing us to prove H^1 stability for the closed-loop system. It also shows an interesting connection between backstepping kernels for coupled parabolic and hyperbolic problems.