Galerkin Approximation for H^∞-Control of Parabolic Systems under Neumann Boundary Control

In this paper, we investigate the state feedback control for the H<sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sup> disturbance-attenuation problem in the context of general parabolic systems. The control mechanism employed is Neumann boundary control, which permits disturbances to enter the system through any channel. We employ the Galerkin approximation method, which generates a sequence of finite-dimensional systems. Our key finding is that the solutions to the corresponding finite-dimensional algebraic Riccati equations converge in norm to the solution of the infinite-dimensional operator algebraic Riccati equation. Furthermore, we establish that the state feedback controls derived from these finite-dimensional algebraic Riccati equations are γ-admissible for the original system, thereby validating the effectiveness of our approximation approach.