Galerkin approximation for H∞-control of the stable parabolic system under Dirichlet boundary control

In this paper, we explore state feedback control for the $H^{\infty }$ disturbance-attenuation problem in stable parabolic systems with in-domain distributed disturbances under Dirichlet boundary control. Calculating the state feedback control involves solving an operator algebraic Riccati equation, which poses challenges in finding an analytic solution. A practical approach is to seek an approximate solution via finite-dimensional approximation. Specifically, we employ the Galerkin approximation, which generates a sequence of finite-dimensional systems that approximate the original infinite-dimensional system. All corresponding finite-dimensional disturbance-attenuation problems are solvable, and it is demonstrated that the sequence of solutions to the associated finite-dimensional algebraic Riccati equations converges in norm to the solution of the infinite-dimensional operator algebraic Riccati equation. Furthermore, the state feedback controls derived from the finite-dimensional algebraic Riccati equations are proven to be $\gamma$-admissible controls for the original system.