Covergence rates for the approximations of the solutions to algebraic Riccati equations with unbounded coefficients: Case of analytic semigroups (Q1195914)

Algebraic Riccati equations are considered in connection with boundary control problems and boundary feedback stabilization. The input operator is unbounded and the dynamics of the system are described by an analytic semigroup. A general abstract theory is developed. Optimal rates of convergence of the Riccati operators and of the feedback dynamics which would reflect the regularity properties of the original continuous problem are established. This is the main novelty of the paper. The arguments leading to nonoptimal rates of convergence (or optimal rates for bounded input operators) are essentially the same as the convergence arguments from the author's joint paper with \textit{R. Triggiani} [Math. Comput. 57, No. 196, 639-662, Suppl. 13-37 (1991; Zbl 0735.65043)]. In the case of an input unbounded operator the optimal rates require much more delicate analysis. Applications are given to the heat equation with boundary controls. The elliptic operator is approximated separately for the Neumann and Dirichlet problems. Since the Dirichlet problem does not admit a natural variational formulation special attention is payed to the approximation of the boundary conditions. The Nitsche elliptic approximation of the Poisson operator is used for the Dirichlet problem while a standard Galerkin approximation is sufficient for the Neumann one.