Optimal convergence rates for Galerkin approximation of operator Riccati equations

In this paper we consider the problem of determining optimal convergence rates of Galerkin approximations to infinite dimensional operator Riccati equations (OREs). Optimal rates are obtained for a class of abstract distributed parameter systems evolving in an infinite dimensional Hilbert space. These general results are then applied to systems modeled by partial differential equations that generate compact and analytic semigroups. The estimates apply to distributed control and observation of classical parabolic equations and to certain vibration problems with sufficiently strong damping. The ORE is formulated as an equivalent operator-valued Bochner integral equation and the Brezzi-Rappaz-Raviart theorem is used to obtain convergence rates. First we establish smoothing property and bounds for the solutions of the infinite dimensional ORE. Then it is shown that, under sui\ assumptions on the coefficients and domain geometry, the hp-finite element approximations of the classical solution converges on the order of O(h(k+1)). Furthermore, these optimal error bounds are shown to hold for the functional gains that define observer and control gain operators. We provide numerical examples that corroborate the theoretical convergence rates.