Convergence rates for the approximations of the solutions to algebraic Riccati Equations with unbounded coefficients: case of analytic semigroups


Approximations schemes for the solutions of the Algebraic Riccati Equations will be considered. We shall concentrate on the case when the input operator is unbounded and the dynamics of the system is described by an analytic semigroup. The main goal of the paper is to establish the optimal rates of convergence of the underlying approximations. By optimal, we mean such approximations which would reconstruct the optimal regularity of the original solutions as well as the 'best' approximation properties of the finite-dimensional subspaces. It turns out that, if one aims to obtain the optimal rates in the case of unbounded input operators, the choice of the approximations to the generator, as well as to the control operator, is very critical. While the convergence results hold with any 'consistent' approximations, the optimal rates require a careful selection of the approximating schemes. Our theoretical results will be illustrated by several examples of boundary/point control problems where the optimal rates of convergence are achieved with the appropriate numerical algorithms. © 1992 Springer-Verlag.

Publication Title

Numerische Mathematik