Numerical Approximations of Algebraic Riccati Equations for Abstract Systems Modelled by Analytic Semigroups , and Applications

This paper provides a numerical approximation theory of algebraic Riccati operator equations with unbounded coefficient operators A and B , such as arise in the study of optimal quadratic cost problems over the time interval [0, oo] for the abstract dynamics y = Ay + Bu . Here, A is the generator of a strongly continuous analytic semigroup, and B is an unbounded operator with any degree of unboundedness less than that of A . Convergence results are provided for the Riccati operators, as well as for all the other relevant quantities which enter into the dynamic optimization problem. The present numerical theory is the counterpart of a known continuous theory. Several examples of partial differential equations with boundary/point control, where all the required assumptions are verified, illustrate the theory. They include parabolic equations with L2"Dirichlet control, as well as plate equations with a strong degree of damping and point control.