ABSTRACT
This chapter considers finite dimensional approximations of Algebraic Riccati equations with unbounded inputs such as those arising in hyperbolic boundary control problems. It studies the approximations of Riccati Equations in the case of bounded control operators. Convergence results, and in some cases rates of convergence, for approximations of Riccati solutions arising in optimization problems with unbounded controls are given in the case of analytic semigroups and in the general hyperbolic case. All these convergence results depend upon the verification of certain control theoretic properties such as a “uniform cost condition” and a “uniform trace condition” (in the case of unbounded control) to hold uniformly in the parameter of discretization. The chapter proposes an algorithm which combines a regularization and an approximation (e.g. Finite Element Method) procedure. It proves that this algorithm produces a convergent approximation to the solution of the Riccati Equation. The theoretical results are illustrated by an example of the wave equation with boundary control.
