CONVERGENCE OF GALERKIN APPROXIMATIONS FOR OPERATOR RICCATI-EQUATIONS - A NONLINEAR EVOLUTION EQUATION APPROACH

We develop an approximation and convergence theory for Galerkin approximations to infinite dimensional operator Riccati differential equations formulated in the space of Hilbert-Schmidt operators on a separable Hilbert space. We treat the Riccati equation as a nonlinear evolution equation with dynamics described by a nonlinear monotone perturbation of a strongly coercive linear operator. We prove a generic approximation result for quasi-autonomous nonlinear evolution systems involving accretive operators which we then use to demonstrate the Hilbert-Schmidt norm convergence of Galerkin approximations to the solution of the Riccati equation. We illustrate the application of our results in the context of a linear quadratic optimal control problem for a one dimensional heat equation. © 1991.