On the Approximation of Operator-Valued Riccati Equations in Hilbert Spaces

TL;DR

This paper develops an abstract framework for approximating operator-valued Riccati equations in Hilbert spaces, providing error bounds and optimal convergence results for finite element methods, with applications to control systems.

Contribution

It introduces a unified theory linking semigroup approximation errors to Riccati equation solutions and derives optimal error estimates for finite element approximations in control applications.

Findings

Error of approximate Riccati solutions bounded by semigroup approximation error

Optimal convergence rates predicted for finite element methods

Computational results confirm theoretical error estimates

Abstract

In this work, we present an abstract theory for the approximation of operator-valued Riccati equations posed on Hilbert spaces. It is demonstrated here that the error of the approximate solution to the operator-valued Riccati equation is bounded above by the approximation error of the governing semigroup, under the assumption of boundedness on the semigroup and compactness on the coefficient operators. One significant outcome of this result is the correct prediction of optimal convergence for finite element approximations of the operator-valued Riccati equations for when the governing semigroup involves parabolic, as well as hyperbolic processes. We derive the abstract theory for the time-dependent and time-independent operator-valued Riccati equations in the first part of this work. In the second part, we derive optimal error estimates for the finite element approximation of the functional gain associated with model weakly damped wave and thermal LQR control systems. These theoretical claims are then corroborated with computational evidence.