Approximation of Algebraic Riccati Equations with Generators of Noncompact Semigroups

TL;DR

This paper proves well-posedness and convergence of Galerkin and finite element approximations for algebraic Riccati equations associated with noncompact semigroups, with applications to wave control systems.

Contribution

It establishes the well-posedness of AREs without compactness assumptions and demonstrates optimal finite element approximations for wave semigroup control problems.

Findings

Galerkin approximations converge to the infinite-dimensional solution.

Finite element methods achieve optimal convergence rates.

Numerical examples confirm theoretical convergence results.

Abstract

In this work, we demonstrate that the Bochner integral representation of the Algebraic Riccati Equations (ARE) are well-posed without any compactness assumptions on the coefficient and semigroup operators. From this result, we then are able to determine that, under some assumptions, the solution to the Galerkin approximations to these equations are convergent to the infinite dimensional solution. Going further, we apply this general result to demonstrate that the finite element approximation to the ARE are optimal for weakly damped wave semigroup processes in the $H^1(\Omega) \times L^2(\Omega)$ norm. Optimal convergence rates of the functional gain for a weakly damped wave optimal control system in both the $H^1(\Omega) \times L^2(\Omega)$ and $L^2(\Omega)\times L^2(\Omega)$ norms are demonstrated in the numerical examples.