An operatorial approach of the well-posedness of an algebraic Riccati equation

TL;DR

This paper investigates the existence of solutions to an algebraic Riccati equation using an operatorial approach in Hilbert-Schmidt spaces, with implications for $H^{ ablafty}$-optimal control and numerical methods.

Contribution

It introduces a direct operatorial method for solving the algebraic Riccati equation in Hilbert-Schmidt spaces, providing proofs and a constructive approach for numerical solutions.

Findings

Existence of solutions under certain operator conditions.

Development of a constructive method for numerical approximation.

Application to a parabolic PDE with Hardy potential.

Abstract

Finding the state feedback control in an $% H^{\infty }$-optimal control problem involves a challenging approach of the associated algebraic Riccati equation of the generic form $A^{\ast }P+PA+P\Gamma P=F$. In view of this objective, we explore in this paper the existence of the solution to this algebraic Riccati equation by a direct operatorial approach in the space of Hilbert-Schmidt operators. The proofs are provided, under certain assumptions on the operators $\Gamma $ and $F,$ for the cases with $A$ coercive and $A\geq 0,$ respectively. They develop a constructive approach, possibly indicating a method for finding the numerical solution. Next, relying on the existence of the solution to the Riccati equation, we provide then a result concerning the associated $% H^{\infty }$-optimal control problem. An example regarding the application of the existence proof for the solution to the Riccati equation is given for a parabolic equation with a singular potential of Hardy type.