An operatorial approach of the well-posedness of an algebraic Riccati equation
Gabriela Marinoschi
TL;DR
This work analyzes the well-posedness of the operatorial algebraic Riccati equation $A^{*}P+PA+P\Gamma P=F$ in the Hilbert–Schmidt framework for infinite-dimensional systems. It develops a direct operatorial approach to prove existence of a self-adjoint, positive solution $P$ in both the coercive case ($A$ coercive) and the noncoercive case ($A\ge0$), employing monotone operator theory and Yosida approximations; the approach is constructive, suggesting numerical pathways. Relying on the Riccati solution, the paper then proves that the feedback $F=-B_{2}^{*}P$ attains the $H^{\infty}$-optimal control objective, with exponential stability of the closed-loop system and an $L^{2}$-gain bound $\|z\|_{L^{2}}\le \gamma\|w\|_{L^{2}}$. An illustrative parabolic equation with a Hardy-type potential demonstrates applicability and the constructive nature of the proofs, pointing to practical numerical schemes for robust PDE stabilization.
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 $A^{\ast }P+PA+PΓP=F$. 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 $Γ$ 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.