{"ID":2889881,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2507.20171","arxiv_id":"2507.20171","title":"An operatorial approach of the well-posedness of an algebraic Riccati equation","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.","short_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...","url_abs":"https://arxiv.org/abs/2507.20171","url_pdf":"https://arxiv.org/pdf/2507.20171v3","authors":"[\"Gabriela Marinoschi\"]","published":"2025-07-27T08:22:46Z","proceeding":"math.OC","tasks":"[\"math.OC\",\"math.AP\"]","methods":"[]","has_code":false}