An operatorial approach of the well-posedness of an algebraic Riccati equation
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866914409029304320 |
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| author | Marinoschi, Gabriela |
| author_facet | Marinoschi, Gabriela |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_20171 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | An operatorial approach of the well-posedness of an algebraic Riccati equation Marinoschi, Gabriela Optimization and Control Analysis of PDEs 93B36, 93B52, 93B35, 35K90 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. |
| title | An operatorial approach of the well-posedness of an algebraic Riccati equation |
| topic | Optimization and Control Analysis of PDEs 93B36, 93B52, 93B35, 35K90 |
| url | https://arxiv.org/abs/2507.20171 |