Finding the state feedback control in an <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$H^{\infty }$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>H</mi> <mi>∞</mi> </msup> </math></EquationSource> </InlineEquation>-optimal control problem involves a challenging approach of the associated algebraic Riccati equation of the generic form <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$A^{*}P+PA+P\Gamma P=F$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>A</mi> <mrow> <mrow /> <mo>∗</mo> </mrow> </msup> <mi>P</mi> <mo>+</mo> <mi>P</mi> <mi>A</mi> <mo>+</mo> <mi>P</mi> <mi mathvariant="normal">Γ</mi> <mi>P</mi> <mo>=</mo> <mi>F</mi> </mrow> </math></EquationSource> </InlineEquation>. 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 <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\Gamma $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Γ</mi> </math></EquationSource> </InlineEquation> and F,  for the cases with A coercive and <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$A\ge 0,$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>A</mi> <mo>≥</mo> <mn>0</mn> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> respectively. They develop a constructive approach, possibly indicating a method for finding the numerical solution. Next, relying on the existence of the solution to the Riccati equation, we provide then a result concerning the associated <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$H^{\infty }$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>H</mi> <mi>∞</mi> </msup> </math></EquationSource> </InlineEquation>-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.

An Operatorial Approach of the Well-Posedness of an Algebraic Riccati Equation

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发表日期：

2026年3月5日

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Gabriela Marinoschi

摘要

Finding the state feedback control in an $H^{\infty }$ $H^{\infty}$ -optimal control problem involves a challenging approach of the associated algebraic Riccati equation of the generic form $A^{*}P+PA+P\Gamma P=F$ $A^{*} P + P A + P Γ P = F$ . 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 $ $Γ$ and F, for the cases with A coercive and $A\ge 0,$ $A \geq 0,$ 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 $H^{\infty }$ $H^{\infty}$ -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.