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<p>This paper concerns a class of <i>linear-state</i> optimal control problems and noncooperative differential games. Deterministic and stochastic systems are considered, as well as finite- and infinite-horizon problems. We give conditions under which these systems have <i>degenerate</i> feedback optimal controls so that the optimal control actions <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10957_2025_2657_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="96" /> </InlineMediaObject> <EquationSource Format="TEX">\(a(t,x) \equiv a(t)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>a</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>,</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>≡</mo> <mi>a</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> are independent of the state variable <i>x</i>. As a consequence, open-loop and feedback (or Markov) optimal controls coincide, the value (or optimal objective) function is linear in the state <i>x</i>, and the <i>certainty equivalence</i> principle is satisfied.</p>

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Linear–State Control Problems and Differential Games: Deterministic and Stochastic Systems

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年4月2日

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  • José E. Márquez–Prado,
  • Onésimo Hernández–Lerma

摘要

This paper concerns a class of linear-state optimal control problems and noncooperative differential games. Deterministic and stochastic systems are considered, as well as finite- and infinite-horizon problems. We give conditions under which these systems have degenerate feedback optimal controls so that the optimal control actions \(a(t,x) \equiv a(t)\) a ( t , x ) ≡ a ( t ) are independent of the state variable x. As a consequence, open-loop and feedback (or Markov) optimal controls coincide, the value (or optimal objective) function is linear in the state x, and the certainty equivalence principle is satisfied.