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<p>In this research paper, we examine an optimal control problem involving a dynamical system governed by a nonlinear Caputo fractional time-delay state equation. The primary objective of this study is to obtain the necessary conditions for optimality, both first and second order, for the Caputo fractional time-delay optimal control problem. We derive the first-order necessary condition for optimality for the given fractional time-delay optimal control problem. Moreover, we focus on a case where the Pontryagin maximum principle degenerates, meaning that it is satisfied in a trivial manner. Consequently, we proceed to derive the second-order optimality conditions specific to the problem under investigation. At the end, illustrative examples are provided.</p>

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Necessary First and Second Order Optimality Conditions for a Fractional Order Differential Equation with State Delay

  • 期刊论文

  • 发表日期：

    2026年2月11日

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  • Jasarat J. Gasimov,
  • Nazim I. Mahmudov

摘要

In this research paper, we examine an optimal control problem involving a dynamical system governed by a nonlinear Caputo fractional time-delay state equation. The primary objective of this study is to obtain the necessary conditions for optimality, both first and second order, for the Caputo fractional time-delay optimal control problem. We derive the first-order necessary condition for optimality for the given fractional time-delay optimal control problem. Moreover, we focus on a case where the Pontryagin maximum principle degenerates, meaning that it is satisfied in a trivial manner. Consequently, we proceed to derive the second-order optimality conditions specific to the problem under investigation. At the end, illustrative examples are provided.