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

Gasimov J. J., Mahmudov N.

Journal of Optimization Theory and Applications, vol.208, no.3, 2026 (SCI-Expanded, Scopus) identifier identifier

  • Publication Type: Article / Article
  • Volume: 208 Issue: 3
  • Publication Date: 2026
  • Doi Number: 10.1007/s10957-025-02919-7
  • Journal Name: Journal of Optimization Theory and Applications
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus, ABI/INFORM, Compendex, INSPEC, MathSciNet, zbMATH
  • Keywords: Constant delay, Optimal control, Pontryagin maximum principle, Retarded differential equation
  • Azerbaijan State University of Economics (UNEC) Affiliated: Yes

Abstract

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.