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Necessary first and second order optimality conditions for a fractional order differential equation with state delay

Jasarat J. Gasimov, Nazim I. Mahmudov

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 the 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 tivial manner. Consequently, we proceed to derive the second order optimality conditions specific to the problem under investigation. At the end illustrative examples are provided.

Necessary first and second order optimality conditions for a fractional order differential equation with state delay

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 the 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 tivial manner. Consequently, we proceed to derive the second order optimality conditions specific to the problem under investigation. At the end illustrative examples are provided.
Paper Structure (5 sections, 11 theorems, 85 equations)

This paper contains 5 sections, 11 theorems, 85 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Representation formula
  4. Statement of main result
  5. Conclusion

Key Result

Lemma 2.1

(29) For any $y(t)\in AC^{\alpha}_{\infty}([0,T],Y),$ the value $( ^{C}D^{\alpha}_{0+}y)(t)$ is correctly defined for a.e. $t\in [0,T].$ Furthermore, the inclusion $( ^{C}D^{\alpha}_{0+}y)(\cdot)\in L^{\infty}([0,T],Y)$ holds (i.e., there exists $\varphi(\cdot)\in L^{\infty}([0,T],Y)$ such that $\va

Theorems & Definitions (24)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Lemma 3.1
  • proof
  • ...and 14 more