Optimal Control Problems Involving Combined Fractional Operators with General Analytic Kernels

Faïçal Ndaïrou,Delfim F M Torres

doi:10.3390/math9192355

Abstract

Fractional optimal control problems via a wide class of fractional operators with a general analytic kernel are introduced. Necessary optimality conditions of Pontryagin type for the considered problem are obtained after proving a Gronwall type inequality as well as results on continuity and differentiability of perturbed trajectories. Moreover, a Mangasarian type sufficient global optimality condition for the general analytic kernel fractional optimal control problem is proved. An illustrative example is discussed.

Highlights

Fractional Calculus, as a generalization of the traditional calculus through derivation and integration of an arbitrary order, is a rapidly growing field of mathematical research
Since this seminal work of 2019, several interesting results appeared, e.g., determination of source terms for fractional Rayleigh–Stokes equations with random data [2], new analytic properties of tempered fractional calculus [3], simulation of nonlinear dynamics with fractional neural networks arising in the modeling of cognitive decision making processes [4], new numerical methods for variable order fractional nonlinear quadratic integro-differential equations [5], and analysis of impulsive φ-Hilfer fractional differential equations [6]
We prove several important results: continuity of solutions to optimal control problems (Lemma 8), which is an application of our Gronwall’s inequality (Theorem 1); differentiability of the perturbed trajectories (Corollary 1); and a necessary optimality condition of Pontryagin type to problem (8) (Theorem 2), which happens to be an application of the results on continuity, differentiability, and integration by parts

Summary

Introduction

Fractional Calculus, as a generalization of the traditional calculus through derivation and integration of an arbitrary order, is a rapidly growing field of mathematical research. This is important in the sense that with a general framework of operators it might be possible to establish a mathematical theory for this general formalism, rather than considering specific models with particular results In this direction, Fernandez, Özarslan and Baleanu proposed in 2019 a fractional integral operator, based on a general analytic kernel, that includes a number of existing and known operators [1]. We investigate, for the first time in the literature, optimal control problems that involve a combined Caputo fractional derivative with a general analytic kernel in the sense of Fernandez, Özarslan and Baleanu.

Preliminaries

Main Results

Gronwall’s Inequality

Applications

Sufficient Condition for Global Optimality

Conclusions