Operator theoretic approach in fractional‐order delay optimal control problems

Abstract

This paper investigates the fractional optimal control problem with a single delay in the state by using the operator theoretic approach. In this approach, we first reduce the delay fractional dynamical system into an equivalent operator equation, and then, by providing sufficient conditions to the operators, the existence of an optimal pair is proved for the abstract system. The optimality system for the quadratic cost functional is derived by using the Frechet derivative. Then we relate the operator theoretic approach optimality system to a Hamiltonian system of Pontryagin's minimum principle. The primary goal of this article is to demonstrate the existence of an optimal pair for the fractional‐order delay dynamical system by using a functional minimization theorem in functional analysis. Likewise, the optimality systems for the fractional‐order delay dynamical system are derived by using the functional (either Gateaux or Frechet) derivatives. Finally, we provide two numerical examples that support our theoretical findings.