Fractional optimal control problem for differential system with delay argument

G Mohamed Bahaa

doi:10.1186/s13662-017-1121-6

Abstract

In this paper, we apply the classical control theory to a fractional differential system in a bounded domain. The fractional optimal control problem (FOCP) for differential system with time delay is considered. The fractional time derivative is considered in a Riemann-Liouville sense. We first study the existence and the uniqueness of the solution of the fractional differential system with time delay in a Hilbert space. Then we show that the considered optimal control problem has a unique solution. The performance index of a FOCP is considered as a function of both state and control variables, and the dynamic constraints are expressed by a partial fractional differential equation. The time horizon is fixed. Finally, we impose some constraints on the boundary control. Interpreting the Euler-Lagrange first order optimality condition with an adjoint problem defined by means of a right fractional Caputo derivative, we obtain an optimality system for the optimal control. Some examples are analyzed in detail.

Highlights

An optimal control problem (OCP) deals with the problem of finding a control law for a given dynamic system that minimizes a performance index in terms of the state and control variables
Various optimization problems associated with the integer optimal control of differential systems with time delays were studied recently by many authors; see for instance [ – ] and the references therein)
The OCP reduces to a fractional optimal control problem (FOCP) when either the performance index or the dynamic constraints or both include at least one fractional-order derivative term

Summary

Introduction

An optimal control problem (OCP) deals with the problem of finding a control law for a given dynamic system that minimizes a performance index in terms of the state and control variables. Various optimization problems associated with the integer optimal control of differential systems with time delays were studied recently by many authors; see for instance [ – ] and the references therein). The OCP reduces to a fractional optimal control problem (FOCP) when either the performance index or the dynamic constraints or both include at least one fractional-order derivative term. Bahaa Advances in Difference Equations (2017) 2017:69 optimal control problem in which either the performance index or the differential equations governing the dynamics of the system or both contain at least one fractional derivative term. The fractional derivative was defined in the Riemann-Liouville sense and the formulation was obtained by means of fractional variation principle and the Lagrange multiplier technique. Fractional optimal control problems for fractional dynamic systems with deviating argument are presented.

Some basic definitions

Conclusions