Fractional optimal control problem for infinite order system with control constraints

G Mohamed Bahaa

doi:10.1186/s13662-016-0976-2

Abstract

In this article, we study a homogeneous infinite order Dirichlet and Neumann boundary fractional equations in a bounded domain. The fractional time derivative is considered in a Riemann-Liouville sense. Constraints on controls are imposed. The existence results for equations are obtained by applying the classical Lax-Milgram Theorem. The performance functional is in quadratic form. Then we show that the optimal control problem associated to the controlled fractional equation has a unique solution. Interpreting the Euler-Lagrange first order optimality condition with an adjoint problem defined by means of the right fractional Caputo derivative, we obtain an optimality system. The obtained results are well illustrated by examples.

Highlights

The fractional calculus is nowadays an excellent mathematical tool which opens the gates for finding hidden aspects of the dynamics of the complex processes which appear naturally in many branches of science and engineering
The recent trend in the mathematical modeling of several phenomena indicates the popularity of fractional calculus modeling tools due to the nonlocal characteristic of fractional-order differential and integral operators, which are capable of tracing the past history of many materials and processes; see, for instance, [ – ] and the references therein
The fractional derivatives were defined in the Riemann-Liouville sense

Summary

Introduction

The fractional calculus is nowadays an excellent mathematical tool which opens the gates for finding hidden aspects of the dynamics of the complex processes which appear naturally in many branches of science and engineering. The recent trend in the mathematical modeling of several phenomena indicates the popularity of fractional calculus modeling tools due to the nonlocal characteristic of fractional-order differential and integral operators, which are capable of tracing the past history of many materials and processes; see, for instance, [ – ] and the references therein. We consider here a different type of evolution equations, namely, fractional partial differential equations involving infinite order operators see Bahaa and Kotarski [ ] and papers therein. Such an infinite order system can be treated as a generalization of the mathematical model for a plasma control process. In Section , we formulate the fractional Dirichlet problem for infinite order equations.

Sobolev spaces with infinite order and fractional derivatives

Conclusions