A global optimization approach to fractional optimal control

Enkhbat Rentsen; J. Zhou; K. L. Teo

doi:10.3934/jimo.2016.12.73

Article Contents

2016, Volume 12, Issue 1: 73-82. Doi: 10.3934/jimo.2016.12.73

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A global optimization approach to fractional optimal control

1.
Institute of Mathematics, National University of Mongolia, Ulaanbaatar
2.
Department of Mathematics and Statistics, Curtin University, Perth, Western Australia, WA 6845

Received: January 31, 2014

Revised: October 31, 2014

Published: December 2015

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Abstract

In this paper, we consider a fractional optimal control problem governed by system of linear differential equations, where its cost function is expressed as the ratio of convex and concave functions. The problem is a hard nonconvex optimal control problem and application of Pontriyagin's principle does not always guarantee finding a global optimal control. Even this type of problems in a finite dimensional space is known as NP hard. This optimal control problem can, in principle, be solved by Dinkhelbach algorithm [10]. However, it leads to solving a sequence of hard D.C programming problems in its finite dimensional analogy. To overcome this difficulty, we introduce a reachable set for the linear system. In this way, the problem is reduced to a quasiconvex maximization problem in a finite dimensional space. Based on a global optimality condition, we propose an algorithm for solving this fractional optimal control problem and we show that the algorithm generates a sequence of local optimal controls with improved cost values. The proposed algorithm is then applied to several test problems, where the global optimal cost value is obtained for each case.

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Mathematics Subject Classification: Primary: 49B; Secondary: 49K, 65K.

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