A global optimization approach to fractional optimal control

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.