A global optimization approach to fractional optimal control
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This is a pre-copy-editing, author-produced PDF of an article accepted for publication in Journal of Industrial and Management Optimization (JIMO) following peer review. The definitive publisher-authenticated version "Rentsen, E. and Zhou, J. and Teo, K.L. 2016. A global optimization approach to fractional optimal control. Journal of Industrial and Management Optimization (JIMO). 12 (1): pp. 73-82." is available online at: http://doi.org/10.3934/jimo.2016.12.73
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 . 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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