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A Central Difference Numerical Scheme for Fractional Optimal Control Problems

Department of Mathematics and Computer Sciences, Faculty of Arts and Sciences, Çankaya University, 06530, Ankara, Turkey, dumitru{at}cankaya.edu.tr

Ozlem Defterli

Department of Mathematics and Computer Sciences, Faculty of Arts and Sciences, Çankaya University, 06530, Ankara, Turkey

Om P. Agrawal

Department of Mechanical Engineering, Southern Illinois University, Carbondale, IL 62901, USA

This paper presents a modified numerical scheme for a class of fractional optimal control problems where a fractional derivative (FD) is defined in the Riemann—Liouville sense. In this scheme, the entire time domain is divided into several sub-domains, and a FD at a time node point is approximated using a modified Grünwald—Letnikov approach. For the first-order derivative, the proposed modified Grünwald— Letnikov definition leads to a central difference scheme. When the approximations are substituted into the fractional optimal control equations, it leads to a set of algebraic equations which are solved using a direct numerical technique. Two examples, one time-invariant and the other time-variant, are considered to study the performance of the numerical scheme. Results show that 1) as the order of the derivative approaches an integer value, these formulations lead to solutions for the integer-order system, and 2) as the sizes of the sub-domains are reduced, the solutions converge. It is hoped that the present scheme would lead to stable numerical methods for fractional differential equations and optimal control problems.

Key Words: Fractional calculus • Riemann-Liouville fractional derivatives • modified Grünwald-Letnikov approximation • fractional optimal control.

This version was published on April 1, 2009

Journal of Vibration and Control, Vol. 15, No. 4, 583-597 (2009)
DOI: 10.1177/1077546308088565


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