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An extended formulation of calculus of variations for incommensurate fractional derivatives with fractional performance index

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Abstract

In this paper, we consider the main problem of variational calculus when the derivatives are Riemann–Liouville-type fractional with incommensurate orders in general. As the most general form of the performance index, we consider a fractional integral form for the functional that is to be extremized. In the light of fractional calculus and fractional integration by parts, we express a generalized problem of the calculus of variations, in which the classical problem is a special case. Considering five cases of the problem (fixed, free, and dependent final time and states), we derive a necessary condition which is an extended version of the classical Euler–Lagrange equation. As another important result, we derive the necessary conditions for an optimization problem with piecewise smooth extremals where the fractional derivatives are not necessarily continuous. The latter result is valid only for the integer order for performance index. Finally, we provide some examples to clarify the effectiveness of the proposed theorems.

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Authors and Affiliations

  1. Intelligent Control Systems Laboratory, School of Electrical and Computer Engineering, Tarbiat Modares University, P.O. Box 14115-194, Tehran, Iran

    Abolhassan Razminia & Vahid Johari Majd

  2. Department of Engineering Science, School of Engineering, University of Tehran, Tehran, Iran

    Ahmad Feyz Dizaji

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  1. Abolhassan Razminia
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  2. Vahid Johari Majd
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  3. Ahmad Feyz Dizaji
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Correspondence to Vahid Johari Majd.

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Razminia, A., Majd, V.J. & Feyz Dizaji, A. An extended formulation of calculus of variations for incommensurate fractional derivatives with fractional performance index. Nonlinear Dyn 69, 1263–1284 (2012). https://doi.org/10.1007/s11071-012-0345-y

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