An extended formulation of calculus of variations for incommensurate fractional derivatives with fractional performance index
| Autor: | Ahmad Feyz Dizaji, Abolhassan Razminia, Vahid Johari Majd |
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| Rok vydání: | 2012 |
| Předmět: |
Applied Mathematics
Mechanical Engineering Multivariable calculus Mathematical analysis Aerospace Engineering Ocean Engineering Fundamental lemma of calculus of variations Time-scale calculus Fractional calculus Fractional programming Control and Systems Engineering Legendre–Clebsch condition Applied mathematics Integration by parts Calculus of variations Electrical and Electronic Engineering Mathematics |
| Zdroj: | Nonlinear Dynamics. 69:1263-1284 |
| ISSN: | 1573-269X 0924-090X |
| DOI: | 10.1007/s11071-012-0345-y |
| Popis: | 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. |
| Databáze: | OpenAIRE |
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