Nonlinear Dynamics

, Volume 69, Issue 3, pp 1263–1284 | Cite as

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

  • Abolhassan Razminia
  • Vahid Johari Majd
  • Ahmad Feyz Dizaji
Original Paper


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.


Fractional calculus Fractional optimization Calculus of variations Extended Euler–Lagrange equation 


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Copyright information

© Springer Science+Business Media B.V. 2012

Authors and Affiliations

  • Abolhassan Razminia
    • 1
  • Vahid Johari Majd
    • 1
  • Ahmad Feyz Dizaji
    • 2
  1. 1.Intelligent Control Systems Laboratory, School of Electrical and Computer EngineeringTarbiat Modares UniversityTehranIran
  2. 2.Department of Engineering Science, School of EngineeringUniversity of TehranTehranIran

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