Nonlinear Dynamics

, Volume 53, Issue 3, pp 215–222 | Cite as

Fractional conservation laws in optimal control theory

  • Gastão S. F. Frederico
  • Delfim F. M. TorresEmail author
Original Paper


Using the recent formulation of Noether’s theorem for the problems of the calculus of variations with fractional derivatives, the Lagrange multiplier technique, and the fractional Euler–Lagrange equations, we prove a Noether-like theorem to the more general context of the fractional optimal control. As a corollary, it follows that in the fractional case the autonomous Hamiltonian does not define anymore a conservation law. Instead, it is proved that the fractional conservation law adds to the Hamiltonian a new term which depends on the fractional-order of differentiation, the generalized momentum and the fractional derivative of the state variable.


Fractional derivatives Optimal control Noether’s theorem Conservation laws Symmetry 


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Copyright information

© Springer Science+Business Media B.V. 2007

Authors and Affiliations

  • Gastão S. F. Frederico
    • 1
  • Delfim F. M. Torres
    • 2
    Email author
  1. 1.Department of Science and TechnologyUniversity of Cape VerdeSantiagoCape Verde
  2. 2.Department of MathematicsUniversity of AveiroAveiroPortugal

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