18 research outputs found
Conservation laws for invariant functionals containing compositions
The study of problems of the calculus of variations with compositions is a
quite recent subject with origin in dynamical systems governed by chaotic maps.
Available results are reduced to a generalized Euler-Lagrange equation that
contains a new term involving inverse images of the minimizing trajectories. In
this work we prove a generalization of the necessary optimality condition of
DuBois-Reymond for variational problems with compositions. With the help of the
new obtained condition, a Noether-type theorem is proved. An application of our
main result is given to a problem appearing in the chaotic setting when one
consider maps that are ergodic.Comment: Accepted for an oral presentation at the 7th IFAC Symposium on
Nonlinear Control Systems (NOLCOS 2007), to be held in Pretoria, South
Africa, 22-24 August, 200
Fractional conservation laws in optimal control theory
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.Comment: The original publication is available at http://www.springerlink.com
Nonlinear Dynamic