Gastao S. F. Frederico

A formulation of Noether's theorem for fractional problems of the calculus of variations

Abstract Fractional (or non-integer) differentiation is an important concept both from theoretical and applicational points of view. The study of problems of the calculus of variations with fractional derivatives is a rather recent subject, the main result being the fractional necessary optimality condition of Euler–Lagrange obtained in 2002. Here we use the notion of Euler–Lagrange fractional extremal to prove a Noether-type theorem. For that we propose a generalization of the classical concept of conservation law, introducing an appropriate fractional operator.

Fractional conservation laws in optimal control theory

Abstract 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 Noether’s theorem in the Riesz–Caputo sense

Abstract We prove a Noether’s theorem for fractional variational problems with Riesz–Caputo derivatives. Both Lagrangian and Hamiltonian formulations are obtained. Illustrative examples in the fractional context of the calculus of variations and optimal control are given.

Constants of motion for non-differentiable quantum variational problems

We extend the DuBois-Reymond necessary optimality condition and Noethers symmetry theorem to the scale relativity theory setting. Both Lagrangian and Hamiltonian versions of Noethers theorem are proved, covering problems of the calculus of variations with functionals defined on sets of non-differentiable functions, as well as more general non-differentiable problems of optimal control. As an application we obtain constants of motion for some linear and nonlinear variants of the Schrodinger equation.

NOETHER'S SYMMETRY THEOREM FOR VARIATIONAL AND OPTIMAL CONTROL PROBLEMS WITH TIME DELAY

We extend the DuBois-Reymond necessary optimality condition and Noethers symmetry theorem to the time delay variational setting. Both Lagrangian and Hamiltonian versions of Noethers theorem are proved, covering problems of the calculus of variations and optimal control with delays.

Noether's Theorem for Nonsmooth Extremals of Variational Problems with Time Delay

We obtain a non-smooth extension of Noether’s symmetry theorem for variational problems with delayed arguments. The result is proved to be valid in the class of Lipschitz functions, as long as the delayed Euler–Lagrange extremals are restricted to those that satisfy the DuBois–Reymond necessary optimality condition. The important case of delayed variational problems with higher order derivatives is considered as well.

Noether's theorem for fractional optimal control problems

Abstract We begin by reporting on some recent results of the authors (Frederico and Torres, 2006), concerning the use of the fractional Euler-Lagrange notion to prove a Noether-like theorem for the problems of the calculus of variations with fractional derivatives. We then obtain, following the Lagrange multiplier technique used in (Agrawal, 2004), a new version of Noethers theorem to fractional optimal control systems.

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. §Presented orally at the 7th IFAC Symposium on Nonlinear Control Systems (NOLCOS 2007), in Pretoria, South Africa, 22–24 August, 2007.

Fractional isoperimetric noether's theorem in the riemann-liouville sense

We prove Noether-type theorems for fractional isoperimetric variational problems with Riemann–Liouville derivatives. Both Lagrangian and Hamiltonian formulations are obtained. Illustrative examples, in the fractional context of the calculus of variations, are discussed.

Fractional Noether's theorem with classical and Riemann-Liouville derivatives

We prove a Noether type symmetry theorem to fractional problems of the calculus of variations with classical and Riemann-Liouville derivatives. As result, we obtain constants of motion (in the classical sense) that are valid along the mixed classical/fractional Euler-Lagrange extremals. Both Lagrangian and Hamiltonian versions of the Noether theorem are obtained. Finally, we extend our Noethers theorem to more general problems of optimal control with classical and Riemann-Liouville derivatives.