Fractional conservation laws in optimal control theory

By Gastão S. F. Frederico, Delfim F. M. Torres. 2007

272 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics, Modeling and Simulation, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/0faba2d8b7149d800443e2fa18e07c333aa3cb08

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In Applied Mathematics

In Modeling and Simulation

In Statistical and Nonlinear Physics

Citations in the corpus

Fractional actionlike variational problems (2008, methodology): “Different methods were used to obtain the fractional Euler-Lagrange equations and the corresponding Hamiltonian canonical equations [1, 3, 4, 5 , 6, 14, 22, 23, 28, 29, 33].”
Conservation laws for time-fractional subdiffusion and diffusion-wave equations (2014, background): “The nonlinear self-adjointness of these equations are established, and corresponding formal fractional Lagrangians have been found.”
Fractional Optimal Control Problems with Several State and Control Variables (2010, background): “Recently symmetries and fractional conservation laws for FOCPs with Riemann–Liouville fractional derivatives (RLFDs) were studied by Frederico and Torres (2008a) and FOCPs in the sense of Caputo were addressed by Frederico and Torres (2008b).”
Fractional calculus of variations for a combined Caputo derivative (2011, background): “, [2, 6, 11, 20, 23, 24, 26, 36, 42] and many others in this journal “Fract.”
Necessary and sufficient conditions for the fractional calculus of variations with Caputo derivatives (2010, methodology): “ention [7], were necessaryand suﬃcient conditions of optimality for functionals containing fractional integrals and fractional derivatives are presented. For results on fractional optimal control see [2,15]. In the present paper we work with the Caputo fractional derivative. For problems of calculus”
Existence of a weak solution for fractional Euler-Lagrange equations (2012, background): “Consequently, numerous studies have been devoted to numerical schemes for (ELα) (see [3, 9]) and to the statement of fractional Noether’s theorems providing constants of motion (see [8, 14, 15]).”
Fractional Optimal Control in the Sense of Caputo and the Fractional Noether's Theorem (2007, background): “18 44 Mathematics Subject Classification: 49K05, 26A33 Keywords: Fractional Optimal Control, Caputo Derivatives, RiemannLiouville Derivatives, Noether’s Theorem”
A fractional calculus of variations for multiple integrals with application to vibrating string (2010, background): “An interesting open question consists to generalize the fractional Noether-type theorems obtained in [7, 21, 22] to the case of several independent variables.”
NECESSARY OPTIMALITY CONDITIONS FOR FRACTIONAL DIFFERENCE PROBLEMS OF THE CALCULUS OF VARIATIONS (2010, background): “The study of fractional problems of the Calculus of Variations and respective Euler-Lagrange equations is a fairly recent issue – see [1, 2, 5, 7, 8, 11, 13, 15, 16, 21] and references therein – and include only the continuous case.”
FRACTIONAL h-DIFFERENCE EQUATIONS ARISING FROM THE CALCULUS OF VARIATIONS (2011, methodology): “The most common used fractional derivative is the Riemann– Liouville [1, 13, 16, 27].”
Fractional order optimal control problems with free terminal time (2013, unknown): “, [1, 3, 4, 8, 9, 23, 24]).”