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We present an extension of some results of higher order calculus of variations and optimal control to generalized functions. The framework is the category of generalized smooth functions, which includes Schwartz distributions, while sharing…
We study in optimal control the important relation between invariance of the problem under a family of transformations, and the existence of preserved quantities along the Pontryagin extremals. Several extensions of Noether theorem are…
A simple proof of Noether's first theorem involves the promotion of a constant symmetry parameter $\epsilon$ to an arbitrary function of time, the Noether charge $Q$ is then the coefficient of $\dot\epsilon$ in the variation of the action.…
The study of fuzzy fractional variational problems in terms of a fractional Liouville-Caputo derivative is introduced. Necessary optimality conditions for problems of the fuzzy fractional calculus of variations with free end-points are…
Fractional Pontryagin's systems emerge in the study of a class of fractional optimal control problems but they are not resolvable in most cases. In this paper, we suggest a numerical approach for these fractional systems. Precisely, we…
In this PhD thesis we introduce a generalized fractional calculus of variations. We consider variational problems containing generalized fractional integrals and derivatives, and study them using standard (indirect) and direct methods. In…
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…
We generalize the fractional Caputo derivative to the fractional derivative ${^CD^{\alpha,\beta}_{\gamma}}$, which is a convex combination of the left Caputo fractional derivative of order $\alpha$ and the right Caputo fractional derivative…
We prove necessary optimality conditions, in the class of continuous functions, for variational problems defined with Jumarie's modified Riemann-Liouville derivative. The fractional basic problem of the calculus of variations with free…
The study of fractional variational problems in terms of a combined fractional Caputo derivative is introduced. Necessary optimality conditions of Euler-Lagrange type for the basic, isoperimetric, and Lagrange variational problems are…
Consideration of the Noether variational problem for any theory whose action is invariant under global and/or local gauge transformations leads to three distinct theorems. These include the familiar Noether theorem, but also two equally…
We prove optimality conditions for different variational functionals containing left and right Caputo fractional derivatives. A sufficient condition of minimization under an appropriate convexity assumption is given. An Euler-Lagrange…
In order to solve fractional variational problems, there exist two theorems of necessary conditions: an Euler-Lagrange equation which involves Caputo and Riemann-Liouville fractional derivatives, and other Euler-Lagrange equation that…
We prove a Noether-type symmetry theorem and a DuBois-Reymond necessary optimality condition for nabla problems of the calculus of variations on time scales.
In this work we prove a weak Noether type theorem for a class of variational problems which include broken extremals. We then use this result to prove discrete Noether type conservation laws for certain classes of finite element…
We generalize the fractional Caputo derivative to the fractional derivative ${{^CD}^{\alpha,\beta}_{\gamma}}$, which is a convex combination of the left Caputo fractional derivative of order $\alpha$ and the right Caputo fractional…
We introduce a discrete-time fractional calculus of variations. First and second order necessary optimality conditions are established. Examples illustrating the use of the new Euler-Lagrange and Legendre type conditions are given. They…
We give a proper fractional extension of the classical calculus of variations. Necessary optimality conditions of Euler-Lagrange type for variational problems containing both classical and fractional derivatives are proved. The fundamental…
This chapter presents some numerical methods to solve problems in the fractional calculus of variations and fractional optimal control. Although there are plenty of methods available in the literature, we concentrate mainly on approximating…
The Noether theorem connecting symmetries and conservation laws can be applied directly in a Hamiltonian framework without using any intermediate Lagrangian formulation. This requires a careful discussion about the invariance of the…