Related papers: Necessary condition for an Euler-Lagrange equation…
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…
The fundamental problem of the calculus of variations on time scales concerns the minimization of a delta-integral over all trajectories satisfying given boundary conditions. In this paper we prove the second Euler-Lagrange necessary…
We deal with direct and inverse problems of the calculus of variations on arbitrary time scales. Firstly, using the Euler-Lagrange equation and the strengthened Legendre condition, we give a general form for a variational functional to…
We obtain Euler-Lagrange and transversality optimality conditions for higher-order infinite horizon variational problems on a time scale. The new necessary optimality conditions improve the classical results both in the continuous and…
We prove the Euler-Lagrange delta-differential equations for problems of the calculus of variations on arbitrary time scales with delta-integral functionals depending on higher-order delta derivatives.
We introduce a discrete-time fractional calculus of variations on the time scale $h\mathbb{Z}$, $h > 0$. First and second order necessary optimality conditions are established. Examples illustrating the use of the new Euler-Lagrange and…
We prove a necessary optimality condition of Euler-Lagrange type for variational problems on time scales involving nabla derivatives of higher-order. The proof is done using a new and more general fundamental lemma of the calculus of…
We prove a version of the Euler-Lagrange equations for certain problems of the calculus of variations on time scales with higher-order delta derivatives.
We establish Euler-Lagrange equations for a problem of Calculus of variations where the unknown variable contains a term of delay on a segment.
This paper provides necessary and sufficient conditions of optimality for variational problems that deal with a fractional derivative with respect to another function. Fractional Euler--Lagrange equations are established for the fundamental…
We study more general variational problems on time scales. Previous results are generalized by proving necessary optimality conditions for (i) variational problems involving delta derivatives of more than the first order, and (ii) problems…
We prove Euler-Lagrange type equations and transversality conditions for generalized infinite horizon problems of the calculus of variations on time scales. Here the Lagrangian depends on the independent variable, an unknown function and…
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 use a modification of the parameterization method to study invariant manifolds for difference equations. We establish existence, regularity, smooth dependence on parameters and study several singular limits, even if the difference…
We study problems of the calculus of variations and optimal control within the framework of time scales. Specifically, we obtain Euler-Lagrange type equations for both Lagrangians depending on higher order delta derivatives and…
We introduce a discrete-time fractional calculus of variations on the time scales $\mathbb{Z}$ and $(h\mathbb{Z})_a$. First and second order necessary optimality conditions are established. Some numerical examples illustrating the use of…
We prove necessary optimality conditions of Euler-Lagrange type for generalized problems of the calculus of variations on time scales with a Lagrangian depending not only on the independent variable, an unknown function and its delta…
We consider problems of the calculus of variations on unbounded time scales. We prove the validity of the Euler-Lagrange equation on time scales for infinite horizon problems, and a new transversality condition.
We prove a necessary optimality condition of Euler-Lagrange type for quantum variational problems involving Hahn's derivatives of higher-order.
In calculus of variations on general time scales, an integral Euler-Lagrange equation is usually derived in order to characterize the critical points of non shifted Lagrangian functionals, see e.g. [R.A.C. Ferreira and co-authors,…