Natália Martins

Calculus of variations on time scales with nabla derivatives

Abstract We prove a necessary optimality condition of the 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 variations on time scales.

Transversality conditions for infinite horizon variational problems on time scales

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.

Noether’s symmetry theorem for nabla problems of the calculus of variations

Abstract 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.

Generalizing the variational theory on time scales to include the delta indefinite integral

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 derivative, but also on a delta indefinite integral that depends on the unknown function. Such kinds of variational problems were considered by Euler himself and have been recently investigated in [J. Gregory, Generalizing variational theory to include the indefinite integral, higher derivatives, and a variety of means as cost variables, Methods Appl. Anal. 15 (4) (2008) 427-435]. Our results not only provide a generalization to previous results, but also give some other interesting optimality conditions as special cases.

Higher-order Hahn’s quantum variational calculus

Abstract We prove a necessary optimality condition of Euler–Lagrange type for quantum variational problems involving Hahn’s derivatives of higher-order.

Generalized transversality conditions for the Hahn quantum variational calculus

We prove optimality conditions for generalized quantum variational problems with a Lagrangian depending on the free end-points. Problems of calculus of variations of this type cannot be solved using the classical theory.

Symmetric differentiation on time scales

Abstract We define a symmetric derivative on an arbitrary nonempty closed subset of the real numbers and derive some of its properties. It is shown that real-valued functions defined on time scales that are neither delta nor nabla differentiable can be symmetric differentiable.

Variational problems of Herglotz typewith time delay: DuBois--Reymond condition and Noether's first theorem

We extend the DuBois--Reymond necessary optimality condition and Noethers first theorem to variational problems of Herglotz type with time delay. Our results provide, as corollaries, the DuBois--Reymond necessary optimality condition and the first Noether theorem for variational problems with time delay recently proved in [Numer. Algebra Control Optim. 2 (2012), no. 3, 619--630]. Our main result is also a generalization of the first Noether-type theorem for the generalized variational principle of Herglotz proved in [Topol. Methods Nonlinear Anal. 20 (2002), no. 2, 261--273].

Higher-Order Variational Problems of Herglotz Type

We obtain a generalized Euler–Lagrange differential equation and transversality optimality conditions for Herglotz-type higher-order variational problems. Illustrative examples of the new results are given.

A Symmetric Quantum Calculus

We introduce the α,β-symmetric difference derivative and the α,β-symmetric Norlund sum. The associated symmetric quantum calculus is developed, which can be seen as a generalization of the forward and backward h-calculus.