2,286 research outputs found
Optimality conditions for the calculus of variations with higher-order delta derivatives
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.Comment: Submitted 26/Jul/2009; Revised 04/Aug/2010; Accepted 09/Aug/2010; for
publication in "Applied Mathematics Letters
Discrete-Time Fractional Variational Problems
We introduce a discrete-time fractional calculus of variations on the time
scale , . 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 show that solutions
to the considered fractional problems become the classical discrete-time
solutions when the fractional order of the discrete-derivatives are integer
values, and that they converge to the fractional continuous-time solutions when
tends to zero. Our Legendre type condition is useful to eliminate false
candidates identified via the Euler-Lagrange fractional equation.Comment: Submitted 24/Nov/2009; Revised 16/Mar/2010; Accepted 3/May/2010; for
publication in Signal Processing
The contingent epiderivative and the calculus of variations on time scales
The calculus of variations on time scales is considered. We propose a new
approach to the subject that consists in applying a differentiation tool called
the contingent epiderivative. It is shown that the contingent epiderivative
applied to the calculus of variations on time scales is very useful: it allows
to unify the delta and nabla approaches previously considered in the
literature. Generalized versions of the Euler-Lagrange necessary optimality
conditions are obtained, both for the basic problem of the calculus of
variations and isoperimetric problems. As particular cases one gets the recent
delta and nabla results.Comment: Submitted 06/March/2010; revised 12/May/2010; accepted 03/July/2010;
for publication in "Optimization---A Journal of Mathematical Programming and
Operations Research
A unified approach to the calculus of variations on time scales
In this work we propose a new and more general approach to the calculus of
variations on time scales that allows to obtain, as particular cases, both
delta and nabla results. More precisely, we pose the problem of minimizing or
maximizing the composition of delta and nabla integrals with Lagrangians that
involve directional derivatives. Unified Euler-Lagrange necessary optimality
conditions, as well as sufficient conditions under appropriate convexity
assumptions, are proved. We illustrate presented results with simple examples.Comment: 7 pages, presented at the 2010 Chinese Control and Decision
Conference, Xuzhou, China, May 26-28, 201
The Variational Calculus on Time Scales
The discrete, the quantum, and the continuous calculus of variations, have
been recently unified and extended by using the theory of time scales. Such
unification and extension is, however, not unique, and two approaches are
followed in the literature: one dealing with minimization of delta integrals;
the other dealing with minimization of nabla integrals. Here we review a more
general approach to the calculus of variations on time scales that allows to
obtain both delta and nabla results as particular cases.Comment: 15 pages; Published in: Int. J. Simul. Multidisci. Des. Optim. 4
(2010), 11--2
Higher-Order Calculus of Variations on Time Scales
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.Comment: Corrected minor typo
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