19,740 research outputs found
Delta-Nabla Optimal Control Problems
We present a unified treatment to control problems on an arbitrary time scale
by introducing the study of forward-backward optimal control problems.
Necessary optimality conditions for delta-nabla isoperimetric problems are
proved, and previous results in the literature obtained as particular cases. As
an application of the results of the paper we give necessary and sufficient
Pareto optimality conditions for delta-nabla bi-objective optimal control
problems.Comment: Preprint version of an article submitted 28-Nov-2009; revised
02-Jul-2010; accepted 20-Jul-2010; for publication in Journal of Vibration
and Contro
The Second Euler-Lagrange Equation of Variational Calculus on Time Scales
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
optimality condition for optimal trajectories of variational problems on time
scales. As an example of application of the main result, we give an alternative
and simpler proof to the Noether theorem on time scales recently obtained in
[J. Math. Anal. Appl. 342 (2008), no. 2, 1220-1226].Comment: This work was partially presented at the Workshop in Control,
Nonsmooth Analysis and Optimization, celebrating Francis Clarke's and Richard
Vinter's 60th birthday, Porto, May 4-8, 2009. Submitted 26-May-2009; Revised
12-Jan-2010; Accepted 29-March-2010 in revised form; for publication in the
European Journal of Contro
Leitmann's direct method for fractional optimization problems
Based on a method introduced by Leitmann [Internat. J. Non-Linear Mech. {\bf
2} (1967), 55--59], we exhibit exact solutions for some fractional optimization
problems of the calculus of variations and optimal control.Comment: Submitted June 16, 2009 and accepted March 15, 2010 for publication
in Applied Mathematics and Computation
Noether's symmetry theorem for nabla problems of the calculus of variations
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.Comment: Submitted 20/Oct/2009; Revised 27/Jan/2010; Accepted 28/July/2010;
for publication in Applied Mathematics Letter
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.Comment: Submitted 6-October-2009; Accepted 19-March-2010 in revised form; for
publication in "Optimization Letters"
The Hahn Quantum Variational Calculus
We introduce the Hahn quantum variational calculus. Necessary and sufficient
optimality conditions for the basic, isoperimetric, and Hahn quantum Lagrange
problems, are studied. We also show the validity of Leitmann's direct method
for the Hahn quantum variational calculus, and give explicit solutions to some
concrete problems. To illustrate the results, we provide several examples and
discuss a quantum version of the well known Ramsey model of economics.Comment: Submitted: 3/March/2010; 4th revision: 9/June/2010; accepted:
18/June/2010; for publication in Journal of Optimization Theory and
Application
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
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
Fractional Calculus of Variations for Double Integrals
We consider fractional isoperimetric problems of calculus of variations with
double integrals via the recent modified Riemann-Liouville approach. A
necessary optimality condition of Euler-Lagrange type, in the form of a
multitime fractional PDE, is proved, as well as a sufficient condition and
fractional natural boundary conditions.Comment: Submitted 07-Sept-2010; revised 25-Nov-2010; accepted 07-Feb-2011;
for publication in Balkan Journal of Geometers and Its Applications (BJGA
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