73 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
Fractional Euler-Lagrange differential equations via Caputo derivatives
We review some recent results of the fractional variational calculus.
Necessary optimality conditions of Euler-Lagrange type for functionals with a
Lagrangian containing left and right Caputo derivatives are given. Several
problems are considered: with fixed or free boundary conditions, and in
presence of integral constraints that also depend on Caputo derivatives.Comment: This is a preprint of a paper whose final and definite form will
appear as Chapter 9 of the book Fractional Dynamics and Control, D. Baleanu
et al. (eds.), Springer New York, 2012, DOI:10.1007/978-1-4614-0457-6_9, in
pres
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
Direct and Inverse Variational Problems on Time Scales: A Survey
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 attain a local minimum at a given point of the vector space.
Furthermore, we provide a necessary condition for a dynamic
integro-differential equation to be an Euler-Lagrange equation (Helmholtz's
problem of the calculus of variations on time scales). New and interesting
results for the discrete and quantum settings are obtained as particular cases.
Finally, we consider very general problems of the calculus of variations given
by the composition of a certain scalar function with delta and nabla integrals
of a vector valued field.Comment: This is a preprint of a paper whose final and definite form will be
published in the Springer Volume 'Modeling, Dynamics, Optimization and
Bioeconomics II', Edited by A. A. Pinto and D. Zilberman (Eds.), Springer
Proceedings in Mathematics & Statistics. Submitted 03/Sept/2014; Accepted,
after a revision, 19/Jan/201
Variational Problems Involving a Caputo-Type Fractional Derivative
We study calculus of variations problems, where the Lagrange function depends on the
Caputo-Katugampola fractional derivative. This type of fractional operator is a generalization
of the Caputo and the Caputo–Hadamard fractional derivatives, with dependence on a real
parameter ρ. We present sufficient and necessary conditions of first and second order to
determine the extremizers of a functional. The cases of integral and holomonic constraints are also considered
Fractional variational calculus of variable order
We study the fundamental problem of the calculus of variations with variable
order fractional operators. Fractional integrals are considered in the sense of
Riemann-Liouville while derivatives are of Caputo type.Comment: Submitted 26-Sept-2011; accepted 18-Oct-2011; withdrawn by the
authors 21-Dec-2011; resubmitted 27-Dec-2011; revised 20-March-2012; accepted
13-April-2012; to 'Advances in Harmonic Analysis and Operator Theory', The
Stefan Samko Anniversary Volume (Eds: A. Almeida, L. Castro, F.-O. Speck),
Operator Theory: Advances and Applications, Birkh\"auser Verlag
(http://www.springer.com/series/4850
Time-Fractional Optimal Control of Initial Value Problems on Time Scales
We investigate Optimal Control Problems (OCP) for fractional systems
involving fractional-time derivatives on time scales. The fractional-time
derivatives and integrals are considered, on time scales, in the
Riemann--Liouville sense. By using the Banach fixed point theorem, sufficient
conditions for existence and uniqueness of solution to initial value problems
described by fractional order differential equations on time scales are known.
Here we consider a fractional OCP with a performance index given as a
delta-integral function of both state and control variables, with time evolving
on an arbitrarily given time scale. Interpreting the Euler--Lagrange first
order optimality condition with an adjoint problem, defined by means of right
Riemann--Liouville fractional delta derivatives, we obtain an optimality system
for the considered fractional OCP. For that, we first prove new fractional
integration by parts formulas on time scales.Comment: This is a preprint of a paper accepted for publication as a book
chapter with Springer International Publishing AG. Submitted 23/Jan/2019;
revised 27-March-2019; accepted 12-April-2019. arXiv admin note: substantial
text overlap with arXiv:1508.0075
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