5,939 research outputs found
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
An induction theorem and nonlinear regularity models
A general nonlinear regularity model for a set-valued mapping , where and are metric spaces, is considered
using special iteration procedures, going back to Banach, Schauder, Lusternik
and Graves. Namely, we revise the induction theorem from Khanh, J. Math. Anal.
Appl., 118 (1986) and employ it to obtain basic estimates for studying
regularity/openness properties. We also show that it can serve as a
substitution of the Ekeland variational principle when establishing other
regularity criteria. Then, we apply the induction theorem and the mentioned
estimates to establish criteria for both global and local versions of
regularity/openness properties for our model and demonstrate how the
definitions and criteria translate into the conventional setting of a
set-valued mapping .Comment: 28 page
A transverse Hamiltonian variational technique for open quantum stochastic systems and its application to coherent quantum control
This paper is concerned with variational methods for nonlinear open quantum
systems with Markovian dynamics governed by Hudson-Parthasarathy quantum
stochastic differential equations. The latter are driven by quantum Wiener
processes of the external boson fields and are specified by the system
Hamiltonian and system-field coupling operators. We consider the system
response to perturbations of these energy operators and introduce a transverse
Hamiltonian which encodes the propagation of the perturbations through the
unitary system-field evolution. This provides a tool for the infinitesimal
perturbation analysis and development of optimality conditions for coherent
quantum control problems. We apply the transverse Hamiltonian variational
technique to a mean square optimal coherent quantum filtering problem for a
measurement-free cascade connection of quantum systems.Comment: 12 pages, 1 figure. A brief version of this paper will appear in the
proceedings of the IEEE Multi-Conference on Systems and Control, 21-23
September 2015, Sydney, Australi
Backward variational approach on time scales with an action depending on the free endpoints
We establish necessary optimality conditions for variational problems with an
action depending on the free endpoints. New transversality conditions are also
obtained. The results are formulated and proved using the recent and general
theory of time scales via the backward nabla differential operator.Comment: Submitted 17-Oct-2010; revised 18-Dec-2010; accepted 4-Jan-2011; for
publication in Zeitschrift fuer Naturforschung
Scalarization and sensitivity analysis in Vector Optimization. The linear case.
In this paper we consider a vector optimization problem; we present some scalarization techniques for finding all the vector optimal points of this problem and we discuss the relationships between these methods. Moreover, in the linear case, the study of dual variables is carried on by means of sensitivity analysis and also by a parametric approach. We also give an interpretation of the dual variables as marginal rates of substitution of an objective function with respect to another one, and of an objective function with respect to a constraint.Vector Optimization, Image Space, Separation, Scalarization, Shadow Prices
Guidance, flight mechanics and trajectory optimization. Volume 10 - Dynamic programming
Dynamic programming and multistage decision processes in guidance, flight mechanics, and trajectory optimizatio
Constrained Nonsmooth Problems of the Calculus of Variations
The paper is devoted to an analysis of optimality conditions for nonsmooth
multidimensional problems of the calculus of variations with various types of
constraints, such as additional constraints at the boundary and isoperimetric
constraints. To derive optimality conditions, we study generalised concepts of
differentiability of nonsmooth functions called codifferentiability and
quasidifferentiability. Under some natural and easily verifiable assumptions we
prove that a nonsmooth integral functional defined on the Sobolev space is
continuously codifferentiable and compute its codifferential and
quasidifferential. Then we apply general optimality conditions for nonsmooth
optimisation problems in Banach spaces to obtain optimality conditions for
nonsmooth problems of the calculus of variations. Through a series of simple
examples we demonstrate that our optimality conditions are sometimes better
than existing ones in terms of various subdifferentials, in the sense that our
optimality conditions can detect the non-optimality of a given point, when
subdifferential-based optimality conditions fail to disqualify this point as
non-optimal.Comment: A number of small mistakes and typos was corrected in the second
version of the paper. Moreover, the paper was significantly shortened.
Extended and improved versions of the deleted sections on nonsmooth Noether
equations and nonsmooth variational problems with nonholonomic constraints
will be published in separate submission
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