Three improved notions of conjugate points in optimal control

^** Email: jfrl{at}servidor.unam.mx Three widely quoted approaches to conjugacy in optimal control,due to Zeidan and Zezza, Loewen and Zheng and Zeidan, are studied.The three definitions of ‘generalized conjugate points’are improved in the sense that, either the range of applicabilityis enlarged or the conditions defining membership of the differentsets of points are simplified. No strong normality or non-singularityassumptions are imposed. Moreover, the new sets introduced characterizecompletely a second-order condition, a property which has remaineduncertain for the previous sets.     

Weak fractured solutions in the calculus of variations

Necessary conditions and sufficient conditions for a weak minimum of a variational problem over a class of functions which allow for a finite number of fractures (simple discontinuities) in the dependent variables are derived. Our results extend those of Razmadzé (Ref. 1).     

Contributions to the calculus of variations

In this paper, the theory of the calculus of variations for the simplest problem is reviewed. Simpler proofs are given for the classical conditions and a new necessary condition is presented which allows strong, necessary, and sufficient conditions to be stated for the first time.The major part of this research was conducted in the Department of Electrical Engineering, University of the West Indies, St. Augustine, Trinidad, West Indies. The author is indebted to Dr. B. Bhatt for interesting discussions.     

An existence theorem in the calculus of variations

In this paper, a nonparametric variational problem is considered in the setting of the theory of generalized curves. It is assumed that the integrand of the problem does not grow at infinity faster than the norm of the variable , for all values of the other variablest andx (which take their values in a compact product set). It is shown that a generalized curve exists such that the minimum of the functional over an appropriate set is achieved. This generalized curve does not in general have compact support.     

The Legendre condition of the fractional calculus of variations

Fractional operators play an important role in modelling nonlocal phenomena and problems involving coarse-grained and fractal spaces. The fractional calculus of variations with functionals depending on derivatives and/or integrals of noninteger order is a rather recent subject that is currently in fast development due to its applications in physics and other sciences. In the last decade, several approaches to fractional variational calculus were proposed by using different notions of fractional derivatives and integrals. Although the literature of the fractional calculus of variations is already vast, much remains to be done in obtaining necessary and sufficient conditions for the optimization of fractional variational functionals, existence and regularity of solutions. Regarding necessary optimality conditions, all works available in the literature concern the derivation of first-order fractional conditions of Euler–Lagrange type. In this work, we obtain a Legendre second-order necessary optimality condition for weak extremizers of a variational functional that depends on fractional derivatives.     

Stochastic extensions to necessary conditions in the theory of the calculus of variations

A stochastic version of the modified Young's generalized necessary conditions in the calculus of variations is given in this paper. It is based on an extension of Minkowski's theorem on the existence of a flat support for a convex figure, and it generalizes the necessary conditions of Weierstrass and Euler in the classical theory of the calculus of variations to a class of admissible curves which are expressible in terms of a finite number of random parameters. The integrals which we consider here are in the general Denjoy sense, except those with respect to the random parameters, which exist in the Lebesgue sense defined on a probability space. The importance of our stochastic analysis lies in the completion that a minimum not attained in the classical sense may be, and frequently is, attained in the stochastic case.This research was supported in part by the National Science Foundation under Grants Nos. GK-1834X and GK-31229     

On the existence in a problem of the calculus of variations without convexity assumptions

Summary We obtain new sufficient conditions for the existence in a problem of the calculus of variations without convexity assumptions.     

Natural boundary conditions in the calculus of variations

We prove necessary optimality conditions for problems of the calculus of variations on time scales with a Lagrangian depending on the free end‐point. Copyright © 2010 John Wiley & Sons, Ltd.     

Coupled intervals in the discrete calculus of variations: necessity and sufficiency

In this work we study nonnegativity and positivity of a discrete quadratic functional with separately varying endpoints. We introduce a notion of an interval coupled with 0, and hence, extend the notion of conjugate interval to 0 from the case of fixed to variable endpoint(s). We show that the nonnegativity of the discrete quadratic functional is equivalent to each of the following conditions: The nonexistence of intervals coupled with 0, the existence of a solution to Riccati matrix equation and its boundary conditions. Natural strengthening of each of these conditions yields a characterization of the positivity of the discrete quadratic functional. Since the quadratic functional under consideration could be a second variation of a discrete calculus of variations problem with varying endpoints, we apply our results to obtain necessary and sufficient optimality conditions for such problems. This paper generalizes our recent work in [R. Hilscher, V. Zeidan, Comput. Math. Appl., to appear], where the right endpoint is fixed.     

Tangentially positive isometric actions and conjugate points

Let be a complete Riemannian manifold with no conjugate points and a principal -bundle, where is a Lie group acting by isometries and the smooth quotient with the Riemannian submersion metric.

We obtain a characterization of conjugate point-free quotients in terms of symplectic reduction and a canonical pseudo-Riemannian metric on the tangent bundle , from which we then derive necessary conditions, involving and , for the quotient metric to be conjugate point-free, particularly for a reducible Riemannian manifold.

Let , with the Lie Algebra of , be the moment map of the tangential -action on and let be the canonical pseudo-Riemannian metric on defined by the symplectic form and the map , . First we prove a theorem, stating that if is not positive definite on the action vector fields for the tangential action along then acquires conjugate points. (We proved the converse result in 2005.) Then, we characterize self-parallel vector fields on in terms of the positivity of the -length of their tangential lifts along certain canonical subsets of . We use this to derive some necessary conditions, on and , for actions to be tangentially positive on relevant subsets of , which we then apply to isometric actions on complete conjugate point-free reducible Riemannian manifolds when one of the irreducible factors satisfies certain curvature conditions.

     

Optimality conditions for discrete calculus of variations problems

Nonlinear discrete calculus of variations problems with variable endpoints and with equality type constraints on trajectories are considered. We derive new nontrivial first- and second-order necessary optimality conditions.     

Helmholtz's inverse problem of the discrete calculus of variations

We derive the discrete version of the classical Helmholtz's condition. Precisely, we state a theorem characterizing second-order finite difference equations admitting a Lagrangian formulation. Moreover, in the affirmative case, we provide the class of all possible Lagrangian formulations.     

On conjugate points and the Leitmann equivalent problem approach

This article extends the Leitmann equivalence method to a class of problems featuring conjugate points. The class is characterised by the requirement that the set of indifference points of a given problem forms a finite stratification.     

Simple proof of the Lagrange multiplier rule for the fixed endpoint problem in the calculus of variations

A new simple proof of the Lagrange multiplier rule is presented in this paper. The approach used involves simple analytical techniques that are very easy to follow and does not involve theorems on imbeddability in a one-parameter family of curves or matrix-rank analysis as do most of the existing techniques. The proof is here developed for the fixed-endpoint problem in a three-dimensional space.     

Fields of extremals and sufficient conditions for the simplest problem of the calculus of variations

In this paper we present an elementary proof of a classical sufficient condition for free problems in the calculus of variations using an extension of the direct sufficiency method, originating in the work of G. Leitmann (Int. J. NonLin. Mech. 2, 55–59 (1967)), found in D.A. Carlson (J. Optim. Theory Appl. 114, 345–362 (2002)).     

Parameter variation for the solution of two-point boundary-value problems and applications in the calculus of variations

In the parameter variation method, a scalar parameterk, k[0, 1], is introduced into the differential equations. The parameterk is inserted in such a way that, whenk=0, the solution of the boundary-value problem is known or readily calculated and, whenk=1, the problem is identical with the original problem. Thus, bydeforming the solution step-by-step throughk-space fromk=0 tok=1, the original problem may be solved. These solutions then provide good starting values for any convergent, iterative scheme such as the Newton-Raphson method.The method is applied to the solution of problems with various types of boundary-value specifications and is further extended to take account of situations arising in the solution of problems from variational calculus (e.g., total elapsed time not specified, optimum control not a simple function of the variables).     

A multisymplectic framework for classical field theory and the calculus of variations II: space + time decomposition

In a previous paper I laid the foundations of a covariant Hamiltonian framework for the calculus of variations in general. The purpose of the present work is to demonstrate, in the context of classical field theory, how this covariant Hamiltonian formalism may be space + time decomposed. It turns out that the resulting “instantaneous” Hamiltonian formalism is an infinite- dimensional version of Ostrogradski 's theory and leads to the standard symplectic formulation of the initial value problem. The salient features of the analysis are: (i) the instantaneous Hamiltonian formalism does not depend upon the choice of Lepagean equivalent; (ii) the space + time decomposition can be performed either before or after the covariant Legendre transformation has been carried out, with equivalent results; (iii) the instantaneous Hamiltonian can be recovered in natural way from the multisymplectic structure inherent in the theory; and (iv) the space + time split symplectic structure lives on the space of Cauchy data for the evolution equations, as opposed to the space of solutions thereof.     

Entropy-expansiveness of geodesic flows on closed manifolds without conjugate points

In this article, we consider the entropy-expansiveness of geodesic flows on closed Riemannian manifolds without conjugate points. We prove that, if the manifold has no focal points, or if the manifold is bounded asymptote, then the geodesic flow is entropy-expansive. Moreover, for the compact oriented surfaces without conjugate points, we prove that the geodesic flows are entropy-expansive. We also give an estimation of distance between two positively asymptotic geodesics of an uniform visibility manifold.