Existence of variationally defined curves with higher order elliptic Lagrangians. (March 2015)

• Record Type: Journal Article
• Authors: Schrader, Philip
• Is Part Of: Nonlinear analysis. Volume 115(2015)
• Page Start: 1
• Page End: 11
• Publication Date: 2015-03
• Subjects: 58E25 -- 49J05; Palais–Smale condition -- Calculus of variations -- Interpolation on manifolds -- Geometric optimal control -- Conditional extremals -- Riemannian cubics in tension; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2014.11.016
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: We present a method for proving the existence of solutions to a class of one dimensional variational problems, namely those where the Lagrangian is strongly elliptic. The method is demonstrated by some examples of optimal interpolation problems which are motivated by applications to the mechanics and control of rigid bodies. In each case the key step is to show that the variational problem satisfies the Palais–Smale condition. We do so in a general setting, showing that the number of initial conditions required depends on the higher order energy bounding properties of the Lagrangian.