A Sturm-Liouville approach applicable to different notions of conjugacy

K e y w o r d s C a l c u l u s of variations, Generalized conjugate points, Nonsingular extremals, SturmLiouville boundary-value problems. 1. I N T R O D U C T I O N Suppose we are g iven an in te rva l T := [t0,Q] in R , two poin ts ~0, ~1 in R n, and a func t ion L m a p p i n g T x R n x R ~ to R . Le t X be the space of p iecewise -C 1 func t ions m a p p i n g T to R ~, let X~ be the set of func t ions x E X sat isfying x(to) = ~0 and x ( t l ) = ~1, and consider t he func t iona l I : X --~ R g iven by I(x) : = ftt~o L(t, x( t ) , 2 ( t ) ) d r . T h e p r o b l e m we shall deal w i t h ( the s imple f ixed-endpo in t p rob l em in t he calculus of var ia t ions) , which we label (P), is t h a t of m in imiz ing I over X~. E l e m e n t s of X are cal led trajectories, and x E X~ solves (P) if I(x) < I(y) for all y E X~. For any t r a j e c t o r y x we use t h e n o t a t i o n (~(t)) to r ep resen t (t, x ( t ) , 2 ( t ) ) , and we shall assume t h r o u g h o u t t h a t L is cont inuous , and C 2 w i t h respec t to x and 5. For all x E X consider t he second variation of I along x given by I"(x; y):= ftt~ 2 ~ ( ~ ( t ) ) d t (y E X) where , for all (t, y, 9) E 0893-9659/04/$ see front matter (~) 2004 Elsevier Ltd. All rights reserved. doi: 10.1016/S0893-9659 (04) 00055-2 Typeset by .4AAS-TEX 468 R. BERLANGA AND J. F. ROSENBLUETH