An approximation theory for focal points and focal intervals

Abstract

The theory of focal points and conjugate points is an important part of the study of problems in the calculus of variations and control theory. In previous works we gave a theory of focal points and of focal intervals for an elliptic form J ( x ) J(x) on a Hilbert space A \mathcal {A} . These results were based upon inequalities dealing with the indices s ( σ ) s(\sigma ) and n ( σ ) n(\sigma ) of the elliptic form J ( x ; σ ) J(x;\sigma ) defined on the closed subspace A ( σ ) \mathcal {A}(\sigma ) of A \mathcal {A} , where σ \sigma belongs to the metric space ( Σ , ρ ) (\Sigma ,\rho ) . In this paper we give an approximation theory for focal point and focal interval problems. Our results are based upon inequalities dealing with the indices s ( μ ) s(\mu ) and u ( μ ) u(\mu ) , where μ \mu belongs to the metric space ( M , d ) , M = E 1 × Σ (M,d),M = {E^1} \times \Sigma . For the usual focal point problems we show that λ n ( σ ) {\lambda _n}(\sigma ) , the nth focal point, is a ρ \rho continuous function of σ \sigma . For the focal interval case we give sufficient hypotheses so that the number of focal intervals is a local minimum at σ 0 {\sigma _0} in Σ \Sigma . Neither of these results seems to have been published before (under any setting) in the literature. For completeness an example is given for quadratic problems in a control theory setting.