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 $J(x)$ on a Hilbert space $\mathcal {A}$. These results were based upon inequalities dealing with the indices $s(\sigma )$ and $n(\sigma )$ of the elliptic form $J(x;\sigma )$ defined on the closed subspace $\mathcal {A}(\sigma )$ of $\mathcal {A}$, where $\sigma$ belongs to the metric space $(\Sigma ,\rho )$. 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 $s(\mu )$ and $u(\mu )$, where $\mu$ belongs to the metric space $(M,d),M = {E^1} \times \Sigma$. For the usual focal point problems we show that ${\lambda _n}(\sigma )$, the nth focal point, is a $\rho$ continuous function of $\sigma$. For the focal interval case we give sufficient hypotheses so that the number of focal intervals is a local minimum at ${\sigma _0}$ in $\Sigma$. 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.