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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

An approximation theory for focal points and focal intervals


Author: John Gregory
Journal: Proc. Amer. Math. Soc. 32 (1972), 477-483
MSC: Primary 49F15
DOI: https://doi.org/10.1090/S0002-9939-1972-0296788-6
MathSciNet review: 0296788
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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.


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DOI: https://doi.org/10.1090/S0002-9939-1972-0296788-6
Keywords: Approximation theory, focal points, conjugate points, calculus of variations, control theory, Hilbert space, quadratic forms
Article copyright: © Copyright 1972 American Mathematical Society

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