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 on a Hilbert space . These results were based upon inequalities dealing with the indices and of the elliptic form defined on the closed subspace of , where belongs to the metric space .

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 and , where belongs to the metric space . For the usual focal point problems we show that , the *n*th focal point, is a continuous function of . For the focal interval case we give sufficient hypotheses so that the number of focal intervals is a local minimum at in . 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.

**[1]**John Gregory,*An approximation theory for elliptic quadratic forms on Hilbert spaces: Application to the eigenvalue problem for compact quadratic forms*, Pacific J. Math.**37**(1971), 383–395. MR**0303311****[2]**John Gregory,*A theory of focal points and focal intervals for an elliptic quadratic form on a Hilbert space*, Trans. Amer. Math. Soc.**157**(1971), 119–128. MR**0278147**, https://doi.org/10.1090/S0002-9947-1971-0278147-9**[3]**John Gregory,*A theory of numerical approximation for elliptic forms associated with second order differential systems. Application to eigenvalue problems*, J. Math. Anal. Appl.**38**(1972), 416–426. MR**0322652**, https://doi.org/10.1016/0022-247X(72)90099-6**[4]**Katharine Elizabeth Hazard,*Index theorems for the problem of Bolza in the calculus of variations*, Contributions to the Calculus of Variations, 1938–1941, University of Chicago Press, Chicago, Ill., 1942, pp. 293–356. MR**0006823****[5]**Magnus R. Hestenes,*Applications of the theory of quadratic forms in Hilbert space to the calculus of variations*, Pacific J. Math.**1**(1951), 525–581. MR**0046590****[6]**E. Y. Mikami,*Focal points in a control problem*, Pacific J. Math.**35**(1970), 473–485. MR**0281081**

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Additional Information

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