## A theory of focal points and focal intervals for an elliptic quadratic form on a Hilbert space

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- by John Gregory PDF
- Trans. Amer. Math. Soc.
**157**(1971), 119-128 Request permission

## 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. Hestenes has shown that for many problems this theory may be studied by Hilbert space methods. In a previous paper the author has extended the theory of Hestenes to elliptic quadratic forms $J(x;\sigma )$ defined on $\mathfrak {A}(\sigma )$ where $\sigma$ is a member of the metric space $(\Sigma ,\rho )$ and $\mathfrak {A}(\sigma )$ denotes a closed subspace of $\mathfrak {A}$. A fundamental part of this extension is concerned with inequalities dealing with the signature $s(\sigma )$ and nullity $n(\sigma )$ of $J(x;\sigma )$ on $\mathfrak {A}(\sigma )$ where $\sigma$ is in a $\rho$ neighborhood of a fixed point ${\sigma _0}$ in $\Sigma$. The purpose of this paper is threefold. The first purpose is to show that the extended theory includes the focal point hypotheses of Hestenes. The second purpose is to obtain a focal point theory much like that of Hestenes. It is interesting to note that our theory is based only on the nonnegative integers $s(\lambda )$ and $n(\lambda )$. This will facilitate later work on numerical calculations of focal points. Our final purpose is to obtain an abstract focal interval theory in which the usual focal points are (degenerate) focal intervals. While previous authors have considered specific problems, no general results for the focal interval case seem to be contained in the literature. An expression for the number of focal intervals on a subinterval $(\lambda ’,\lambda '')$ of $[a,b]$ is given. This expression is a key result for our work on approximation of focal intervals (to be published). For completeness we give comparison theorems for focal point problems. In addition an example is given for problems in optimal control theory. The correspondence between our focal intervals and solutions to the differential equations of the example is given.## References

- G. D. Birkhoff and M. R. Hestenes,
*Natural isoperimetric conditions in the calculus of variations*, Duke Math. J.**1**(1935), no. 2, 198–286. MR**1545876**, DOI 10.1215/S0012-7094-35-00118-1
R. F. Dennemeyer, - 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**303311** - 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** - 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**46590**
A. S. Householder, - Marston Morse,
*The calculus of variations in the large*, American Mathematical Society Colloquium Publications, vol. 18, American Mathematical Society, Providence, RI, 1996. Reprint of the 1932 original. MR**1451874**, DOI 10.1090/coll/018 - William T. Reid,
*An Integro-Differential Boundary Value Problem*, Amer. J. Math.**60**(1938), no. 2, 257–292. MR**1507311**, DOI 10.2307/2371292

*Quadratic forms in Hilbert space and second order elliptic differential equations*, Dissertation, University of California, Los Angeles, 1956. J. Gregory,

*An approximation theory for elliptic quadratic forms on Hilbert spaces*, Dissertation, University of California, Los Angeles, 1969.

*The dependence of a focal point upon curvature in the calculus of variations*, Contributions to the Calculus of Variations, 1937, Univ. of Chicago Press, Chicago, Ill. G. C. Lopez,

*Quadratic variational problems involving higher order ordinary derivatives*, Dissertation, University of California, Los Angeles, 1961. E. Mikami,

*Quadratic optimal control problems*, Dissertation, University of California, Los Angeles, 1968.

## Additional Information

- © Copyright 1971 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**157**(1971), 119-128 - MSC: Primary 49.10
- DOI: https://doi.org/10.1090/S0002-9947-1971-0278147-9
- MathSciNet review: 0278147