A theory of focal points and focal intervals for an elliptic quadratic form on a Hilbert space
John A. Gregory · Transactions of the American Mathematical Society · 1971
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.