This paper applies a few results
on quadratic forms in Hilbert space and the theory of focal points from a paper by
Hestenes to a linear control problem with a constraint equation. The abnormality
inherent in this problem allows focal intervals to exist. The main results are, after
assuming the strengthened Clebsch condition, the following: (1) The signature is
equal to the sum of the focal points on the open interval, (2) The focal points
are the discontinuous points of rank and abnormality of the conjugate base
matrix, and (3) The dimension of a maximal linear space of broken transversal
extremal arcs is less than or equal to n − a, where a is the abnormality of the
problem.
