Sufficiency Criteria via Focal Points and via Coupled Points
Vera M. Zeidan · SIAM Journal on Control and Optimization · 1992
Considered in this paper is a general quadratic functional $J(\eta )$ with constraints of the form $\eta (a) = D\eta (b) = 0$, where D is an $r \times n$-matrix $(r \leqq n)$ and of full rank. It is shown that the nonexistence of focal points to b in $[a,b)$ is equivalent to the existence of a solution to the corresponding Riccati equation with associated boundary conditions. Thus, it is equivalent to the positivity of $J(\eta )$. This result generalizes W. A. Coppel, Proceedings of the Royal Society of Edinburgh, 73A, 18, 1974–1975, pp. 271–289; V. B. Haas, Systems and Control Letters, 5 (1984), North-Holland, Amsterdam, pp. 55–57; and W. T. Reid, Academic Press, New York, 1972, in which either $D = I $ or 0 is assumed. Moreover, it is proven that the nonexistence of “coupled points with a” in $(a,b]$ is also equivalent to the positivity of $J(\eta )$, proving that for this problem, the notion of a “coupled point with a” is the one searched for to extend that of a “conjugate point to a” from $D = I $ to a general D. Each of these conditions is proved to be sufficient for optimality in the nonlinear calculus of variations problem with fixed initial state but variable final endpoints.