Positive definiteness of a quadratic functional

In a 1975 paper, Molinari [1] proved that under certain continuity and controllability hypotheses, the infinum of a quadratic linear functional subject to linear differential equations constraints and a linear terminal constraint, is a quadratic function of the initial state. We show here how to constructively find this quadratic form under the addition of a positivity assumption. We also show that if a strengthened generalized Legendre-Clebsch condition holds then there is a linear optimal feedback control law.