Summary: We study nondiagonal and diagonal quadratic forms over continuous and discrete time intervals. In particular a strong connection between the corresponding Hamiltonians is established. The coercivity of the nondiagonal form turns out to be equivalent to the positivity of its corresponding diagonal form. In the continuous time setting, the traditional strengthened Legendre-Clebsch condition is required, while another condition is needed in the discrete case. In the latter, the proof of the result is far more intricate than it is in the continuous case. The accessory problems for nonlinear optimal controls are prototypes for nondiagonal quadratic forms. Thus, the results of this paper are instrumental in obtaining sufficient conditions for optimality in nonlinear controls over either of the two time scale intervals.
For the entire collection see [Zbl 1166.39001].