Iii. Convexity and Sufficiency Ubc M402 Lecture Notes C 2015

The practical effectiveness of Thm. A.1 is roughly commensurate with the difficulty of its proof. That is, the proof is essentially trivial, and the condition is nearly useless. There are very few problems where simple pointwise minimization of the integrand at every instant yields an arc that is even admissible, much less optimal! Even worse, we have met many true minimizers (often in problems where L(t, x, v) is quadratic in (x, v)) for which the pointwise inequality in (∗) is false. Theorem A.1 is just too special to confirm optimality in many reasonable situations. We need theorems that have the same conclusion as above, but weaker hypotheses, so that more true minimizers can be verified.