Generalized needles and second-order conditions in optimal control (Q2317211)

The authors consider a classical optimal problem for systems described by state differential equations with fixed terminal time and endpoint constraints of general form. A notion of a generalized needle variation of the control variable is defined and used to formulate a second order necessary optimality conditions for a strong minimum in the optimal control problem. To proof the conditions the authors follow the line of reasoning used in the derivation of the Pontryagin maximum principle (see [\textit{L. S. Pontryagin} et al., Mathematische Theorie der optimalen Prozesse (Russian). Moskva: Gosudarstv. Izdat. Fiz.-Mat. Lit. (1961; Zbl 0102.31901)]) but three auxiliary lemmas are added, a lemma on a package of needles, a lemma on equations in variations and a special inverse function lemma. The conditions are illustrated by a simple example for which the classical conditions are ineffective.