The simplest nonconvex control problem. The maximum principle and sufficient optimality conditions

The optimal control problem with linear phase system and linear-quadratic functional is considered. The transition from the maximum principle to sufficient optimality conditions is fulfilled with the help of the notion of strongly extremal control. It means that in the problem of maximization of Pontryagin's function phase or conjugate trajectory should be replaced with any admissible trajectory. Sufficient conditions give opportunity to obtain explicit expressions for extremal values of auxiliary problems contained in these conditions. Results are presented in the form of inequalities and equalities for a function with one variable with respect to time segment. A special situation is implemented in the analysis of the combined control with interior and boundary segments with respect to the constraint. At the point of connection of these segments there is a non-standard condition of maximum type.
A positive factor is dual nature of obtained results: it is a pair of symmetrical relations, which operate independently. Their origin is connected with two types of strongly extremal controls with respect to phase or conjugate variables.