Sufficient Optimality Conditions Based on Functional Increment Formulas in Control Problems

A typical optimal control problem with convex terminal function is considered. Sufficient optimality conditions are obtained with the help non-standard functional increment formulas. So far, these formulas didn't apply to construction of numerical methods for successive improvement of auxiliary controls. A notion of strongly extremal control is introduced for each formula. It provides the maximum for Pontryagin's function in regard to some set of trajectories. Strongly extremal controls are optimal ones in linear and quadratic problems. In common case optimality of strongly extremal controls is provided with concavity condition of Pontryagin's function with regard phase variables. Examples of effective realization obtained relations are given.