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.
1. Aksenyushkina E. V., Srochko V. A. Sufficient optimality conditions for one class of nonconvex control problems . Zhurnal vichislitelnoy matematiki i matematicheskoy fiziki, 2015, Vol. 55, №10, pp. 1070–1080. (in Russian)
2. Antipina N. V., Dychta V. A. Linear funtions of Lyapunov – Krotov and sufficient optimality conditions in the form of maximum principle. Izvestiya vuzov. Matematika, 2002, №12, pp. 11—22. (in Russian)
3. Antonik V. G., Srochko V. A. Optimality conditions of the maximum principle type in bilinear control problems. Zhurnal vichislitelnoy matematiki i matematicheskoy fiziki, 2016, Vol. 56, №12, pp. 2054–1064. (in Russian)
4. Baturin V. A., Urbanovich D. E. Approximate optimal control methods based on the extension principle. Novosibirsk, Nauka, 1997, 175 p. (in Russian)
5. Gurman V. I. The extension principle in control problems. Moscow, Nauka, 1985, 288 p. (in Russian)
6. Dychta V. A. The inequality of Lyapunov – Krotov and sufficient conditions in optimal control theory. Itogi nauki i tehniki. Sovremennaya matematika i eyo prilozheniya, 2006, Vol. 110, pp. 76–108.(in Russian)
7. Krotov V. F., Gurman V. I. Methods and problems of optimal control. Moscow, Nauka, 1988, 446 p.(in Russian)
8. Srochko V. A., Antonik V. G. Sufficient optimality conditions of extremal controls based on functional increment formulas. Izvestiya vuzov. Matematika, 2014, №8,pp. 96—102. (in Russian)