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June  2017, 7(2): 191-199. doi: 10.3934/naco.2017013

## Sufficient optimality conditions for extremal controls based on functional increment formulas

 1 Irkutsk State University, K. Marks str., 1, Irkutsk, 664003, Russia 2 Baikal State University, Lenin str., 11, Irkutsk, 664003, Russia

Received  November 2016 Revised  May 2017 Published  June 2017

Fund Project: This paper was prepared at the occasion of The 10th International Conference on Optimization: Techniques and Applications (ICOTA 2016), Ulaanbaatar, Mongolia, July 23-26,2016, with its Associate Editors of Numerical Algebra, Control and Optimization (NACO) being Prof. Dr. Zhiyou Wu, School of Mathematical Sciences, Chongqing Normal University, Chongqing, China, Prof. Dr. Changjun Yu, Department of Mathematics and Statistics, Curtin University, Perth, Australia, and Shanghai University, China, and Prof. Gerhard-Wilhelm Weber, Middle East Technical University, Ankara, Turkey.

Optimal control problem without phase and terminal constraints is considered. Conceptions of strongly extremal controls are introduced on the basis of nonstandard functional increment formulas. Such controls are optimal in linear and quadratic problems. In general case optimality property is guaranteed by concavity condition of the Pontryagin function with respect to phase variables.

Citation: Vladimir Srochko, Vladimir Antonik, Elena Aksenyushkina. Sufficient optimality conditions for extremal controls based on functional increment formulas. Numerical Algebra, Control and Optimization, 2017, 7 (2) : 191-199. doi: 10.3934/naco.2017013
##### References:
 [1] N. V. Antipina and V. A. Dykhta, Linear funtions of Lyapunov-Krotov and sufficient optimality conditions in the form of maximum principle (in Russian), Izvestia vuzov. Matematika, 12 (2002), 11-22. [2] F. H. Clarke, Optimization and Nonsmooth Analysis, New York, John Wiley & Sons Inc, 1983. [3] R. Gabasov and F. M. Kirillova, The Maximum Principle in Optimal Control Theory (in Russian), Moscow, Librokom, 2011. [4] E. N. Khailov, On extremal controls in homogeneous bilinear system (in Russian), Trudy MIAN, 220 (1998), 217-235. [5] V. F. Krotov and V. I. Gurman, Methods and Problems of Optimal Control (in Russian), Moscow, Nauka, 1988. [6] O. L. Mangasarian, Sufficient conditions for the optimal control of nonlinear systems, SIAM J. Control Optim., 4 (1966), 139-152. [7] M. S. Nikolsky, On sufficiency of Pontryagin maximum principle in some optimization problems (in Russian), Vestnik Moskovskogo universiteta. Seria 15, 1 (2005), 35-43. [8] L. S. Pontryagin, V. G. Boltiansky, R. V. Gamkrelidze and E. F. Mischenko, Mathematical Theory of Optimal Proccesses (in Russian), Moscow, Fizmatlit, 1961. [9] V. A. Srochko, Iterative Methods for Solving of Optimal Control Problems (in Russian), Moscow, Fizmatlit, 2000. [10] V. A. Srochko and E. V. Aksenyushkina, Optimal control problems for the bilinear system of special structure (in Russian), Izvestia Irkutskogo universiteta. Seria Matematika, 15 (2016), 78-91. [11] A. Swierniak, Cell cycle as an object of control, Journal of Biological Systems, 1 (1995), 41-54.

show all references

##### References:
 [1] N. V. Antipina and V. A. Dykhta, Linear funtions of Lyapunov-Krotov and sufficient optimality conditions in the form of maximum principle (in Russian), Izvestia vuzov. Matematika, 12 (2002), 11-22. [2] F. H. Clarke, Optimization and Nonsmooth Analysis, New York, John Wiley & Sons Inc, 1983. [3] R. Gabasov and F. M. Kirillova, The Maximum Principle in Optimal Control Theory (in Russian), Moscow, Librokom, 2011. [4] E. N. Khailov, On extremal controls in homogeneous bilinear system (in Russian), Trudy MIAN, 220 (1998), 217-235. [5] V. F. Krotov and V. I. Gurman, Methods and Problems of Optimal Control (in Russian), Moscow, Nauka, 1988. [6] O. L. Mangasarian, Sufficient conditions for the optimal control of nonlinear systems, SIAM J. Control Optim., 4 (1966), 139-152. [7] M. S. Nikolsky, On sufficiency of Pontryagin maximum principle in some optimization problems (in Russian), Vestnik Moskovskogo universiteta. Seria 15, 1 (2005), 35-43. [8] L. S. Pontryagin, V. G. Boltiansky, R. V. Gamkrelidze and E. F. Mischenko, Mathematical Theory of Optimal Proccesses (in Russian), Moscow, Fizmatlit, 1961. [9] V. A. Srochko, Iterative Methods for Solving of Optimal Control Problems (in Russian), Moscow, Fizmatlit, 2000. [10] V. A. Srochko and E. V. Aksenyushkina, Optimal control problems for the bilinear system of special structure (in Russian), Izvestia Irkutskogo universiteta. Seria Matematika, 15 (2016), 78-91. [11] A. Swierniak, Cell cycle as an object of control, Journal of Biological Systems, 1 (1995), 41-54.
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