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Русская версия

**Evgenii Nicolaevich Orel**

Financial university under the Gouvernment of Russian Federation,

professor (Department of data analysis, solution solving and

financial technologies),

doctor of physics and mathematics, professor;

**Olga Evgenyevna Orel**

Moscow Institute of Physics and Technology

associate professor of

the Department of higher mathematics,

PhD, associate professor

Dolgoprudnyi town

We consider a direct method of dynamic optimization for an approximation of global extremum. It is based on splitting the state space into classes (cells) and the construction of piecewise constant functions on the partition. Such an approach leads to a generalization of the Euler polygonal method and using shortest path algorithms on graphs. In the proposed algorithm for each class (cell) a path from an initial point to this class is formed. In addition, the program remembers only the terminal point of the path, functional value along the path and the number of the previous class. If another path with the same boundary conditions but lesser functional value (for the minimization problem) is found, this path becomes the current approximation. It is proved that if the partition is sufficiently small, then we obtain the optimal polygon. The suggested approach is applied to problems of dynamic optimization with incomplete information (differential games, control of systems with unknown dynamics). In addition, some results of numerical solutions of optimal control problems and differential games are given.

- differential games
- Euler's polygonal method
- global extremum
- optimal control
- piecewise constant functions
- splitting into classes
- systems with unknown dynamics

- Orel E. N. [Approximation of Bellman's function by piecewise constant functions. ] USSR Computational Mathematics and Mathematical Physics. Volume 18, Issue 4, 1978, P. 95-107 (In Russ. )
- Orel E. N. [Method for solving of Optimal Control Problems. ] Original Russian Text © E. N. Orel, 1989, published in Doklady Akademii Nauk, 1989, 1; 6, p. 1301-1304 (In Russ. )
- Orel E. N. [Algorithms for Searching of Quasi-Optimal Control Using a Partition of State Space. ] USSR Computational Mathematics and Mathematical Physics. Issue 9, 1990, P. 1283-1294 (In Russ. )
- Orel E. N., Orel O. E. [Optimality Conditions for Central Fields of Trajectories] Differential Equations and Control Processes, 2017, ¹ 1, P. 53-77 (In Russ. ) http://diffjournal.spbu.ru/pdf/orel.pdf
- Moiseev N. N. [Elements of the Theory of Optimas Systems] Moscow, Science, 1975, 528 p. (In Russ. )
- Elsgolts L. E. [Differential Equations and Calculus of Variations], Moscow, Science, 1965, 424 p. (In Russ. )
- Orel E. N. [Construction of Shortest Path on a Graph Using a minorant of the Bellman Function] Automatics ans Telemechanics, 1977, ¹ 2, P. 88-91 (In Russ. )
- Orel E. N. [Correction of a heuristic function during the process of decision making] Published in Doklady Akademii Nauk, 1991, ¹ 4, p. 853-855 (In Russ. )
- Orel E. N. [Heuristic of teaching in search problems] Engineering Cybernetics, 1992, 1; 5, p. 69-81 (In Russ. )
- Orel E. N., Orel O. E. [Central fields of optimal trajectories] Doklady Mathematics, 2014, Vol. 90, No 2, P. 651-653. Original Russian Text © E. N. Orel, O. E. Orel, 2014, published in Doklady Akademii Nauk, 2014, Vol. 458, No. 4, P. 402-405. (In Russ. )
- Orel E. N., Orel O. E. [Central field of trajectories in Problems of Optimal Control and Calculus of Variations. ] Accounting. Analysis. Audit, 2015, ¹ 3, P. 35-42 (In Russ. )
- Orel E. N., Orel O. E. [Optimal Control Process of Manufacturing During Execution of Order to given deadline. ] Economics and Mathematical Methods, 2016, ¹ 2, P 65-77 (In Russ. )
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- Neimark U. I., Kogan N. Ya., Savelyev V. P. [Dynamics Models of Control Theory] Moscow, Science, 1985, 400 P. (In Russ. )
- Michie D., Johnston R. [The Creative Computer. Machine intelligence and human knowledge] Viking, 1984
- Orel E. N. [Discrete Boundary Problems and Optimization Processes on graphs. ] Artificial Intelligence in Engineering Systems. - Moscow, RAN, GosIFTP, P. 3-33 (In Russ. )