ISSN 1817-2172, рег. Эл. № ФС77-39410, ВАК

# Differential Equations and Control Processes (Differencialnie Uravnenia i Protsesy Upravlenia)

## Dynamic Optimization and Piecewise Constant Functions on the State Set

### Author(s):

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;

enorel@fa.ru

Olga Evgenyevna Orel

Moscow Institute of Physics and Technology
associate professor of
the Department of higher mathematics,
PhD, associate professor
Dolgoprudnyi town

Olga_Orel72@mail.ru

### Abstract:

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.

### Keywords

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

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