**10. Findings/Results**

This paper presents a new formulation of the optimal control problem, taking into account the objectively existing uncertainties of the model. The concept of feasibility is introduced, which means that small changes in the model do not lead to a loss of quality. Given the theoretical substantiations (definitions and theorems) that a system of differential equations of the mathematical model is feasible if it obtains, as a one-parametric mapping, a contraction property in the implementation domain. This property is an alternative to Lyapunov stability; it is softer, but sufficient for the development of real stable practical systems. An approach based on the method of synthesized optimal control is proposed, which makes it possible to develop systems that have the property of feasibility.

## **11. Discussion**

According to the method of synthesized optimal control, the stability of the object is first ensured, that is, an equilibrium point appears in the phase space. In the neighbourhood of the stability point, the phase trajectories contract, and this property determines the feasibility of the system. For this, it is necessary to numerically solve the problem of synthesizing the stabilization system in order to obtain expressions for the control and substitute them in the right-hand sides of the object model. The synthesis problem is quite difficult. This paper proposes using numerical methods of symbolic regression to solve it. There are several successful applications, but they are still not very popular due to the complexity of the search area on a non-numerical space of functions where there is no metric. This is the direction for future research.

In the applied method of synthesized optimal control in the second stage we searched positions of equilibrium points as a piece-wise constant function. It is necessary to investigate other types of functions to change the position of the equilibrium point, how many points should be and how often they should be switched.

In further studies it is also necessary to consider solutions of the new optimal control problem for different control objects.

With the numerical solution of the optimal control problem by evolutionary algorithm it was defined that these algorithms can find solutions for complex optimal control problems with static and dynamic phase constraints. It is necessary to continue to research different evolutionary algorithms for the solution of the optimal control problems.

**Author Contributions:** Conceptualization, A.D. and E.S.; methodology, A.D., E.S.; software, A.D., E.S.; validation, V.S., P.Z.; investigation, E.S.; resources, V.S.; writing—original draft preparation, A.D., E.S.; writing—review and editing, E.S.; supervision, V.S., P.Z. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research is supported by the Ministry of Science and Higher Education of the Russian Federation, project No. 075-15-2020-799.

**Conflicts of Interest:** The authors declare no conflict of interest.
