# Approaches to Numerical Solution of Optimal Control Problem Using Evolutionary Computations
## 7 Conclusions  
The optimal control problem of a group of robots with phase constraints was solved using two numerical approaches. Robots had to achieve the terminal points without collisions among themselves or with obstacles in a minimal time. The presence of phase constraints complicated the search. Evolutionary computations were applied to cope with the nonconvex and multimodal quality criterion.  
In the first approach, a robotic group was treated as one object. The optimal control problem was reduced to the nonlinear programming problem. In the second approach, the problem of control synthesis was first solved separately for each robot relative to the points in the state space. To solve the synthesis problem, the multilayer network operator method was used, although this could also be achieved using other symbolic regression methods that can derive mathematical expressions.  
Then, the optimal control problem was considered in the original statement, where the coordinates of the stabilization points of robots were used as control. A search was carried out for three stabilization points for each of the robots. Switching between points occurred at a specified time interval. A search of the coordinates of stability points was performed using evolutionary algorithms (genetic algorithm, particle swarm optimization, bee algorithm, and gray wolf optimizer) and a random search.  
After a series of 10 tests for each algorithm, they were evaluated by the average value of functional for the best solution identified. Experiments showed that, on average, the PSO algorithm was the most effective in terms of the search of parameters in the direct method and coordinates of stabilization points in synthesized control.  
With respect to the application of evolutionary algorithms to the compared approaches, a synthesized control approach was proven to perform better. In the direct approach, only GA and PSO gave satisfactory results on average, whereas, in the second approach, all the evolutionary algorithms performed well and gave, on average, approximately the same results.  
Currently, there are no universal numerical methods for solving optimal control problems. It is proposed to continue the study of evolutionary algorithms and consider other new evolutionary algorithms, including hybrid ones. To use evolutionary algorithms in optimal control problems, it is necessary to include phase constraints in the functional, discretize the problem, and reduce it to a nonlinear programming problem. For this purpose, the time of the control process is divided into intervals in which the control function is approximated by polynomials depending on a finite number of parameters. Furthermore, the search for the values of the parameters of the approximating function can be carried out by evolutionary algorithms.  
A distinctive feature of the application of evolutionary algorithms for solving optimal control problems is that, when calculating the value of the quality criterion, it is necessary to integrate a system of differential equations that describe the mathematical model of the control object with an approximating control function.  
Conceptualization, A.D.; methodology, A.D.; investigation, A.D., E.S., S.K.; software, A.D., E.S., S.K.; supervision, A.D.; writing--original draft preparation, A.D., E.S., writing--review and editing, E.S., S.K. All authors have read and agreed to the published version of the manuscript.  
The research was supported by the Ministry of Science and Higher Education of the Russian Federation, project No. 075-15-2020-799.  
Not applicable.  
The authors thank the unknown reviewers for their useful comments.  
The authors declare no conflict of interest.