# Approaches to Numerical Solution of Optimal Control Problem Using Evolutionary Computations
## 2 Optimal Control Problem for Group of Robots  
The optimal control problem for a group of mobile robots is considered. Robots should move from given initial states to terminal states in a minimum time. They must not collide with each other or with static obstacles. The coordinates of obstacles in the state space are given. An essential feature of this problem is the mandatory presence of phase constraints that significantly complicate the search for its solution. To solve the optimal control problem with phase constraints, due to the lack of convexity and unimodality of the quality function, it is suggested to use mainly evolutionary algorithms. The second obstacle here is the feasibility of the obtained solution. It is obvious that the found optimal control function as a time function cannot be applied to a real object. All researchers in the field of optimal control argue that, to implement the found optimal solution, it is necessary to develop a stabilization system for the optimal program path. In this paper, we propose to solve the optimal control problem using the synthesized control method, which is widely used in practice. Formal studies on this method in mathematical scientific papers have not been carried out because this method requires solving the control synthesis problem, which is more complicated than the optimal control problem since it requires finding a solution in the form of a mathematical expression of the control function, the argument of which is the state vector of the control object. Substitution of the found control function into the model of the control object ensures the stability of the control object in the neighborhood of a point in the state space. The approaches to solving control synthesis problems are often limited by analytical methods, backstepping [26], and the analytical design of aggregated regulators [27]. The successful application of these methods depends on the mathematical model of the control object. In this work, the numerical method of symbolic regression, a multilayer network operator, was used to solve the synthesis problem. After solving the synthesis problem, the object was controlled by switching from one stabilization point to another. Evolutionary algorithms were used to implement the synthesized control method. Initially, when solving the synthesis problem, a specialized genetic algorithm was used to find an encoded mathematical expression for the optimal control function. Then, using another evolutionary algorithm, the coordinates of stable equilibrium points for all control objects were found. To analyze the effectiveness of this approach, we also investigated a direct solution to the optimal control problem using various algorithms. Consider the optimal control problem below for a group of \(N\) identical mobile robots. A mathematical model of a mobile robot is given as follows [28]:  
\[ \begin{split}\dot{x}_{1}^{j}&=u_{1}^{j}\cos(x_{3}^{ j}),\\ \dot{x}_{2}^{j}&=u_{1}^{j}\sin(x_{3}^{j}),\\ \dot{x}_{3}^{j}&=u_{2}^{j},\end{split}  \tag{1}\]  
where \(j\) is an index of a robot in a group, \(j=1,\ldots,N\), \(x_{1},\ x_{2}\) are coordinates of the mass center of the robot, \(x_{3}\) is a rotation angle of robot axis, and \(u_{1},\ u_{2}\) are control signals on rotors. The control signals are limited by  
\[u_{i}^{-}\leq u_{i}^{j}\leq u_{i}^{+},\ i=1,2,\ j=\overline{1,N}. \tag{2}\]Initial states of each mobile robot \(j\) are given as  
\[x_{i}^{j}(0)=x_{i}^{0,j},\ i=1,2,3,\ j=\overline{1,N}. \tag{3}\]  
Static phase constraints in the state space are  
\[\beta_{i}(\mathbf{x}^{j})=r_{i}^{2}-\left(x_{1}^{*,i}-x_{1}^{j}\right)^{2}- \left(x_{2}^{*,i}-x_{2}^{j}\right)^{2}\leq 0,\ \ j=\overline{1,N},i=\overline{1,B}, \tag{4}\]  
where \(r_{i}\) represents distances between the center of static phase constraint and center of the robot that cannot be violated, \(x_{1}^{*,i},\ x_{2}^{*,i}\) are coordinates of the center of static phase constraint \(i\), and \(B\) is the number of static constraints.