# Approaches to Numerical Solution of Optimal Control Problem Using Evolutionary Computations
###### Abstract  
Two approaches to the numerical solution of the optimal control problem are studied. The direct approach is based on the reduction of the optimal control problem to a nonlinear programming problem. Another approach is so-called synthesized optimal control, and it includes the solution of the control synthesis problem and stabilization at some point in the state space, followed by the search of stabilization points and movement of the control object along these points. The comparison of these two approaches  
Although there is no exact mathematical theory available today for studying the unimodality of quality criteria, it is intuitively obvious that, for example, collision avoidance of two robots moving on a plane is possible using at least two methods, i.e., bypassing the robot to the right or left. Both methods will provide the local minima of the quality criterion. This shows that the quality criterion is not unimodal. It follows from the above that it is more expedient to develop numerical methods based on the direct approach for solving the optimal control problem, whereby phase constraints are included in the quality criterion as penalties. In this case, global optimization methods, including evolutionary algorithms, can be used to solve the optimal control problem [16]. An indirect approach based on the Pontryagin maximum principle requires the functional to be convex, which cannot be ensured in the presence of phase constraints. The presence of phase constraints requires supplementary variables in addition to the conjugate variables. In this paper, two numerical methods for the solution of the optimal control problem are compared. The first method uses a piecewise linear control approximation as a time function [17]. The second method is synthesized optimal control [18], in which the control synthesis problem is initially solved, and the stability of the control object to some point in the state space is ensured. Then, the search of the optimal location of stabilization points is performed. The points are switched at a given time interval, which ensures that the object reaches the goal with the optimal value of the quality criterion. As a result of solving the optimal control problem using the second method, we obtain the control function as a function of the coordinates of the state space and a piecewise constant function of time that ensures switching of the stabilization points. The application of the second method of synthesized optimal control is caused by the fact that the solution of the optimal control problem in the form of time function cannot be directly used in a real object. Such a solution is an open-loop control system, and any deviations from the optimal trajectory caused by external influences or inaccuracies in the model will lead to terminal conditions being missed. To implement the control system, it is necessary to develop a stabilization system to move the object along the optimal trajectory. The presence of the stabilization system changes the mathematical model of the control object. Thus, we can obtain different mathematical models of the control object for the optimal control problem and its implementation. On the other hand, the stabilization system relative to the programmed path depends on the path itself; therefore, it cannot be included in the mathematical model when solving the optimal control problem. The lack of unimodality of the quality criterion is caused by either phase constraints or the properties of the quality criterion itself or the model. Ensuring the stability of the control object before solving the optimal control problem is an obvious stage in the development of control systems, which is often used in practice, especially in robotics. Usually, this stage involves the study of the properties of the control object and is based on the experience and intuition of the developer of control systems.