The problem of multiobjective synthesis of optimal control is considered. It is necessary to find the control as a function of problem space coordinates. Initial values of the control object are in closed bounded domain. Functionals of control quality include multiple integrals over domain of initial values. To solve the problem of multiobjective synthesis the numerical method was developed. This method is based on genetic algorithm and special data structure, network operator. The numerical method allows automatic search of mathematical equations that describe dependence of control from the problem space coordinates. The solution of the synthesis problem for space vehicle descent with indeterminate initial conditions at entry in the dense layers of atmosphere is given.

Optimization problems and their solution by symbolic regression methods are considered. The search is performed on non-Euclidean space. In such spaces it is impossible to determine a distance between two potential solutions and, therefore, algorithms using arithmetic operations of multiplication and addition are not used there. The search of optimal solution is performed on the space of codes. It is proposed that the principle of small variations of basic solution be applied as a universal approach to create search algorithms. Small variations cause a neighborhood of a potential solution, and the solution is searched for within this neighborhood. The concept of inheritance property is introduced. It is shown that for non-Euclidean search space, the application of evolution and small variations of possible solutions is effective. Examples of using the principle of small variation of basic solution for different symbolic regression methods are presented.

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 was carried out as the solution of the optimal control problem as a time function cannot be directly used in the control system, although the obtained discretized control can be embedded. The control object was a group of interacting mobile robots. Dynamic and static constraints were included in the quality criterion. Implemented methods were evolutionary algorithms and a random parameter search of piecewise linear approximation and coordinates of stabilization points, along with a multilayer network operator for control synthesis.

In this paper we use symbolic regression methods for control system synthesis. We compare three methods: network operator method, genetic programming and analytical programming. We developed variational versions of genetic programming and analytical programming to improve the search process efficiency. All the methods perform search over the set of the small variations of the given basic solution. Search efficiency depends on the basic solution. We give an example of control system synthesis for the unmanned vehicle with the state constraints over the set of the initial states using these methods.

In the known methods of symbolical regression by search of the solution with the help of a genetic algorithm, there is a problem of crossover. Genetic programming performs a crossover only in certain points. Grammatical evolution often corrects a code after a crossover. Other methods of symbolical regression use excess elements in a code for elimination of this shortcoming. The work presents a new method of symbolic regression on base of binary computing trees. The method has no problems with a crossover. Method use a coding in the form of a set of integer numbers like analytic programming. The work describes the new method and some examples of codding for mathematical expressions.