Search for a solution to a multi-extremal optimal control problem based on the evolutionary method

When searching for solutions to nonlinear optimal control problems, one may encounter difficulties related to the presence of local extremes. The use of traditional optimization methods is effective in the case of convex problems with the property that the found local extremum is global. Therefore, it is important to develop methods and algorithms for solving multi-extremal optimal control problems. Since the operation of most optimization methods depends on the choice of the initial values of the optimized parameters, it is proposed to apply the method of differential evolution. This method optimizes a set of possible solutions in the range of acceptable values of the desired parameters, the initial values of which are set randomly. The aim of the work is to develop an evolutionary algorithm for finding a solution to a multi-extremal optimal control problem. Overcoming the stuck solution in the local optimum is possible by maintaining population diversity. If the solution falls into the region of the local extremum with an insufficient set number of iterations of the algorithm, an incorrect solution can be obtained. Therefore, in order to dislodge a population from the area of the local extremum, a modification of the differential evolution method is proposed – a dynamic population size. If the population is drawn into the region of a local extremum, then its average fitness changes slightly. In this case, the vectors-individuals with the lowest fitness are removed and new individuals are added. Computational experiments have been carried out on a model optimal control problem with a non-convex reachability domain. The work of the developed evolutionary algorithm is compared with the method of variations in the control space and the algorithm of differential evolution with a constant population size. The effectiveness of the developed evolutionary algorithm in solving a multi-extremal optimal control problem is demonstrated.