# Approaches to Numerical Solution of Optimal Control Problem Using Evolutionary Computations
## 5 Evolutionary Methods  
Evolutionary methods form a class of modern optimization algorithms that work with a set of randomly created possible solutions and apply certain modifications of these solutions. All evolutionary methods have the following common features:  
* The search process is iterative. For each search iteration, a set of possible solutions is considered. * Each solution takes part in avoiding local extrema and navigating to promising areas of search space. * Not only the best solution but also other information about the search space is preserved over the iterations and used to form new possible solutions. * The evolution of solutions is based on the inheritance property, in which better solutions have a greater chance of being included in next search iteration. The efficiency of evolutionary methods is due to the right balance between exploration and exploitation search. Exploration allows investigating the whole search space for the promising areas. This part of the search process should be applied to the search space as broadly as possible. On the other hand, exploitation involves a local search in promising areas to ensure higher accuracy of the solution. The methods of performing exploration and exploitation searches and the balance between them differ from one evolutionary method to another. The first and the most well-known of the evolutionary methods is the genetic algorithm (GA) [20]. This method was inspired by [PERSON] evolution concepts. In the genetic algorithm, operators of crossover and mutation were introduced to modify a set of possible solutions in search for a best one. In this method, each possible solution is encoded with a Gray binary code and called a chromosome, whereas each search iteration is called a generation. Analogous to [PERSON]'s theory, better solutions have a higher probability of participating in generating new chromosomes for the next generation. However, taking into account only the best solution may lead to premature convergence to some local extremum. To prevent this, similar solutions should not be used during crossover. The mutation operator also helps not to get stuck in a local extremum. The probability of mutation, the probability of crossing depending on the quality of solutions, and their proximity are the main tuning parameters of the genetic algorithm. Another well-known and popular evolutionary method is particle swarm optimization (PSO) [21; 22; 23]. As the method's name suggests, it mimics the social behavior of some creatures in nature which exist in swarms. PSO was proposed in 1995 and gained popularity due to its effective way of combining exploration and exploitation searches during the solution modification. In this method, possible solutions are called particles. These particles are distributed in the search space and seek better positions by taking into account the best particle and the best position in previous iterations. This technique can be seen in flocks of birds or in schools of fish and is called social intelligence. The movement toward the global best solution is part of the exploitation search. The inertial movement and the movement toward the best position is part of the exploration search. The balance among these three components of the particle movement is determined by the corresponding tuning parameters of the method. The observation of the collective behavior of bees in finding nectar sources prompted the creation of the bee algorithm (BA) [24]. In this evolutionary method, the exploration part of the search is represented by a random search of the whole search space. The analogous process among bees is the process of looking for promising places with high nectar concentration by scout bees. The found solutions can be divided into highly interesting, interesting, and not interesting. Not interesting solutions are substituted by new randomly created ones. The subdomains of given radii of highly interesting and interesting solutions are investigated more intensively.