Pashko S. Mathematical methods for choosing optimal solutions in systems consisting of rational agents.

The thesis is devoted to the construction and research of mathematical methods for choosing optimal solutions in systems consisting of autonomous objects that have a common goal and act in an optimal way for the sake of achieving it. Elements of such systems are usually called rational agents. In the thesis, the main types of activities related to systems of rational agents are considered: the formation of a system of agents (cooperation), planning and coordination of agents' action plans, the placement of the system, recognition. For these types of activities, systems have been selected that have an theoretical and practical value, and methods for selecting optimal solutions have been developed and studied for these systems. Also, the effectiveness of recognition procedures that can be used by rational agents is investigated. The Bayesian recognition procedure error is estimated depending on the training sample size and other parameters. The suboptimality of the Bayesian approach is proved and the complexity of classes of recognition problems is found. The effectiveness of known recognition procedures has been studied. The problems of pursuit and evasion are considered, in which for each evader a group of pursuers is formed. Theorem on NP-difficulties of optimization problem of pursuit groups is proved. The variants of branch and boundary methods and random search with local optimization of the solution of such problems are constructed. The problems of pursuit and evasion are investigated, in which several agents pursue one by using the parallel approach strategy. The optimal escape strategy is constructed, its properties are studied, the maximum time of pursuit is found. Linear programming problems are formulated that allow constructing optimal or close to optimal strategies. Using the maximum pursuit time for a parallel approach strategy as a Lyapunov function, a new pursuit strategy is built that surpasses the parallel approach strategy due to more coordinated actions of agents. The problems of the optimal placement of sensors for collective recognition are solved. An algorithm for detecting an underwater threat using a system of acoustic sensors, as well as extreme sensor placement problems, is described. A method for solving such problems is developed, and a theorem on the asymptotic optimality of the constructed layouts of sensors is proved.