Stochastic multi-step saddle point process minimax control parametric synthesis
Authors: Krutskikh P.P. , Tsarik O.V. 
Published: November 2021

The actual research problem of operations is development of methods of increase of a management efficiency by processes of the conflict nature. Article is devoted to development of methods of increase of a management efficiency by such processes. The purpose of article is the substantiation of the approach to parametrical synthesis of optimum control by multi-step stochastic minimax processes and procedures of the numerical analysis of likelihood dynamic characteristics of process. Formalization of process consists in definition of its type, a vector of phase coordinates and corresponding restrictions, the task of set of the actions sold by each the parties, efficiency of each action, control parameters (varied parameters of process) which task of values each of the parties influences a course of process, control restrictions, criteria of efficiency of the parties expressed through elements of a vector of phase coordinates. Discrete final stochastic process is considered. Change of phase coordinates occurs during the discrete moments of time, named steps of process. Phase coordinates depend on values of two groups of control parameters (controls of the counteracting parties). Within the limits of the modern theory of optimization of stochastic systems procedure of synthesis of optimum control is realized two-phase. At the first stage with use of analytical methods the structure of optimum control is determined. For these purposes the simplified determined model of process can to be used. At the second stage parametrical control optimization with use of algorithmic methods and computing procedures statistical linearization is carried out. Dynamics of process is described vectorial finite-difference equation. It is necessary to distinguish cases when there is saddle a point and when saddle the point is absent. Parametrical synthesis of optimum control is possible only in the first case. It is considered three basic variants of the equation: the linear equation; the nonlinear equation with optimum controls on border of a range of definition; the nonlinear equation with optimum controls inside of a range of definition. For the first variant there is an effective algorithm of parametrical synthesis of optimum control. For the second variant of synthesis of optimum control it is possible, but the algorithm is not effective. For the third variant to determine optimum managements it is not possible. Procedure statistical linearization is offered. Procedure consists in generation of set of realizations of the casual process set by the vector equation, calculation of optimum control for each concrete realization and the further statistical processing of the received results. The process described by the piecewise linear vector equation, is a special case of nonlinear process. At that it keeps property of independence of optimum control from coordinates of process. It provides expansion of a scope of effective computing procedure of synthesis of optimum control on a new class of piecewise linear processes. Property of a constancy process Hamiltonian can be used as criterion of correctness of calculation of optimum control in concrete cases. Application of the offered procedure provides use of methods of statistical modelling for the decision of tasks of the analysis of dynamics of the conflict and synthesis of optimum control in view of nonlinearity of functions of losses of the parties, dependence of efficiency of means used by them on the random factors formalized in the form of stochastic functions with various likelihood distributions, and also uncertainty concerning actions of the opponent.