Algorithm for solving the maximin evasion-approach problem

We consider the problem of transferring the trajectory of a dynamical system controlled by two players whose goals are opposite from a point in a given set to a terminal set. The class of controls for the first player consists of a set of piecewise constant functions and vectors, while the class of controls for the second player consists of a class of impulsive functions - piecewise constant functions with known (possible) discontinuity points. The given set is described by the control parameters of the first player. This problem is solved using a modular maximin control problem called the “mean deviation minimization” maximization problem. For the equivalent problem of a modular maximin control problem, the concept of a dual problem is introduced. Theorems establishing the connection between these problems are proven. Using these connections, an algorithm for solving the problem under consideration has been developed. The algorithm is based on comparing the values of the quality criterion of the problem dual to the linear maximin control problem on special classes of controls. The main research tool is the concept of support. The maximum value of the quality criterion for the problem dual to the linear maximin control problem is determined using a finite number of supports. To solve the problem under consideration, it is sufficient to find controls on special classes for which the value of the quality criterion for the problem dual to the linear maximin control problem is positive. This property is taken into account when developing the algorithm. An illustrative example is given.