Numerical encoding of sampled controls and an approximation metric criterion for the solvability of a guidance game problem

A game problem of guaranteed guidance in the class of positional strategies is considered for a conflict-controlled system with affine scalar controls in the equation of the system. Simplified sampled analogs of quasi-strategies, i.e., of nonanticipating program responses of the first player to the controls of the second player, are introduced. The nonanticipating property is characterized in metric terms with the use of numerical images (codes) of argument controls and response controls. A class of nonanticipating transformations is introduced that is approximately equivalent by the criterion of the solvability of the game problem to the class of positional strategies. The elements of this class as transformations of the numerical codes of controls are characterized by the 1-Lipschitzness property. A numerical algorithm for checking the solvability of the problem in this class is described. The complexity order of the algorithm is close to that of the approximation variant of the classical program construction.