Iterations of stability and the evasion problem with a constraint on the number of switchings of the formed control

This paper is concerned with one variant of the programmed iterations method used for solving the differential game of guidance-evasion. The proposed procedure is connected with iterations on the basis of the property of stability of sets introduced by N.N. Krasovskii. A relationship is established between the resulting iteration procedure and the solution to the evasion problem under a constraint on the switching number of the formed control: the stability iterations define the set of successful solvability of the above-mentioned problem. It is proved that the guaranteed realization of evasion is possible if and only if (guaranteed) strong evasion (namely, the evasion with respect to neighborhoods of sets defining the guidance-evasion game) is realizable. A representation of strategies guaranteeing the evasion with constraints on the switching number is presented. The concrete operation of every such strategy consists in the formation of constant control extruding the trajectory from the set corresponding to the next iteration on the basis of the stability operator. The duration of operation of the above-mentioned control is defined in terms of the result of the multifunctional employment defined on the trajectory space; the values of this multifunctional are nonempty subsets of the remaining time interval. Attention is given to problems involved in the convergence, in the sense of Hausdorff metric, of fragments of sets which are realized by the iteration procedure. On this basis, conditions for convergence (in the Hausdorff metric) for the sets-iterations themselves are obtained.