The evasion problem under uncertainty is considered for discrete-time systems with an initially linear structure and state constraints, where controls u, U, and v act; u and v enter additively, and U enters into the system matrix. In the considered control synthesis problem, which we call the enhanced evasion problem, the aim of v is either to avoid the trajectory to hit a given terminal set at a given final moment, as well as a sequence of sets specified at previous moments, or to violate at least one of the state constraints, whatever the admissible realizations of u and U. The presence of U introduces nonlinearity into the systems and leads to bilinear type systems. It is assumed that the terminal and intermediate sets are parallelepipeds, the controls u and v are constrained by parallelotope-valued constraints, U by interval constraints, and the state constraints are specified in the form of zones. A polyhedral method for synthesizing controls v is developed using polyhedral (parallelepiped-valued) tubes, which can be found from recurrence relations using explicit formulas. To obtain a solution to the problem under consideration, a solution to an auxiliary one-step polyhedral evasion problem with bilinearity is found. Its connections with the problems of interval analysis concerning the so-called sets of quantifier solutions to interval equations are noted. Examples illustrating the efficiency of the method are given.
E. K. Kostousova (Fri,) studied this question.