Abstract
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.
