On the method of numerical simulation of limit reachable sets for linear discrete-time systems with bounded control

The paper considers the issues of numerical modeling of the limit reachable sets for linear discrete-time systems with convex control constraints. The method based on the principle of contraction mappings has been developed. This method is designed to construct an external estimate of the limit reachable set, which is a significant problem in control theory and analysis of dynamical systems. The application of the principle of contraction mappings makes it possible to obtain an estimate with an arbitrary order of accuracy in the sense of the Hausdorff distance. Moreover, the limit point up to the closure must coincide with the limit reachable set. The value of the compression ratio depends on the choice of the norm in the vector space, which, accordingly, influences the Hausdorff distance in the compact space, as well as the operator norm of the system matrix. To demonstrate the capabilities of the proposed method, a three-dimensional system with real eigenvalues is presented as an example. Additionally, an example for constructing the limit reachable set in the damping system of a high-rise structure located in a seismic zone is provided.