On Some Properties of Sets of Bounded Controllability for Stationary Linear Discrete Systems with Total Control Constraints

We consider the problem of constructing reachable sets, i.e., sets of terminal states into which a system can be transferred from the origin in a fixed time, and 0-controllability sets, i.e., sets of initial states, from which a system can be transferred to the origin in a fixed time, for stationary linear discrete systems with a total control constraint. The representation of reachable and 0-controllable sets as linear transformations of superellipsoidal sets of finite and infinite dimensions is proved. A constructive method for describing the desired sets based on the apparatus of supporting half-planes is proposed, including for the limit sets of reachability and controllability. In the case of Euclidean spaces, the description is obtained explicitly. Examples are given. For a three-dimensional satellite motion control system in a near-circular orbit, reachability sets are modeled.