Description of Limit Reachable and Null-Controllable Sets for Linear Discrete-Time Systems and Integral Second-Order Constraints

A discrete-time linear system with second-order integral control constraints is considered. The properties of the limit reachable and null-controllable sets of the system are investigated. Necessary and sufficient conditions for the boundedness of these sets are obtained, including for uncontrollable systems. It is proved that these sets are ellipsoids, when they are bounded, and their support function is constructed explicitly. Numerical simulations are performed.