Null Controllability — Continuous-Time Systems

This chapter studies null controllability of continuous-time linear systems with bounded controls. Null controllability of a system refers to the possibility of steering its state to the origin in a finite time by an appropriate choice of the admissible control input. If the system is linear and is controllable, then any state can be steered to any other location, for example, the origin, in the state space in a finite time by a control input. This implies that any controllable linear system is null controllable from any point in the state space. The control inputs that are used to drive certain states to the origin, however, might have to be large in magnitude. As a result, when the control input is limited by the actuator saturation, a linear controllable system might not be globally null controllable. In this situation, it is important to identify the set of all the states that can be steered to the origin with the bounded controls delivered by the actuators. This set is referred to as the null controllable region and is denoted as C