Admissible controls and controllable sets for a linear time-varying ordinary differential equation (Q2001568)

This paper is on the ordinary differential linear system \[ y'(t) = A(t)y(t) + B(t)u(t) \quad (t \ge 0) \tag{1} \] with measurable scalar control $u(t)$ satisfying a control constraint $|u(t)| \le r.$ The matrix functions $A(t)$ and $B(t)$ are locally $L^\infty.$ The problem is driving an initial condition $y_0$ to the origin; if we denote by $y(t, 0, y_0, u)$ the trajectory of (1) satisfying $y(0) = y_0$ then the control $u(\cdot)$ is admissible if, for some $T > 0$ we have \[ y(T, 0, y_0, u) = 0.\tag{2} \] As the authors point out, this problem is related (but not equivalent) to controllability, where there are no bounds on the control. In the present linear-convex setting, the existence of an admissible control $u(t)$ is equivalent to the existence of a control that drives $y_0$ to $0$ in minimum time. The authors give a criterion (in five steps) to check whether a given $y_0$ can be driven to the origin in some time $T$ (depending on $y_0)$ by an admissible control.