Controllability in Linear Autonomous Systems with Positive Controllers

This paper presents results on the controllability of the autonomous linear control system in $R^n $, $\dot x = Ax + Bu$, where $u \in \Omega \subset R^m $ without the assumption that the origin in $R^m $ is interior to $\Omega $. Necessary and sufficient conditions are given for null-controllability (controllability of each point in some neighborhood of the origin to the origin) and global null-controllability with uniformly bounded controllers. This paper extends some results of Saperstone and Yorke who considered the problem of the controllability of the above system with $m = 1$ and $\Omega = [0,1]$ and obtained necessary and sufficient conditions for controllability for this system. Corollaries to the main result include existence of time-optimal controllers and controllability of nonlinear systems. An example of control of an economic system is presented.