Simultaneous exact controllability and some applications

We study the exact controllability of two systems by means of a common fini te-dimensional input function, a property called simultaneous exact control lability. Most of the time we consider one system to be infinite-dimensiona l and the other finite-dimensional. In this case we show that if both syste ms are exactly controllable in time T-0 and the generators have no common e igenvalues, then they are simultaneously exactly controllable in any time T > T-0. Moreover, we show that similar results hold for approximate control lability. For exactly controllable systems we characterize the reachable su bspaces corresponding to input functions of class H-1 and H-2. We apply our results to prove the exact controllability of a coupled system composed of a string with a mass at one end. Finally, we consider an example of two in finite-dimensional systems: we characterize the simultaneously reachable su bspace for two strings controlled from a common end. The result is obtained using a recent generalization of a classical inequality of Ingham.