Can state feedback stabilize a chaotic orbit uniformly and asymptotically in the sense of orbital stability?

In this Letter, we show that it is impossible to uniformly and asymptotically stabilize a chaotic orbit (in the sense of the classical orbital stability) by using state feedback control, for both discrete and continuous autonomous systems, if the autonomous chaotic system has a continuous vector field and the controller is a continuous function of the system state. We draw this conclusion from our new result stating that once a bounded orbit is uniformly and asymptotically orbitally stabilized, it must become periodic or converge to a periodic orbit.