Controlling Transport and Synchronization in Non-identical Inertial Ratchets

In this paper, transport control and synchronization are investigated between two periodically driven non-identical, inertial ratchets that are able to exhibit directed transport. One of the two ratchets is acting as a drive system, while the other one represents the response system. Based on the Lyapunov stability theorem, the essential conditions, under which the error nonlinear system is transformed into an equivalent passive system and globally asymptotically stabilized at equilibrium points, are established. With these results, synchronization, not only between two non-identical ratchets with known parameters but also between two different uncertain ratchets, are realized via adaptive passive controllers and parameter update algorithm. The direction of transporting particles can be dominated along expected one and it is useful to control the motion of tiny particles, ratchetlike devices in nanoscience.