Given any finite , during the time

Given any finite , during the time period , apply the backstepping-based nonlinear feedback which follows an application of the procedure in section 3 to the original –system in (1) using as defined in (27), instead of (3). Then, form a differential inequality similar to (24), we concluded that the –state of (1) cannot blow up during the time period if the design parameters are properly selected. Since , at , we switch the control inputs and into (3) and (23), respectively.We are now ready to state the main theorem of our paper.Theorem: For system (1), under the assumptions 1 and 2, if the control strategy (3 ) and (23 ) are applied, with an appropriate choice of the design parameters, the globally asymptotic stability of the closed loop system is achieved and the state as long as the initial condition is nonzero is regulated at the origin asymptotically. Moreover, if the above switching control strategy (27) is applied to (1), the state is regulated at the origin asymptotically when the initial condition is zero.