Designated-time stabilizing strategy for a class of time-varying nonlinear systems and its application to pendulum system

In this paper, a new scheme is proposed to stabilize a class of nonlinear systems within a designated time, mainly motivated by two unsolved problems: the singularity caused by infinity control magnitudes at the prescribed time instant and the lack of formulating/manipulating state responses after the prescribed time. Under the proposed scheme, an exquisite controller is constructed by entailing both the state scaling approach and a finite-time stabilizing procedure. It is shown that the obtained control is well defined with bounded magnitude on the whole-time horizon and ensures the existence of the solutions of the closed-loop system. Superior to the existing prescribed time stabilization results, the proposed method is capable of ensuring that the states of the closed-loop system converge to a small compact set within the designated time, stay inside it thereafter and perform finite-time convergence eventually. Simulation examples are presented to illustrate the effectiveness of the proposed approach.