Prescribed-Time Stabilization of Nonlinear Systems via Impulsive Regulation

This article studies the problem of prescribed-time stabilization for nonlinear systems, in which impulses are cautiously regulated not only to stabilize the system in a finite-time sense but also to adjust the settling time to fit the prescribed terminal time. By fetching and utilizing positive effect of impulses, constrains on continuous flows are effectively relaxed for prescribed-time stabilization of nonlinear systems. Meanwhile, different from traditional prescribed-time approaches, where continuous flows of the system are generally required to be globally stable, it shows in this article that even systems involving unstable flows can be prescribed-time stabilized via the proposed impulsive regulation, but as a tradeoff, a constrained initial condition is needed for controller design. Thus, the main functions of impulsive regulation in this article are balancing unstable dynamics of the system, and regulating the settling time to fit the prescribed terminal time. As an application, the proposed impulsive regulation scheme is exploited in prescribed-time stabilization of affine dynamical systems involving uncertain input noises. Two examples, including the one focusing on the spin stabilization of spacecrafts, are given to verify the results.