Prescribed-Instant Stabilization for Second-Order Systems With Unmatched Uncertainties

This brief investigates the stabilization of second-order systems subject to mismatched uncertainties. We present prescribed-instant stability, a unique form of prescribed-time stability, where the uncertain second-order system is driven to the origin exactly at the prescribed time instant$T_{p}$. A strict Lyapunov method proof is presented on two sides, i.e., the system converges to and reaches zero not after nor before$T_{p}$. The former is very common in existing articles about prescribed-time stability. The latter is usually ignored, but it is very important to obtain the prescribed-instant stability in this brief. We present the controllers in two cases based on different mismatched uncertainties. Simulations validate the effectiveness of the proposed method.