Strict Lyapunov functions for homogeneous finite-time second-order systems

In this note we propose some families of differentiable strict Lyapunov functions (LFs) (whose derivative along the system's solutions is negative definite) for two classes of second-order systems. One of these classes arises in finite-time controlled systems, while the second class arises as observation error dynamics of finite-time observers. The design of the LFs is inspired by the fact that, for the considered systems, the construction of a weak (or non-strict) LF (whose derivative along the system's solutions is negative semidefinite) is relatively simple, and the LaSalle's conditions always hold to prove asymptotic stability at the origin. Then, the construction of strict LFs consists in transforming a weak LF into a strict one.