On robustness of a class of homogeneous continuous finite time controllers

This paper gives a Lyapunov based proof of robustness of a class of finite time controllers applied to the double integrator system. The literature of continuous finite time stabilisation contains the proof of finite time stability when continuous disturbances with a Lipschitz upper bound appear in the system dynamics. It is also known that continuous finite time controllers render the trajectories ultimately bounded for persisting disturbances. However, proving robustness of continuous finite time controllers to continuous disturbances with a non-Lipschitz upper bound is challenging. The main contribution of the paper is that it identifies a C¹ Lyapunov function to prove uniform asymptotic stability as well as uniform finite time stability in the presence of a class of disturbances that have non-Lipschitz upper bound.