On the fixed-time stability and stabilization of autonomous dynamical systems. Application to some academic examples

This paper discusses a new characterization of the fixed-time stability of autonomous dynamical systems. We prove that the combination of the finite-time stability with the polynomial stability leads to fixed-time stabilities provided that both stabilities are obtained by the same Lyapunov function. We apply this result to the construction of new fixed-time stabilizing feedbacks for the famous double integrator problem; One is continuous while the other is discontinuous. This allows the construction of observers of double integrator in fixed-time followed by its fixed-time output feedbacks.