Finite-time stabilization of affine nonlinear systems

The finite-time stabilization problem deals with the enforcement of a quantitative bound on the state trajectory of a dynamical system along a finite-time horizon. Sufficient conditions have been provided to solve this problem for different classes of nonlinear systems. However, by implying a certain degree of conservatism, these conditions are only sufficient. This work contributes to fill this gap, by giving a necessary and sufficient condition for the finite-time stabilization of affine nonlinear systems. First a necessary and sufficient condition available in literature to check finite-time stability (FTS) of nonlinear systems is extended to the case of continuously differentiable Lyapunov-like functions. Thanks to the obtained smoothness of the Lyapunov function and by a proper definition of a FTS Control Lyapunov Function, we show that it is possible to apply the Sontag’s universal formula to derive a necessary and sufficient condition to solve the stabilization problem for affine nonlinear systems.