A finite-time stability concept and conditions for finite-time and exponential stability of controlled nonlinear systems

This paper introduces a finite-time stability concept for nonlinear systems and the corresponding notion of stabilizability of controls. The concept generalizes previous finite-time stability as it requires that the state trajectory satisfies a given bound at some time instant in a certain interval, whereas previous notions considers that the system stays within that bound over the entire interval. Here, the bound on the trajectory may represent a region arbitrarily close to the origin, and thus it is in tune with situations where contractive trajectories are required. This feature allows us to relate the proposed concept with the usual exponential stability concept, and simultaneously, with the previous finite-time concepts, thus clarifying the relations among them. As regards to controlled systems, we derive a sufficient condition for stabilizability, with the interpretation that the size of the time interval demanded by the control to drive the trajectory into a specified region is in inverse proportion with the size of the region. Moreover, we present a simple moving horizon implementation for a stabilizing (in the new sense) control that provides an exponentially stable controlled system. For stationary controls we can connect stabilizability in the sense here with the classical exponential sense. We show that the control is exponentially stabilizing whenever it is finite-time stabilizing and stationary. An illustrative example is included.