| While non-Lipschitzian effects such as Coulomb friction abound in nature, most of the available techniques for feedback stabilization yield closed-loop systems with Lipschitzian dynamics. The convergence in such systems is at best exponential with infinite settling time. In this dissertation, we are interested in finite-settling-time behavior, that is, finite-time stability. The object of this dissertation is to provide a rigorous foundation for the theory of finite-time stability of continuous autonomous systems and motivate a closer examination of finite-time stability as a possible objective in control design.;Accordingly, the notion of finite-time stability is precisely formulated and properties of the settling-time function are studied. Lyapunov and converse Lyapunov results involving scalar differential inequalities are obtained. It is shown that, under certain conditions, finite-time-stable systems possess better disturbance rejection and robustness properties.;As an application of these ideas, we consider the finite-time stabilization of the translational and rotational double integrators. In the case of the rotational double integrator, the topology of the cylindrical state space renders continuous global stabilization impossible and hence the closed-loop system possesses saddle points. Because of the non-Lipschitzian character of the feedback law, these saddle points are, in fact, finite-time repellers--equilibria from which solutions can spontaneously and unpredictably depart.;It is generally believed that models obtained from classical dynamics are completely deterministic. We present a counterexample to this widely held notion. The counterexample consists of a particle moving along a nonsmooth (once, but not twice, differentiable) constraint in a uniform gravitational field. The equation of motion obtained by applying classical Lagrangian dynamics contains non-Lipschitzian terms and, consequently, the dynamics of the system exhibit a finite-time saddle. The system possesses multiple solutions starting at the finite-time saddle. Every such solution corresponds to the particle spontaneously departing from the equilibrium position.;Homogeneity is introduced as a tool for analyzing finite-time stability of higher dimensional systems. For this purpose a coordinate-free notion of homogeneity is formulated. Our main result is that a homogeneous system is finite-time stable if and only if it is asymptotically stable and has negative degree of homogeneity. Under the assumption of homogeneity, results related to finite-time stability can be strengthened. Finally, homogeneity is exploited to obtain a finite-time stabilizing controller for a chain of integrators and hence prove that every controllable linear control system can be finite-time stabilized through a continuous feedback law. |
