Abstract. Finite-time stability is defined for equilibria of continuous but non-Lipschitzian autonomous systems. Continuity, Lipschitz continuity, and Hölder continuity of the settling-time function are studied and illustrated with several examples. Lyapunov and converse Lyapunov results involving scalar differential inequalities are given for finite-time stability. It is shown that the regularity properties of the Lyapunov function and those of the settling-time function are related. Consequently, converse Lyapunov results can only assure the existence of continuous Lyapunov functions. Finally, the sensitivity of finite-time-stable systems to perturbations is investigated.Key words. stability, finite-time stability, non-Lipschitzian dynamics AMS subject classifications. 34D99, 93D99PII. S03630129973213581. Introduction. The object of this paper 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.Classical optimal control theory provides several examples of systems that exhibit convergence to the equilibrium in finite time [17]. A well-known example is the double integrator with bang-bang time-optimal feedback control [2]. These examples typically involve dynamics that are discontinuous. Discontinuous dynamics, besides making a rigorous analysis difficult (see [9]), may also lead to chattering [10] or excite high frequency dynamics in applications involving flexible structures. Reference [8] considers finite-time stabilization using time-varying feedback controllers. However, it is well known that the stability analysis of time-varying systems is more complicated than that of autonomous systems. Therefore, with simplicity as well as applications in mind, we focus on continuous autonomous systems.Finite [21] suggests, based on a scalar example, that systems with finite-settling-time dynamics possess better disturbance rejection and robustness properties. However, no precise results exist for multidimensional systems. This paper attempts to fill these gaps.In section 2, we define finite-time stability for equilibria of continuous autonomous systems that have unique solutions in forward time. Continuity and forward uniqueness render the solutions continuous functions of the initial conditions, so that the

This paper studies properties of homogeneous systems in a geometric, coordinate-free setting. A key contribution of this paper is a result relating regularity properties of a homogeneous function to its degree of homogeneity and the local behavior of the dilation near the origin. This result makes it possible to extend previous results on homogeneous systems to the geometric framework. As an application of our results, we consider finite-time stability of homogeneous systems. The main result that links homogeneity and finite-time stability is that a homogeneous system is finite-time stable if and only if it is asymptotically stable and has a negative degree of homogeneity. We also show that the assumption of homogeneity leads to stronger properties for finite-time stable systems.

A class of bounded continuous time-invariant finite-time stabilizing feedback laws is given for the double integrator. Lyapunov theory is used to prove finite-time convergence. For the rotational double integrator, these controllers are modified to obtain finite-time-stabilizing feedbacks that avoid "unwinding."

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This paper examines finite-time stability of homogeneous systems. The main result is that a homogeneous system is finite-time stable if and only if it is asymptotically stable and has a negative degree of homogeneity.

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We derive a continuous nonlinear control law for spacecraft attitude tracking of arbitrary continuously differentiable attitude trajectories based on rotation matrices. This formulation provides almost global stabilizability, that is, Lyapunov stability of the desired equilibrium of the error system as well as convergence from all initial states except for a subset for which the complement is open and dense. This controller thus overcomes the unwinding phenomenon associated with continuous controllers based on attitude representations, such as quaternions, that are not bijective and without resorting to discontinuous switching. The controller requires no inertia information, no information on constant-disturbance torques, and only frequency information for sinusoidal disturbance torques. For slew maneuvers (that is, maneuvers with a setpoint command in the absence of disturbances), the controller specializes to a continuous, nonlinear, proportional-derivative-type, almost globally stabilizing controller, in which case the torque inputs can be arbitrarily bounded a priori. For arbitrary maneuvers, we present an approximate saturation technique for bounding the control torques.

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