Advances in converse and control Lyapunov functions.

摘要

Lyapunov's second method states that if a Lyapunov function strictly decreases along trajectories of a differential equation, then the equilibrium point is asymptotically stable. The great utility of this method lies in not having to explicitly solve the differential equation in order to demonstrate stability properties for equilibria. However, there is no general method for finding Lyapunov functions. Consequently, the converse of Lyapunov's second method is of interest; that is, under what conditions is it possible to construct an appropriate so-called converse Lyapunov function?; The primary contributions of this dissertation are a collection of results on the existence of converse Lyapunov functions within various contexts. We first consider differential inclusions and stability with respect to two measures. This subsumes the standard notions of asymptotic stability such as uniform asymptotic stability. When considering differential inclusions, we typically will have multiple solutions and, consequently, we present converse results for two different notions of stability; one corresponding to the behavior of all solutions and one corresponding to the behavior of at least one solution. These are referred to as strong and weak stability notions, respectively. We present converse results for both notions.; As an application of our result on the existence of a converse Lyapunov function for weak asymptotic stability, we consider what have come to be known as control Lyapunov functions by examining controlled differential equations, which corresponds to the weak asymptotic stability of an appropriate differential inclusion. Then one wishes to find a function such that, with an appropriate control, the function acts as a Lyapunov function. We demonstrate that, under a mild controllability condition, there always exists a locally Lipschitz control Lyapunov function. Furthermore we show how such a control Lyapunov function is used to construct a robust state feedback stabilizer.; Finally, we parallel the results for differential inclusions by considering difference inclusions and stability with respect to two measures. We state and prove several new theorems related to the existence of discrete-time converse Lyapunov functions. Again, we consider strong and weak notions of stability and we present converse results for both notions.