Stability and stabilizability

Abstract

Three types of stability and stabilizability are studied: exponential, asymptotic and Lyapunov. Discussions are based on linearization and Lyapunov’s function approaches. When analysing a relationship between controllability and stabilizability topological methods are used.

Bibliographical notes

The main stability test is taken from E.A. Coddington and N. Levinson [18]. The content of Theorem 7.4 is sometimes called the second Lyapunov method of studying stability. The first one is the linearization. Theorem 7.7 is due to M.A. Krasnoselski and P.P. Zabreiko [59]; see also J. Zabczyk [114]. A similar result was obtained independently by R. Brockett, and Theorem 7.11 is due to him, see R.W. Brockett [13]. He applied it to show that, in general, for nonlinear systems, local controllability does not imply asymptotic stabilizability. A converse to the Brockett theorem is studied in a recent paper [42] by R. Gupta, F. Jafari, R.J. Kipka and B.S. Mordukhovich. Theorem 7.9 is taken from J. Zabczyk [114], as is Theorem 7.10. Theorem 7.13 is due to M. Szafrański [92], see D. Aeyels and M. Szafrański [1] for some extensions, and Exercise 7.9 to H. Sussmann.