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IMA Journal of Mathematical Control and Information 2003 20(1):105-127; doi:10.1093/imamci/20.1.105
© 2003 by Institute of Mathematics and its Applications
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Lyapunov stability of abstract nonlinear dynamic system in Banach space

Gen-Qi Xu1 and Siu Pang Yung2

1 Department of Mathematics of Shanxi University, TaiYuan 030006, People's Republic of China 2 Department of Mathematics, University of Hong Kong, People's Republic of China

The Lyapunov stability theory for nonlinear time-varying dynamic system in Banach space is given in this paper. The Lyapunov stable theorem and the Barbashin–Krasovskii–LaSalle invariant set principle in classical theory are extended to infinite-dimensional Banach space. Under the assumptions of the existence of solution and the additive property of motions, sufficient and necessary conditions for uniform stability and uniform asymptotic stability are obtained, and the Lyapunov functions are explicitly constructed. This extension can be used as a criterion of stability for continuous and discontinuous systems.

Keywords: Banach space; abstract nonlinear dynamic equation; Lyapunov stability.


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