Systems and Control
Broad in scope, this text shows the multidisciplinary role of dynamics and control, presents neural networks, fuzzy systems, and genetic algorithms, and provides a self-contained introduction to chaotic systems.
TABLE OF CONTENTS Systems and Control
4.8: Stability of Nonlinear Systems
4.8 Stability of Nonlinear Systems
In this section we will be concerned with stability analysis of nonlinear time-varying dynamical system models described by (4.1) that is, models of the form
where 
is the state vector and f is a vector-valued function.
To proceed, we need the following lemma. Let B q( 0) denote an open ball with the center at 0 and radius q, and let 
be a closed ball, a compact set, that contains the origin x = 0 in its interior.
Lemma 4.2
For a continuously differentiable function 
, where V ( 0) = 0, we have V ( x) > 0 for all 
, if and only if there is a continuous, strictly increasing function 
such that ?(0) = 0, ?( r) > 0 for r > 0, and V ( x) ? ?( r), where r = ? x ? and r ? R.
| Proof | Sufficiency ( ?) is clear. To prove necessity ( ?), we define, following Willems [300, p. 21], the function  Note that the function ? is well-defined because the set  is compact and V is continuous. Therefore, by the Weierstrass theorem on page 650, the function V achieves its infimum over the set (4.48). We will now show that ? is continuous on |
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