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Unformatted text preview: Lyapunov stability In mathematics , the notion of Lyapunov stability occurs in the study of dynamical systems . In simple terms, if all solutions of the dynamical system that start out near an equilibrium point x e stay near x e forever, then x e is Lyapunov stable . More strongly, if all solutions that start out near x e converge to x e , then x e is asymptotically stable . The notion of exponential stability guarantees a minimal rate of decay, i.e., an estimate of how quickly the solutions converge. The idea of Lyapunov stability can be extended to infinitedimensional manifolds, where it is known as structural stability , which concerns the behaviour of different but "nearby" solutions to differential equations. Contents 1 Definition for continuoustime systems 2 Definition for iterated systems 3 Lyapunov stability theorems o 3.1 Lyapunov second theorem on stability 4 Stability for linear state space models 5 Stability for systems with inputs 6 Example 7 Barbalat's lemma and stability of timevarying systems 8 References 9 See also Definition for continuoustime systems Consider an autonomous nonlinear dynamical system , where denotes the system state vector, an open set containing the origin, and continuous on . Without loss of generality, we may assume that the origin is an equilibrium. 1. The origin of the above system is said to be Lyapunov stable , if, for every > 0 , there exists a = ( ) > 0 such that, if , then , for every . 2. The origin of the above system is said to be asymptotically stable if it is Lyapunov stable and if there exists > 0 such that if , then . 3. The origin of the above system is said to be exponentially stable if it is asymptotically stable and if there exist , , > 0 such that if , then , for . Conceptually, the meanings of the above terms are the following: 1. Lyapunov stability of an equilibrium means that solutions starting "close enough" to the equilibrium (within a...
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This note was uploaded on 10/16/2011 for the course MECHATRONI 111 taught by Professor Jung during the Spring '11 term at Hanyang University.
 Spring '11
 Jung

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