Lyapunov functions for quasi-Hamiltonian systems |
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Authors: | ZL Huang XL Jin WQ Zhu |
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Affiliation: | 1. Department of Mechanics, Zhejiang University, Hangzhou, 310027, PR China;2. State key Laboratory of Fluid Power Transmission and Control, Zhejiang University, Hangzhou, 310027, PR China |
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Abstract: | A procedure for constructing the Lyapunov functions and studying their asymptotic Lyapunov stability with probability one for quasi-Hamiltonian systems is proposed. For quasi-non-integrable Hamiltonian systems, the Hamiltonian (the total energy) is taken as the Lyapunov function. For quasi-integrable and quasi-partially-integrable Hamiltonian systems, the optimal linear combination of the independent first integrals in involution is taken as the Lyapunov function. The derivative of the Lyapunov function with respect to time is obtained by using the stochastic averaging method for quasi-Hamiltonian systems. The sufficient condition for the asymptotic Lyapunov stability with probability one of quasi-Hamiltonian systems is determined based on a theorem due to Khasminskii and compared with the corresponding necessary and sufficient condition obtained by using the largest Lyapunov exponent. Three examples are worked out to illustrate the proposed procedure and its effectiveness. |
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Keywords: | Stochastic stability Lyapunov function Stochastic averaging method |
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