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Nonlinear Dynamics

, Volume 49, Issue 1–2, pp 31–47 | Cite as

Response of quasi-integrable Hamiltonian systems with delayed feedback bang–bang control

  • W. Q. ZhuEmail author
  • Z. H. Liu
Original Article

Abstract

The response of quasi-integrable Hamiltonian systems with delayed feedback bang–bang control subject to Gaussian white noise excitation is studied by using the stochastic averaging method. First, a quasi-Hamiltonian system with delayed feedback bang–bang control subjected to Gaussian white noise excitation is formulated and transformed into the Itô stochastic differential equations for quasi-integrable Hamiltonian system with feedback bang–bang control without time delay. Then the averaged Itô stochastic differential equations for the later system are derived by using the stochastic averaging method for quasi-integrable Hamiltonian systems and the stationary solution of the averaged Fokker–Plank–Kolmogorov (FPK) equation associated with the averaged Itô equations is obtained for both nonresonant and resonant cases. Finally, two examples are worked out in detail to illustrate the application and effectiveness of the proposed method and the effect of time delayed feedback bang–bang control on the response of the systems.

Keywords

Delayed feedback bang–bang control Gaussian white noise Quasi-integrable Hamiltonian system Stationary solution Stochastic averaging method 

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Copyright information

© Springer Science + Business Media B.V. 2007

Authors and Affiliations

  1. 1.Department of Mechanics, State Key Laboratory of Fluid Power Transmission and ControlZhejiang UniversityHangzhouP.R. China
  2. 2.Department of PhysicsZhejiang UniversityHangzhouP.R. China

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