WIAS Preprint No. 181, (1995)

Stochastic stability of structures under active control with distributed time delays


  • Karmeshu, Prof.
  • Schurz, Henri

2010 Mathematics Subject Classification

  • 60H20 65C20 70K20 70Q05 82C31 86A15


  • Stochastic stability, Lyapunov exponents, Exit frequencies, Weeak and strong time delay, Seismic excitation, Active control, Stochastic differential equations, Implicit numerical methods, Numerical mean square and almost sure stability




The pathwise behaviour of a single degree of freedom (SDOF) system with symmetric nonlinearity and distributed delays is investigated under the presence of seismic excitation and multiplicative noise. Besides distributed time delays and finite build-up time of control force are taken into consideration. The system is modelled as stochastic integro-differential equation with exponential type kernels. Interpreting stochastic equations in Stratonovich sense, stochastic stability is analyzed in terms of Lyapunov exponents. Estimates of frequencies with which sample paths of displacement of SDOF system cross certain critical values are also obtained. Studies of stochastic linear and nonlinear systems are carried out by resorting to numerical techniques for the solution of (ordinary) stochastic differential equations.

Appeared in

  • Applications of Statistics and Probability, M. Lemaire, J. L. Favre, A. Mebarki, eds., A.A. Balkema Publishers, Rotterdam, 1995, pp. 1111--1119

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