Stochastic stability of viscoelastic systems under Gaussian and Poisson white noise excitations.

*(English)*Zbl 1398.74041Summary: As the use of viscoelastic materials becomes increasingly popular, stability of viscoelastic structures under random loads becomes increasingly important. This paper aims at studying the asymptotic stability of viscoelastic systems under Gaussian and Poisson white noise excitations with Lyapunov functions. The viscoelastic force is approximated as equivalent stiffness and damping terms. A stochastic differential equation is set up to represent randomly excited viscoelastic systems, from which a Lyapunov function is determined by intuition. The time derivative of this Lyapunov function is then obtained by stochastic averaging. Approximate conditions are derived for asymptotic Lyapunov stability with probability one of the viscoelastic system. Validity and utility of this approach are illustrated by a Duffing-type oscillator possessing viscoelastic forces, and the influence of different parameters on the stability region is delineated.

##### MSC:

74D10 | Nonlinear constitutive equations for materials with memory |

93E15 | Stochastic stability in control theory |

93D20 | Asymptotic stability in control theory |

##### Keywords:

stochastic stability; Gaussian and Poisson noise; Lyapunov function; stochastic averaging; viscoelastic system
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\textit{X. Li} et al., Nonlinear Dyn. 93, No. 3, 1579--1588 (2018; Zbl 1398.74041)

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