Stochastic stability of some mechanical systems with a multiplicative white noise

Pages: 91 - 99, Issue Special, July 2003

 Abstract        Full Text (175.3K)              

Boris P. Belinskiy - Department of Mathematics, University of Tennessee at Chattanooga, 615 McCallie Avenue, Chattanooga, TN 37403-2598, United States (email)
Peter Caithamer - U.S. Military Academy, West Point, NY10996, United States (email)

Abstract: We discuss the behavior, for large values of time, of a class of linear mechanical systems with a multiplicative white noise in its parameters. The initial conditions may be random as well but are independent of white noise. It is well known that a deterministic linear mechanical system with viscous damping is stable, i.e., its energy approaches zero as time increases. We calculate the expected energy and check that this behavior takes place in the case when the initial conditions are random but the parameters are not. When the parameters contain a random noise the expected energy may be infinite, approach zero, remain bounded, or increase with no bound. We give necessary and sufficient conditions for stability of the systems considered in terms of the roots of an auxiliary equation.

Keywords:  stochastic partial differential equation, expected energy, Gaussian noise, stability.
Mathematics Subject Classification:  Primary: 60H15, 34F05, 60H10; Secondary: 35B35.

Received: June 2002;      Revised: March 2003;      Published: April 2003.