Undamped nonlinear beam excited by additive [formula omitted]-regular noise

Abstract

Stability of a nonlinear elastic beam excited by a space-time dependent nonrandom force or a L 2 -regular random noise f ( x , t ) (white in time, with general spatial covariance) is considered. It is supposed that the transversal displacement y of the beam with length l is governed by the quasilinear partial differential equation (PDE) ρ 0 h y t t + D y x x x x − [ N 0 + ε ∫ 0 l ( y x ′ ) 2 d x ′ ] y x x + k y = f ( x , t ) , 0 < x < l , t ≥ 0 . As a main result we show that its expected energy is linearly bounded at time t in spite of the presence of additive L 2 -regular space-time noise and cubic-type nonlinearities. Appropriate partial-implicit discretization with similar qualitative behavior, consistency and stability are discussed as well. The technique of Lyapunov-type functionals processing the information on the related energy E is exploited. We compare our results to those for the linear beam. In both cases, the expected energy E [ E ] ( t ) is nondecreasing and growing linearly in time t . In fact, for all nonrandom t ≥ s and additive L 2 -regular noise f with covariance operator Q ( f ) with finite trace, it satisfies the trace formula E [ E ] ( t ) = E [ E ] ( s ) + Trace ( Q ( f ) ) ( t − s ) / 2 .