On the Problem of Semiinfinite Beam Oscillation with Internal Damping*

We study the Cauchy problem for the equation of the form (*) ü(t) + (αA+ B)u̇(t) + (A+G)u(t) = 0 where A, B, and G are operators in a Hilbert space H with A selfadjoint, σ(A) = [0,∞), B > 0 bounded, and G symmetric and Asubordinate in a certain sense. Spectral properties of the corresponding operator pencil L(λ) := λI +λ(αA+B) + A+G are studied, and existence and uniqueness of generalized and classical solutions of the Cauchy problem are proved. Equations of the type (*) include, e.g., an abstract model for the problem of semiinfinite beam oscillations with internal damping. This article is based on the lecture delivered at VII Crimean Autumn Mathematical School-Symposium on Spectral and Evolutionary Problems, Crimea, Ukraine, September 18–29, 1996. Introduction The aim of the present article is to study some class of differential equations and corresponding operator pencils in a Hilbert space, which provide abstract models for many problems in elasticity theory, hydrodynamics, control theory etc. Consider, for example, a visco-elastic semiinfinite beam placed in viscous external medium. Its small transverse oscillations are described in dimensionless coordinates by the equation (cf. [P1]) (1) α ∂u ∂t∂x4 + ∂u ∂x4 + ∂ ∂x ( g(x) ∂u ∂x ) + β(x) ∂u ∂t + ∂u ∂t2 = 0, x > 0, t > 0. Here u(x, t) is the transverse displacement of the beam at point x and time t; α > 0 is a small parameter specifying internal damping, β(x) determines external damping, and g(x) describes tension force distribution. Suppose for simplicity that the left beam end is clamped, i.e. that u(x, t) satisfies the boundary conditions (2) u(0, t) = ∂u(x, t) ∂x |x=0= 0, *This work was partially supported by Russian Fund of Basic Research, grant No. 96-01-01292. 1991 Mathematics Subject Classification. 47A56, 47N20, 35P05.