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Generating the exponentially stable C0-semigroup in a nonhomogeneous string equation with damping at the end

Rzepnicki Ł.

Opuscula Mathematica

2013

Vol. 33, no. 1

151-162

Small vibrations of a nonhomogeneous string of length one with left end fixed and right one moving with damping are described by the one-dimensional wave equation [formula] where ρ is the density of the string and h is a complex parameter. This equation can be rewritten in an operator form as an abstract Cauchy problem for the closed, densely defined operator B acting on a certain energy space H. It is proven that the operator B generates the exponentially stable C0-semigroup of contractions in the space H under assumptions that Re h > 0 and the density function is of bounded variation satisfying 0 < m ≤ ρ(x) for a.e. x ∈ [0; 1].