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

Abstract

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 $\begin{cases} v_{tt}(x,t) - \frac{1}{\rho}v_{xx}(x,t) = 0, x \in [0,1], t \in [0, \infty),\\ v(0,t) = 0, v_x(1,t) + hv_t(1,t) = 0, \\ v(x,0) = v_0(x), v_t(x,0) = v_1(x),\end{cases}$ where $\rho$ 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 $C_0$-semigroup of contractions in the space $H$ under assumptions that $\text{Re}\; h \gt 0$ and the density function is of bounded variation satisfying $0 \lt m \leq \rho(x)$ for a.e. $x \in [0, 1]$.