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

Łukasz Rzepnicki

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. \(x \in [0, 1]\).

Keywords: nonhomogeneous string, one-dimensional wave equation, exponentially stable \(C_0\)-semigroup, Hilbert space.

Mathematics Subject Classification: 34L99, 47B44, 47D03.

Full text (pdf)