Boundary control, quiet boundaries, super-stability and super-instability
The equation studied here is a copy of the classical wave equation: \(\rho\partial u^2/\partial t^2= EA_0\partial u^2/\partial x^2\), \(0< x< 1\), \(t\geq 0\), with zero initial displacement at \(0\), and \((EA_0/\eta)\partial u(1,t)/\partial x= -\partial u(1,t)/\partial t\), implying that the damping boundary control is proportional to velocity. The speed of compressive stress propagation is given by \(c^2= EA_0/\rho\). Thus the state equation is: \(c^{-2}\partial u^2/\partial t^2= \partial u^2/\partial x^2\). It has the classical solution: \(u(x,t)= e^{i\omega t}v(x)\). Its real part is: \(\partial v^2/\partial x^2+ (\omega c^{-2})v= 0\). The author introduces Green's function for this type of boundary condition. A crucial parameter \(\alpha=-i\tan\omega c^{-1}\) is introduced. For \(\alpha= 1\) all poles of the Laplace transform lie in the left half of the complex plane, and the system is ``super-stable''. For \(\alpha=-1\) the displacement becomes infinite in finite time, which the author calls the ``super-instability'' condition. Because of the boundary control conditions, this equation is not selfadjoint and one has to thread gently in applications of spectral analysis.
scientific article; zbMATH DE number 2175168