State reconstruction of the wave equation with general viscosity and non-collocated observation and control
It is considered the problem of reconstructing the initial condition (state) for the input-state-output \(1D\) system with distributed parameters described by hyperbolic partial differential equation and controlled/observed boundary conditions \[ \displaylines{\displaystyle{y_{tt} - y_{xx} + a(x)y_t = 0\ ;\ 0<x<1\;,\;t>0}\cr \displaystyle{y(0,t) = 0\ ,\ y_x(1,t) = u(t)}\cr O(t) = y_x(0,t)} \] The problem is solved in terms of Luenberger observers, requiring analysis of stability and stabilization of the above system. Stabilization is achieved by local feedback \(u(t)=\alpha y_t(1,t)\) The paper also makes use of the approximation based on the method of lines.
scientific article; zbMATH DE number 7358723
scientific article