Sidewise control of 1-d waves

We analyze the sidewise controllability for the variable coefficients one-dimensional wave equation. The control is acting on one extreme of the string with the aim that the solution tracks a given path at the other free end. This sidewise control problem is also often referred to as nodal profile or tracking control. First, the problem is reformulated as a dual observability property for the corresponding adjoint system. Using sidewise energy propagation arguments the sidewise observability is shown to hold, in a sufficiently large time, in the class of BV-coefficients. We also present a number of open problems and perspectives for further research.