Y. Sarac, E. Zuazua. Sidewise control of 1-d waves. J Optim Theory Appl. Vol. 193, pp. 931–949 (2022). https://doi.org/10.1007/s10957-021-01986-w
Abstract. 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 or profile at the other free end. This sidewise profile control problem is also often referred to as nodal profile or tracking control. The problem is reformulated as a dual observability property for the corresponding adjoint system, which is proved by means of sidewise energy propagation arguments in a sufficiently large time, in the class of BV -coefficients. We also present a number of open problems and perspectives for further research.