Exponential bases in Sobolev spaces in control and observation problems for the wave equation

The Fourier method in control systems reduces the study of controllability/ observability to the study of related exponential families. In this paper w e present examples of such systems, specifically those for which we can pro ve that the related exponential families form a Riesz basis in correspondin g appropriately defined Sobolev spaces. This makes it possible to choose 'n atural' pairs of spaces: the state space observability space and the contro l space state space, depending on whether an observation or a control probl em is studied, respectively, so that the observation and control operators are isomorphisms.