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 we present examples of such systems, specifically those for which we can prove that the related exponential families form a Riesz basis in corresponding appropriately defined Sobolev spaces. This makes it possible to choose ‘natural’ pairs of spaces: the state space observability space and the control space state space, depending on whether an observation or a control problem is studied, respectively, so that the observation and control operators are isomorphisms.