  We are motivated by the problem of control for a non-homogeneous elastic
string with memory. We reduce the problem of controllability to a non-standard
moment problem. The solution of the latter problem is based on an auxiliary
Riesz basis property result for a family of functions quadratically close to
the nonharmonic exponentials. This result requires the detailed analysis of an
integro-differential equation and is of interest in itself for Function Theory.
Controllability of the string implies observability of a dual system.
