Treatises on vibrations devote large space to study the dynamical behavior of
an elastic system subject to known external tractions. In fact, usually a
"system" is not an isolated body but it is part of a chain of mechanisms which
disturb the "system" for example due to the periodic rotation of shafts. This
kind of problem has been rarely studied in control theory. In the specific case
we shall study, the case of a viscoelastic string, the effect of such external
action is on the horizontal component of the traction, and so it affects the
coefficients of the corresponding wave type equation, which will be time
dependent. The usual methods used in controllability are not naturally adapted
to this case. For example at first sight it might seem that moment methods can
only be used in case of coefficients which are constant in time. Instead, we
shall see that moment methods can be extended to study controllability of a
viscoelastic string subject to external traction and in particular we shall
study a controllability problem which is encountered in the solution of the
inverse problem consisting in the identification of a distributed disturbance
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