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Treatises on vibrations devote large space to study the dynamical behavior ofan 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 whichdisturb the "system" for example due to the periodic rotation of shafts. Thiskind of problem has been rarely studied in control theory. In the specific casewe shall study, the case of a viscoelastic string, the effect of such externalaction is on the horizontal component of the traction, and so it affects thecoefficients of the corresponding wave type equation, which will be timedependent. The usual methods used in controllability are not naturally adaptedto this case. For example at first sight it might seem that moment methods canonly be used in case of coefficients which are constant in time. Instead, weshall see that moment methods can be extended to study controllability of aviscoelastic string subject to external traction and in particular we shallstudy a controllability problem which is encountered in the solution of theinverse problem consisting in the identification of a distributed disturbancesource.

Boundary controllability and source reconstruction in a viscoelastic string under external traction