Scanning control for the string equation

It is well known that for pointwise control problems we generally have a lack of robustness with respect to the location of the actuator. More precisely, any open subset of the considered domain ([0, π] in our case) contains points for which controllability fails, see, for instance, [1] and references therein. A remedy which has been proposed in Berggren [2] is to consider an actuator which moves according to a prescribed law. To describe the problem introduced in [2], let α, β and ω be positive numbers and define the given equation. We consider the initial and boundary value problem with a given equation where, for every real a, δa stands for the Dirac measure at a. The above system describes the linear vibrations of an elastic string with a pointwise scanning actuator. This means that, for very t ⩾ 0 the actuator is positioned at ϱ(t) at instant t. It is easy to see that if α/π ∈ Q and β = 0 (i.e., for some fixed actuators) the above system is not approximately controllable.