Control of vibrating structures with internal damping

We begin our talk with explanations of various types of controllability for distributed parameter systems and approaches to solving this problem. Then we consider a vibrating beam and ask the question: can one apply an external force in a way that brings the beam to rest in a finite time interval? In the absence of friction, the answer is well known and positive, in other words the system is “null-controllable”. However, the answer is much less well understood in the important case where there is a friction known as internal or structural damping. The intensity of the damping is typically modelled by two parameters, which we label ρ, α, and the beam is described by the equation

u_tt + ∆^2 u + ρ ∆^α u_t = Bf, x ∈ (0, l), t > 0,

with a positive constant ρ, α ∈ [0, 2], and where B is the control operator. We consider distributed and boundary controls and prove null-controllability for α < 3/2 and any ρ. Our results sharply improve, in one dimension, on previous works and in some ways can be viewed as definitive.