Exponential stabilization of a Rayleigh beam using collocated control

We consider a hinged elastic beam described by the Rayleigh beam equation on the interval [0, pi]. We assume the presence of two sensors: one measures the angular velocity of the beam at a point xi is an element of [0, pi] and the other measures the bending (curvature) of the beam at the same point. (If xi = 0 or xi = pi, then the second output is not needed.) The corresponding operator semigroup is unitary on a suitable Hilbert state space. These two measurements are advantageous because they make the open-loop system exactly observable, regardless of the point xi. We design the actuators and the feedback law in order to exponentially stabilize this system. Using the theory of collocated static output feedback developed in our recent paper [6], we design the actuators such that they are collocated, meaning that B = C*, where B is the control operator and C is the observation operator. It turns out that if xi is an element of (0, pi), then the actuators cause a discontinuity of the bending exactly at xi (this is the price, in this example, of having collocated actuators and sensors). This obliges us to use an extension of C to define the output signal in terms of the left and right limit of the bending at xi. We prove that, for all static output feedback gains in a suitable finite range, the closed-loop system is well posed and exponentially stable. This follows from the general theory in our paper [6], whose main points are recalled here.