Output-based stabilization for a one-dimensional wave equation with distributed input delay in the boundary control

In this paper, we consider the stabilization problem for a wave equation with distributed controller delay. Suppose that the output of the boundary controller is of the form alpha u(t) + beta u(t - tau) + integral(0)(-tau) g(eta) u(t + eta) d eta, where u(t) is the controller input, alpha, beta is an element of R are two constants of the controller, tau > 0 is the maximal time delay and g(eta) is an element of L-2[-tau, 0] which does not equal zero. Using the tricks of the Luenberger observer and partial state predictor, the delayed system can be transformed to a system which has no time delay. Based on this system with no time delay, we deduce a kind of dynamic feedback control law that can exponentially stabilize the delayed system.