A time delay present in the observation represents a mathematical challenge in output feedback stabilization for linear infinite dimensional systems. It is well known that for a linear hyperbolic system, a stabilizing output feedback control may become unstable when the observation has a time delay. For fixed time delay in observation, the problem for 1-d PDEs has been solved by observer-based feedback in the time interval where the observation is available and the predictor where the observation is not available. However, the generalization to multi-dimensional partial differential equation systems has been a long standing unsolved problem. In this paper, we investigate the problem from operator point of view for abstract first-order equation setting of infinite-dimensional systems. We formulate the problem in the framework of thewell-posed and regular linear systems and solve it in the operator form. The result is then applied to the stabilization of a multi-dimensional Schr\"{o}dinger equation.

In this paper, we study the output feedback exponential stabilization of a one-dimensional wave equation with control matched nonlinear disturbance. The well-posedness of the open-loop system which is essentially important for the designs of observer and controller, is verified. By designing an infinite-dimensional unknown input state observer in place of the conventional extended state observer, we estimate the total disturbance in real time. Moreover, we prove the convergence of the disturbance estimator. Based on the disturbance estimator, we design a state observer by compensating the total disturbance, and an estimated state feedback controller to exponentially stabilize the original system by means of the Riesz basis approach. Some numerical simulations are presented.

In this paper, we are concerned with dynamic stabilisation for a one-dimensional Euler–Bernoulli beam equation with boundary moment control and matched nonlinear uncertain disturbance. In the case of no disturbance, we show that a boundary feedback control law exponentially stabilises the system and Riesz basis generation holds for the closed-loop system. The well-posedness of the system in the sense of Salamon-Weiss, which is essentially important for the design of observer, is verified. We design an infinite-dimensional disturbance estimator, which doesn't need slow variation or high gain or boundedness of the derivation of the disturbance, to estimate the total disturbance. Based on the disturbance estimator, we design an output feedback control law. The Riesz basis generation and exponential stability of a couple system including the original equation is proved. Moreover, the boundedness of the closed-loop system is verified. Some numerical simulations are presented to illustrate the results.