Output Feedback Stabilization for a Class of First-Order Equation Setting of Collocated Well-Posed Linear Systems With Time Delay in Observation

A time delay present in the observation represents a mathematical challenge in an 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 the fixed time delay in observation, the problem for one-dimensional partial differential equations (PDEs) has been solved by the 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 multidimensional PDE 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 the well-posed and regular linear systems and solve it in the operator form. The result is then applied to the stabilization of a multidimensional Schrödinger equation.