Well-posedness and regularity of partial differential equation control systems

Wellposed and regular linear systems are a quite general class of linear infinite-dimensional systems, which cover many control systems described by partial differential equations with actuators and sensors supported at isolated points, sub-domain, or on a part of the boundary of the spatial region. This class of infinite-dimensional systems, although the input and output operators are allowed to be unbounded, possess many properties that parallel in many ways to finite-dimensional systems. In this talk, I shall introduce briefly the development of this theory with exemplification of one-dimensional vibrating system control. The relations among well-posedness, exact controllability, and exponential stability under the proportional feedback control for the abstract first order and second order collocated systems are specially emphasized. The focus will be on the abstract formulation, verification of well-posedness and regularity of multi-dimensional Schrodinger equation, wave equation, plate equation, and coupled both weakly and strongly wave equations with variable coefficients. Finally, the significance of well-posedness is also illustrated by non-collocated control of multi-dimensional wave equations.