Triggiani R., Lasiecka I. Control theory for partial differential equations: continuos and approximation theories. Vol. 1

Cambridge: Cambridge University Press, 2000. — 674 p.This is the first volume of a comprehensive and up-to-date three-volume treatment of quadratic optimal control theory for partial differential equations over a finite or infinite time horizon and related differential (integral) and algebraic Riccati equations. Both continuous theory and numerical approximation theory are included. An abstract space, operator theoretic treatment is provided, which is based on semi group methods, and which is unifying across a few basic classes of evolution. A key feature of this treatise is the wealth of concrete multidimensional PDE illustrations, which fit naturally into the abstract theory, with no artificial assumptions imposed, at both the continuous and numerical level.
Throughout these volumes, emphasis is placed on unbounded control operators or on unbounded observation operators as they arise in the context of various abstract frameworks that are motivated by partial differential equations with boundary/point control. Relevant classes of PDEs include: parabolic or parabolic-like equations, hyperbolic and Petrowski-type equations (such as plate equations and the SchrOdinger equation), and hybrid systems of coupled PDEs of the type that arise in modern thermo-elastic and smart material applications. Purely PDE dynamical properties are critical in motivating the various abstract settings and in applying the corresponding theories to concrete PDEs arising in mathematical physics and in other recent technological applications.
Volume I covers the abstract parabolic theory, including both the finite and infinite horizon optimal control problems, as well as the corresponding min-max theory with nondefinite quadratic cost. A lengthy chapter presents many multidimensional PDE illustrations with boundary/point control and observation. These include not only the traditional parabolic equations, such as the heat equation, but also second-order equations with structural ("high") damping, as well as thermo-elastic plate equations. Recently discovered, critical dynamical properties are provided in detail. Many of these new results are appearing here in print for the first time.