Mathematical control theory of coupled PDEs (Q2778312)

The two earliest papers on control theory [\textit{C. Maxwell}, On governors, Proc. R. Soc. Lond. 16, 270-283 (1868; JFM 01.0337.01), \textit{J. Vyshnegradski}, Sur la théorie générale des régulateurs, C. R. Acad. Sci. Paris 83, 318-321 (1876; JFM 08.0606.01)] were on stabilization by feedback of regulators, in particular of the Watts centrifugal governor that in some steam engines tended to settle in an (undesired) periodic regime rather than in a steady state. This pioneering work was extended by the work of Hermite, Routh and Hurwitz on stability of polynomials. For almost a century, control theory dealt mainly with stabilization by feedback. In the late 1940s it began to formulate clearly ideas as yet implicit such as controllability, observability and optimality of control according to given performance indices, with special emphasis on the linear-quadratic model (quadratic performance index, linear dynamics) and the associated matrix Riccati equation. NEWLINENEWLINENEWLINEAbout a decade later, control theory ``went infinite dimensional'' with the object of modeling systems described by partial (rather than ordinary) differential equations. The objectives were the same (stabilization, optimality, the exact or approximate attainment of certain target states), but the level of complexity increased exponentially; to mention one subject, existence, uniqueness and good behavior under perturbation of solutions of the model (well posedness, usually automatic in finite-dimensional models) became a fundamental prerequisite and a field of its own, especially in view of the possibility of modeling with states and controls in many different function spaces. NEWLINENEWLINENEWLINEFor systems described by partial differential equations, feedback can take various forms depending on where in the interior or the boundary of the domain sensors and actuators are placed. Boundary feedback and sensing are usually the most realistic setup. The finite-dimensional equivalence between stabilization and placement of spectra in the left half of the complex plane may break down, and stabilization may or may not be exponential. NEWLINENEWLINENEWLINEThis monograph deals with control systems described by partial differential equations, in particular with (a) well posedness, (b) stabilization and stability, and (c) the closely related linear-quadratic problem (although nonlinear dynamics occupies a prominent place). Stabilization is achieved with boundary feedback controls or damping terms on the boundary. The Riccati equation is now an operator equation, and attention must be paid to the right choice of spaces, both in terms of solvability and of the correct modeling of realistic physical situations. The equations under study are coupled hyperbolic-parabolic equations that arise in many applications, for instance stabilization of membranes and plates, structural acoustic interactions, feedback noise control, thermoelasticity, etc. NEWLINENEWLINENEWLINEThis book is a fundamental contribution to the modeling of physical phenomena by partial differential equations. Most, if not all the results were available to date only in research papers, and some are new in any form. The book will be of use to experts and students in control theory alike, but not only to them. As most outstanding works in control theory, it throws light on existing results and presents many new ones even in the classical (uncontrolled) case.