Analysis and control of infinite dimensional systems

The interactions with physical dynamic systems as resulting from control laws are mainly intended to assign a certain desired behavior to the closed-loop system, where it is reasonable, at the stage of the controller design, to be aware of the (physical) restrictions implied by the plant. In this context the subject of the present work is constituted by a system- and control-theoretical analysis for the class of (coupled) lumped- and distributed-parameter systems, where the evolution in time is allowed to be governed by both (nonlinear) ordinary and/or partial differential equations. By means of the combined application of differential geometrical and functional analytical methods a formal approach is presented to investigate the considered class of dynamic systems. Although the interrogations as well as the approach for solving the discussed problems are equivalent in their nature to those of finite-dimensional systems, differences in their handling will be mostly inevitable if infinite-dimensional are treated, which lead to the requirement of a much more elaborate and adequate mathematical machinery. At first a formal description of dynamic systems is introduced by using differential geometrical methods, where special consideration is bestowed upon the distributed parameter case and particular issues regarding the system class. Systems are related to suitable geometric objects, which reflect their associated dynamics and allow for a covariant description. The principal part of the work is dedicated to the investigation of qualitative properties of systems and their solutions, respectively, where especially the observability, the reachability and the accessibility along a fixed trajectory are discussed, which play a key role for the controller design. Based on the intrinsic picture of systems an approach via transformation groups, which emerge as an appropriate tool to formalize and synthesize the adduced considerations and ideas into adequate criteria in order to (dis-)prove certain concepts, is proposed. It is shown how (Iocal) conditions for the discussed qualitative properties can be achieved by applying an infinitesimal invariance criterion. Further, a comparative discussion on lumped-parameter and distributed-parameter systems illustrates the analogies and disparities of the chosen approach for these system classes.