Nonlinear infinite-dimensional systems are nonlinear dynamical systems whose state components lie in an infinite-dimensional space, typically a function space. Such systems, which are also called distributed parameter systems, are ubiquitous in real-life since they are able to model many physical processes, going from conservative mechanical systems to dissipative phenomena. A lot of questions may arise when dealing with such classes of systems. For instance, the well-posedness in terms of existence and uniqueness of solutions as well as the study of the equilibria, their stability and their control are paramount steps when studying these dynamical systems. On the basis of the existing literature, we pay a particular attention to the existence, the uniqueness and the stability of equilibria and the control of nonlinear distributed parameter systems. In particular, as main contributions, we extend the classical approach that allows to deduce the stability of equilibria for a nonlinear system based on the stability of a corresponding linearized version of it. Using a new concept of differentiability for nonlinear operators which takes another space as the state space into account, we show how to guarantee local exponential stability or instability of the equilibria for the original nonlinear system. This is applied to the determination of the stability of the equilibria of a nonlinear plug-flow tubular reactor model with axial dispersion for which the temperature and the reactant concentration are considered as state variables. From a control point of view, the previous results are extended to the stabilization of equilibria of nonlinear infinite-dimensional systems. Thanks to this extension, we are able to identify a class of optimally controlled systems for which the required assumptions hold. As another contribution, we study the field of tracking control, and especially funnel control, which constitutes an appropriate tool for the output of a system to track a class of reference signals. As a main contribution on this topic, we extend the available results that allow to consider linear infinite-dimensional systems as internal dynamics to the nonlinear setting. We prove that a general class of nonlinear infinite-dimensional systems that satisfy some standard assumptions admits a differential relation between the input and the output that is conducive for funnel control. A large number of theoretical results in this thesis are illustrated by means of examples and numerical simulations, especially in the field of process control. The considered applications are related to chemical reactor models, damped wave equations and damped Sine-Gordon equations.
|Date of Award
|30 Mar 2022
|Joseph Winkin (Supervisor), Denis Dochain (Co-Supervisor), Timoteo Carletti (President), Hans Zwart (Jury), Laurent Lefèvre (Jury) & Kirsten Morris (Jury)