Control in finite and infinite dimension

TL;DR

This book introduces mathematical control theory in finite and infinite dimensions, covering controlled ODEs and PDEs, emphasizing the extension of finite-dimensional results to infinite-dimensional systems using functional analysis and semigroup theory.

Contribution

It systematically develops control theory from finite to infinite dimensions, highlighting the extension of concepts like controllability and stabilization through semigroup and operator theory.

Findings

Finite-dimensional control systems are well-understood with classical results.

Infinite-dimensional systems require advanced functional analysis tools.

The Hilbert Uniqueness Method (HUM) extends controllability concepts to infinite dimensions.

Abstract

This short book is the result of various master and summer school courses I have taught. The objective is to introduce the readers to mathematical control theory, both in finite and infinite dimension. In the finite-dimensional context, we consider controlled ordinary differential equations (ODEs); in this context, existence and uniqueness issues are easily resolved thanks to the Picard-Lindel\''of (Cauchy-Lipschitz) theorem. In infinite dimension, in view of dealing with controlled partial differential equations (PDEs), the concept of well-posed system is much more difficult and requires to develop a bunch of functional analysis tools, in particular semigroup theory -- and this, just for the setting in which the control system is written and makes sense. This is why I have splitted the book into two parts, the first being devoted to finite-dimensional control systems, and the second to infinite-dimensional ones. In spite of this splitting, it may be nice to learn basics of control theory for finite-dimensional linear autonomous control systems (e.g., the Kalman condition) and then to see in the second part how some results are extended to infinite dimension, where matrices are replaced by operators, and exponentials of matrices are replaced by semigroups. For instance, the reader will see how the Gramian controllability condition is expressed in infinite dimension, and leads to the celebrated Hilbert Uniqueness Method (HUM). Except the very last section, in the second part I have only considered linear autonomous control systems (the theory is already quite complicated), providing anyway several references to other textbooks for the several techniques existing to treat some particular classes of nonlinear PDEs. In contrast, in the first part on finite-dimensional control theory, there are much less difficulties to treat general nonlinear control systems, and I give here some general results on controllability, optimal control and stabilization. Of course, whether in finite or infinite dimension, there exist much finer results and methods in the literature, established however for specific classes of control systems. Here, my objective is to provide the reader with an introduction to control theory and to the main tools allowing to treat general control systems. I hope this will serve as motivation to go deeper into the theory or numerical aspects that are not covered here.