A short introduction to the control theory in finite-dimensional spaces

TL;DR

This paper provides a concise, detailed introduction to control theory in finite-dimensional spaces, emphasizing linear systems, key theorems, and control techniques, with insights applicable to infinite-dimensional extensions.

Contribution

It offers a clear, detailed overview of finite-dimensional control theory, including proofs and extensions relevant for infinite-dimensional systems, based on lecture material.

Findings

Detailed proofs of control theory results

Extension strategies for infinite-dimensional systems

Comprehensive coverage of key control concepts

Abstract

This is a brief introduction to control theory in finite-dimensional spaces. The material is partly based on my lectures for the Master 1 program in Math\'ematiques et applications at Sorbonne University, delivered over the past few years. The aim is to provide a concise overview of the subject, primarily focusing on the linear setting. Proofs are presented in detail and are selected to allow for extensions to the infinite-dimensional case in many situations. Topics covered include the Kalman rank condition, the Hautus test, observability, stability, detectability and dynamic observers, the Pole Shifting Theorem, the linear test for controllability, linear-quadratic optimal control over finite and infinite horizons, and stabilization via Gramians.