Introduction to control theory of systems

Abstract

This comprehensive course material bridges classical control theory with modern challenges in distributed systems governed by partial differential equations (PDEs). It systematically explores controllability and observability principles for both finite-dimensional systems (ODEs) and infinite-dimensional systems (PDEs), emphasizing rigorous mathematical tools such as the Kalman rank condition, Gramian-based methods, and the Hilbert Uniqueness Method (HUM). Key topics include:  Controllability: From Kalman criteria for linear time-invariant (LTI) systems to boundary control of wave/heat equations.  Observability: Duality principles, spectral techniques, and geometric control conditions (GCC) for PDEs.  Optimal Control: Adjoint methods, PDE-constrained optimization for elliptic, parabolic, and hyperbolic systems. The text integrates theoretical proofs with computational and real-world applications, designed for mathematicians, engineers, and scientists, it equips readers with tools to analyze, control, and optimize spatially distributed systems.