Hafdallah, Abdelhak2025-05-282025-05-282023http//localhost:8080/jspui/handle/123456789/12602This 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.enControllabilityl ; Observability ; Kalman Rank Condition; Gramian; Hilbert Uniqueness Method (HUM); Optimal ControlOther