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This book is designed for undergraduate and graduate students in engineering enrolled in courses on control systems and optimal control. It will also serve as a valuable reference for mathematics students studying control theory. It offers a rigorous and systematic treatment of both finite-dimensional and infinite-dimensional control systems. The volume opens with chapters on essential mathematical foundations, including mathematical modelling, linear algebra, and ordinary differential equations, establishing a solid framework for the study of control theory. Subsequent chapters provide an in-depth treatment of key topics such as controllability, observability, feedback control, state observer, optimal control, constrained control, stability, approximate controllability, and regularized control. The text concludes with comprehensive coverage of discrete-time systems and infinite-dimensional systems. Throughout the book, theoretical developments are supported by detailed mathematical proofs, illustrative examples, solved problems, and end-of-chapter exercises, making it suitable for both classroom use and self-study.