This book provides a comprehensive overview of optimization theory and its applications in control systems. It begins with the fundamentals of optimization, covering key concepts such as convexity, optimality conditions, and duality theory. The text then explores linear and quadratic optimization, including polyhedra, linear programming, and quadratic programming. It also introduces mixed-integer optimization and numerical methods for solving optimization problems, including convergence analysis and algorithms for unconstrained and constrained optimization. The book then delves into the theory of polyhedra and P-collections, discussing set operations, polyhedral representations, and polytopal complexes. The latter part of the book focuses on constrained optimal control of linear systems, covering topics such as controllability, reachability, invariant sets, and robust control. It discusses various control strategies, including receding horizon control, approximate receding horizon control, and on-line control computation. The text also addresses constrained robust optimal control, exploring problem formulations, feasible solutions, and state feedback solutions under different cost functions. The final section of the book examines constrained optimal control of hybrid systems, discussing models of hybrid systems, piecewise affine systems, discrete hybrid automata, and mixed logical dynamical systems. It explores optimal control of hybrid systems, including properties of state feedback solutions, computation via mixed integer programming, and different approaches to state feedback solutions. The book concludes with a review of relevant literature and provides a comprehensive index for reference. The content is structured to provide a thorough understanding of optimization theory and its application in control systems, making it a valuable resource for researchers and practitioners in the field.This book provides a comprehensive overview of optimization theory and its applications in control systems. It begins with the fundamentals of optimization, covering key concepts such as convexity, optimality conditions, and duality theory. The text then explores linear and quadratic optimization, including polyhedra, linear programming, and quadratic programming. It also introduces mixed-integer optimization and numerical methods for solving optimization problems, including convergence analysis and algorithms for unconstrained and constrained optimization. The book then delves into the theory of polyhedra and P-collections, discussing set operations, polyhedral representations, and polytopal complexes. The latter part of the book focuses on constrained optimal control of linear systems, covering topics such as controllability, reachability, invariant sets, and robust control. It discusses various control strategies, including receding horizon control, approximate receding horizon control, and on-line control computation. The text also addresses constrained robust optimal control, exploring problem formulations, feasible solutions, and state feedback solutions under different cost functions. The final section of the book examines constrained optimal control of hybrid systems, discussing models of hybrid systems, piecewise affine systems, discrete hybrid automata, and mixed logical dynamical systems. It explores optimal control of hybrid systems, including properties of state feedback solutions, computation via mixed integer programming, and different approaches to state feedback solutions. The book concludes with a review of relevant literature and provides a comprehensive index for reference. The content is structured to provide a thorough understanding of optimization theory and its application in control systems, making it a valuable resource for researchers and practitioners in the field.