The book "Representation and Control of Infinite Dimensional Systems" is a comprehensive resource on the control theory of infinite-dimensional systems, updated in its second edition. The authors, Alain Bensoussan, Giuseppe Da Prato, Michel C. Delfour, and Sanjoy K. Mitter, have revised and expanded the content to reflect the advancements in the field over the past decade. The book is structured into five parts: 1. **Control of Finite Dimensional Linear Dynamical Systems and Linear Quadratic Two-Person Zero-Sum Differential Games**: This part provides a broad review of finite-dimensional systems, including controllability and observability, and introduces linear quadratic games, which are essential for understanding the infinite-dimensional context. 2. **Representation of Infinite Dimensional Systems**: This part focuses on semigroup theory and variational methods for representing infinite-dimensional systems, such as partial differential equations and delay differential systems. It covers topics like semigroups of linear operators, interpolation theory, and the variational theory of parabolic systems. 3. **Generic Qualitative Properties of Controlled Systems**: This part studies controllability for infinite-dimensional abstract linear dynamical systems, including parabolic and hyperbolic partial differential equations. It discusses exact controllability and its implications for stabilizability. 4. **Quadratic Cost Optimal Control Problem Over a Finite Time Horizon**: This part develops the theory for an abstract dynamical model and applies it to concrete situations using differential equations. It covers boundary control and observation for parabolic and hyperbolic systems, emphasizing the use of dynamic programming and operator Riccati equations. 5. **Quadratic Cost Optimal Control Problem Over an Infinite Time Horizon**: This part explores the concepts of stabilizability and detectability, and the use of dynamic programming and algebraic Riccati equations to solve optimal control problems over infinite time horizons. The book is a valuable reference for researchers and advanced engineers in the field of control theory, providing a detailed and updated treatment of the subject.The book "Representation and Control of Infinite Dimensional Systems" is a comprehensive resource on the control theory of infinite-dimensional systems, updated in its second edition. The authors, Alain Bensoussan, Giuseppe Da Prato, Michel C. Delfour, and Sanjoy K. Mitter, have revised and expanded the content to reflect the advancements in the field over the past decade. The book is structured into five parts: 1. **Control of Finite Dimensional Linear Dynamical Systems and Linear Quadratic Two-Person Zero-Sum Differential Games**: This part provides a broad review of finite-dimensional systems, including controllability and observability, and introduces linear quadratic games, which are essential for understanding the infinite-dimensional context. 2. **Representation of Infinite Dimensional Systems**: This part focuses on semigroup theory and variational methods for representing infinite-dimensional systems, such as partial differential equations and delay differential systems. It covers topics like semigroups of linear operators, interpolation theory, and the variational theory of parabolic systems. 3. **Generic Qualitative Properties of Controlled Systems**: This part studies controllability for infinite-dimensional abstract linear dynamical systems, including parabolic and hyperbolic partial differential equations. It discusses exact controllability and its implications for stabilizability. 4. **Quadratic Cost Optimal Control Problem Over a Finite Time Horizon**: This part develops the theory for an abstract dynamical model and applies it to concrete situations using differential equations. It covers boundary control and observation for parabolic and hyperbolic systems, emphasizing the use of dynamic programming and operator Riccati equations. 5. **Quadratic Cost Optimal Control Problem Over an Infinite Time Horizon**: This part explores the concepts of stabilizability and detectability, and the use of dynamic programming and algebraic Riccati equations to solve optimal control problems over infinite time horizons. The book is a valuable reference for researchers and advanced engineers in the field of control theory, providing a detailed and updated treatment of the subject.