The book "Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations" by Martino Bardi and Italo Capuzzo-Dolcetta provides a comprehensive treatment of the theory of viscosity solutions for first-order partial differential equations of Hamilton-Jacobi type and its applications to optimal deterministic control and differential games. The authors introduce the viscosity solutions approach, which was initiated in the early 1980s, and use it to address various significant model problems in optimal control and differential games. The book is structured into several chapters, each focusing on different aspects of the theory and its applications: 1. **Chapter I** introduces the main ideas through a model problem, the infinite horizon discounted regulator. 2. **Chapter II** covers the basic theory of continuous viscosity solutions, including definitions, properties, and calculus. 3. **Chapter III** specializes the theory to optimal control problems with continuous value functions and unrestricted state spaces. 4. **Chapter IV** deals with problems involving exit times and state constraints. 5. **Chapter V** discusses discontinuous value functions and various notions of discontinuous viscosity solutions. 6. **Chapter VI** addresses approximation and perturbation problems, including discrete time Dynamic Programming and regular perturbations. 7. **Chapter VII** analyzes asymptotic problems such as singular perturbations, penalization, and vanishing discount. 8. **Chapter VIII** introduces the theory of two-person zero-sum differential games and the relevant Hamilton-Jacobi-Isaacs equations. The book also includes two appendices: one on numerical solution methods for Dynamic Programming equations and another on nonlinear \( \mathcal{H}_\infty \) control. The authors aim to provide a self-contained and comprehensive presentation suitable for scientists in optimal control, system theory, and partial differential equations. The book is pedagogically oriented, with exercises at the end of each section, making it suitable for graduate courses in optimal control.The book "Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations" by Martino Bardi and Italo Capuzzo-Dolcetta provides a comprehensive treatment of the theory of viscosity solutions for first-order partial differential equations of Hamilton-Jacobi type and its applications to optimal deterministic control and differential games. The authors introduce the viscosity solutions approach, which was initiated in the early 1980s, and use it to address various significant model problems in optimal control and differential games. The book is structured into several chapters, each focusing on different aspects of the theory and its applications: 1. **Chapter I** introduces the main ideas through a model problem, the infinite horizon discounted regulator. 2. **Chapter II** covers the basic theory of continuous viscosity solutions, including definitions, properties, and calculus. 3. **Chapter III** specializes the theory to optimal control problems with continuous value functions and unrestricted state spaces. 4. **Chapter IV** deals with problems involving exit times and state constraints. 5. **Chapter V** discusses discontinuous value functions and various notions of discontinuous viscosity solutions. 6. **Chapter VI** addresses approximation and perturbation problems, including discrete time Dynamic Programming and regular perturbations. 7. **Chapter VII** analyzes asymptotic problems such as singular perturbations, penalization, and vanishing discount. 8. **Chapter VIII** introduces the theory of two-person zero-sum differential games and the relevant Hamilton-Jacobi-Isaacs equations. The book also includes two appendices: one on numerical solution methods for Dynamic Programming equations and another on nonlinear \( \mathcal{H}_\infty \) control. The authors aim to provide a self-contained and comprehensive presentation suitable for scientists in optimal control, system theory, and partial differential equations. The book is pedagogically oriented, with exercises at the end of each section, making it suitable for graduate courses in optimal control.