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This book presents a deep, wide-ranging introduction to the mathematical theory of the optimal control of processes governed by ODEs and certain types of differential equations with memory. Many examples illustrate the mathematical issues that need to be addressed when using optimal control techniques in diverse areas. By providing a sufficient and rigorous treatment of finite dimensional control problems, the book equips readers with the foundation to deal with other types of control problems, such as those governed by stochastic differential equations, PDEs, and differential games.