8. Optimal Control

This chapter delves into the theory of optimal control, a powerful tool for dynamic optimization that addresses the limitations of calculus of variations. The text begins by introducing the maximum principle, the core concept of optimal control, and explains its conditions for optimality, including the Hamiltonian and co-state variables. It contrasts optimal control with calculus of variations, emphasizing the former's ability to handle constraints along the optimal path. The chapter provides a detailed example of a problem with one control variable, illustrating the step-by-step solution process. It also revisits an economic problem from a societal perspective, demonstrating the application of optimal control in continuous time. Additionally, the text discusses control restrictions and presents a 'bang-bang' optimal control problem, highlighting the practical relevance of optimal control in real-world scenarios. The chapter concludes with an economic interpretation of the maximum principle, providing insights into its significance in dynamic optimization.