9. Further Refinements in Optimal Control

This chapter examines the further refinements in optimal control, focusing on dynamic economic problems that deviate from standard setups. It revisits transversality conditions, which are crucial when endpoints are not precisely specified, introducing new variables and constraints. The text explores three basic cases where endpoints are flexible: free terminal time with a fixed terminal state, fixed terminal time with a free terminal state, and both terminal time and state being free. Each scenario is illustrated with practical examples, such as a firm managing its inventory or a salesman planning his route. The chapter also discusses control parameters, which impose constraints on control variables, and blocked intervals, where constraints prevent the state variable from following the optimal path. Additionally, it delves into current-value Hamiltonians, explaining their advantages in simplifying solutions and interpreting marginal values. The chapter concludes with a comparison of present value and current value formulations, highlighting the benefits of the latter in economic problems. By understanding these advanced techniques, readers gain insights into optimizing economic decisions under complex conditions.