The Maximum Principle: Mixed Inequality Constraints

In many applied problems, we find constraints on control and state variables. In Sect. 3.1, a Lagrangian form of the maximum principle is discussed for models in which there are mixed constraints involving both state and control variables in addition to constraints only on control variables. In Sect. 3.2, we state conditions under which the Lagrangian maximum principle is also sufficient for optimality. Economists frequently analyze optimal control problems involving a discount rate. In that case, it is convenient to use the current-value formulation of the maximum principle as described in Sect. 3.3. It is often the case infinite horizon problems that some restrictions are imposed on the state variables at the end of the horizon. In Sect. 3.4, we discuss the transversality conditions to be satisfied by the adjoint variable in special cases of interest. Section 3.5 is devoted to the study of free terminal time problems where the terminal time itself is a decision variable to be determined. Models with infinite horizons and their stationary equilibrium solutions are covered in Sect. 3.6. Section 3.7 presents a classification of a number of the most important and commonly used kinds of optimal control models, together with a brief description of the forms of their optimal solutions. Several examples are solved throughout the chapter to illustrate the theory. There are many exercises at the end of the chapter.