The Maximum Principle: Pure State and Mixed Inequality Constraints

Often in management science and economics problems, there are inequality constraints only on state variables including constraints that require inventory levels or wealth to remain nonnegative. This chapter considers such pure state constraints that are more difficult to deal with than mixed constraints treated in Chap. 2 as the adjoint functions may be discontinuous at instants where the pure state constraints become tight. Section 4.1 provides a simple example to motivate the necessity of possible jumps in the adjoint functions. Section 4.2 formulates the problem with pure state constraints along with the required assumptions. In Sect. 4.3, we use the direct method for stating the Lagrangian form of the maximum principle. This method associates a Lagrange multiplier directly with each constraint for appending it to the Hamiltonian to form the Lagrangian. Sufficiency conditions are stated in Sect. 4.4. Section 4.5 is devoted to developing the maximum principle for the indirect method, which involves adjoining the first derivative of the pure state constraints to form the Lagrangian function and imposing some additional constraints on the Lagrange multipliers of the resulting formulation. Also, the adjoint variables and the Lagrange multipliers arising in this method will be related to those arising in the direct method. Finally, the current-value form of the maximum principle for the indirect method is described in Sect. 4.6. There are many exercises at the end of the chapter.