Many management science applications involve the control of dynamic systems, i.e., systems that evolve over time. They are called continuous-time systems or discrete-time systems depending on whether time varies continuously or discretely. We will deal with both kinds of systems in this book, although the main emphasis will be on continuous-time systems.

What Is Optimal Control Theory

Optimal control theory has been extensively applied to the solution of economic problems since the early papers that appeared in Shell (1967) and the works of Arrow (1968) and Shell (1969) . The field is too vast to be surveyed in detail here, however.

Applications to Economics

The problems to which the maximum principle derived in the previous chapter was applicable had constraints involving only the control variables. We will see that in many applied models it is necessary to impose constraints involving both control and state variables. Inequality constraints involving control and possibly state variables are called mixed inequality constraints.

The Maximum Principle: Mixed Inequality Constraints

This chapter focuses on constraints in classical theory. In the theory of ordinary maxima and minima, the most interesting and realistic and also the most difficult problems are those in which constraints are present. Constraints also appear in many applications of the calculus of variations. Constraints add to the complexity both of the mathematical theory and of the associated computational methods. This chapter analyzes the isoperimetric constraints. The name isoperimetric arose in the historical development of the calculus of variations as a result of interest in the problem of finding the closed curve of given length (constant perimeter), which encloses the greatest area. The answer is a circle, and it is of some interest that this result seems to have been known since earliest times, long before the development of the calculus of variations. The chapter explores the problem of Lagrange in detail and presents a simplified proof of the multiplier rule for fixed end point.

G. Hadley

Papers

Classical theory — constraints