Introduction to Dynamic Optimization

This chapter gives a brief introduction to the classical Calculus of Variations and to Optimal Control Theory, which may be regarded as a modern generalization of the Calculus of Variations. These two fundamental topics offer a precious tool for treating very general dynamic optimization problems with continuous time. The Calculus of Variations has a long history, which goes back to the works of Euler and Lagrange (eighteenth century), whereas the Optimal Control Theory was created mainly by the Russian mathematician L. S. Pontryagin and his school, in the mid of the last century. This theory has revolutionized the traditional theory concerned with the Calculus of Variations and has attracted attention in several sectors of physics, engineering and also economic theory and financial analysis. Also for these subjects, since a rigorous exposition would require an entire book and since several specific books are available, our exposition is basic and somewhat intuitive. We emphasize that every researcher of modern economic analysis and modern financial analysis should know the main features of the Calculus of Variations and Optimal Control Theory. In the last section we give an idea of another tool of Dynamic Optimization, due to the American mathematician R. Bellman, i.e. the Dynamic Programming and the Optimality Principle of Bellman.