Overview

Dynamic optimization emerged as a distinct field of research in the late 1950’s, to address new kinds of optimization problems, in aerospace, economics and other areas. The distinctive feature of these problems was an underlying dynamic constraint, typically in the form of a controlled differential equation, which placed these problems beyond the scope of earlier variational techniques. Rapid advances were made in the 1970’s and 80’s, with the discovery of the maximum principle and methodologies (dynamic programming) that linked optimal strategies and the Hamilton Jacobi equation. These were the main elements in what, today, is known as the classical theory of dynamic optimization. While classical dynamic optimization was adequate for many applications, deficiencies became apparent, leading to a new body of theory in the 1980’s, including Clarke’s nonsmooth maximum principle and generalized solutions of Hamilton-Jacobi equations, based on techniques of nonsmooth analysis.