The Lagrange Principle for Problems of the Classical Calculus of Variations and Optimal Control Theory

Abstract

The subject of this chapter is clear from its title. Here our primary aim is, on the one hand, to show the similarity between the two versions of the theory, which is stressed by the unified system of notation, and, on the other hand, to elucidate the distinction between the classical and the modern statements of the problem. We begin with necessary conditions for the so-called Lagrange problem to which many other problems of the classical calculus of variations can be reduced. Then we derive the Pontryagin maximum principle, which is one of the most important means of the modern theory of optimal control problems. The rest of the chapter is devoted to some more special classes of problems and to the derivation of consequences of the general theory. Sufficient conditions for an extremum are treated less thoroughly, and we confine ourselves to some particular situations.