The calculus of variations: A historical perspective

Abstract

We begin with an introduction to the historical origin of optimal control theory, the calculus of variations. But it is not our intention to give a comprehensive treatment of this topic. Rather, we introduce the fundamental necessary and sufficient conditions for optimality by fully analyzing two of the cornerstone problems of the theory, the brachistochrone problem and the problem of determining surfaces of revolution with minimum surface area, so-called minimal surfaces. Our emphasis is on illustrating the methods and techniques required for getting complete solutions for these problems. More generally, we use the so-called fixed-endpoint problem, the problem of minimizing a functional over all differentiable curves that satisfy given boundary conditions, as a vehicle to introduce the classical results of the theory: (a) the Euler–Lagrange equation as the fundamental first-order necessary condition for optimality, (b) the Legendre and Jacobi conditions, both in the form of necessary and sufficient second-order conditions for local optimality, (c) the Weierstrass condition as additional necessary condition for optimality for so-called strong minima, and (d) its connection with field theory, the fundamental idea in any sufficiency theory. Throughout our presentation, we emphasize geometric constructions and a geometric interpretation of the conditions. For example, we present the connections between envelopes and conjugate points of a fold type and use these arguments to give a full solution for the minimum surfaces of revolution.