Optimal Control as Programming in Infinite Dimensional Spaces

Abstract

Introduction. The terms “Programming” and “Mathematical Programming” refer here to the usual constrained maximization problem of the type: given a function φ (x) = (φ _A (x),…, φ_k (x)) from Eⁿ (the Euclidean n-dimensional space) into E^k, find a point x̂ ϵ Eⁿ such that φ _A (x̂) is maximized subject to equality and/or inequality constraints on φ₂(x), …, φ_k(x̂). This problem has received considerable attention and has led to interesting results ranging from the initial work of Lagrange to the more recent studies of Kuhn and Tucker. Calculus of Variations and Optimal Control are also concerned with constrained maximization problems but over given sets of continuous curves instead of finite dimensional Euclidean spaces as in Programming. Programming has always extended a strong influ ence on Calculus of Variations and one of the motivating forces behind the creation of Functional Analysis was to build a bridge between these two fields. The methods of Functional Analysis have been used to derive Euler-Lagrange equation but we do not know any previous successful attempts to apply those methods to the deri vation of the Weierstrass-E test and the Multiplier Rule for the pro blem of Bolza. This is indeed the purpose of the present paper.

In Section I we shall study a mathematical programming problem in infinite dimensional spaces. In Section II we prove that the standard optimal control problem (a generalization of the problem of Bolza) can be casted into a problem of the type studied in Section I, and by applying to this problem the results of Section I we obtain a generalization of the Maximum Principle of Pontryagin which is itself a generalization of the classical Weierstrass-E test and of the Multiplier Rule for the problem of Bolza (including the abnormal case).