Sufficient conditions for an optimal control problem in the calculus of variations

Abstract

An arc C is a collection of parameters bp (p = 1, , r) on an open set B and sets of functions y\\x)9 a\\x) (i = 1, , n; h = 1, , m) defined on an interval x1 ̂ x ^ xz with y\\x) con-tinuous and yι{x)9 a\\x) piecewise continuous. The arc is admis-sible if it satisfies the differential equations if = PXx9y,d) ( £ = 1,-- 9n) on x 1 S x ^ x2 and the end conditions x s = X φ \\ y%xs) = Yis(b) (β = 1, 2). The dot denotes differentiation with respect to x. The problem at hand is to find in a class of admissible arcs C, an arc C o which minimizes the integral where P(x, y9 a) and f(x9 y, a) are assumed to be class C " for (x, y, a) in an open set R while g(b), Xs(b), Yis(b) are of class Cπ on B. Under the added assumption that P(x9 y, a) is Lip-schitzian in y and a, the indirect method of Hestenes is used to prove that the necessary conditions for relative minima of the problem above, strengthened in the usual manner, yield a set of sufficient conditions. This problem differs from that of Pontryagin in the choice of (x9 y9 a) to lie in an open set. DEFINITIONS AND NOTATION. The arc C will be denoted by C: δ, y(x), a(x) and the minimizing arc will be called C o. A set of parameters βp and functions ^(x), ah(x) is called a variation 7 and denoted by 7: β, η(x)9 a(x) if η\\x) are continuous and φix), ah(x) are in L2 on x1 ^ x ^ x2. The variation 7 is differentially admissible if