Vol. 17, No. 2, 1966

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

Edwin Haena Mookini

Vol. 17 (1966), No. 2, 263–277

An arc C is a collection of parameters bρ (ρ = 1,,r) on an open set B and sets of functions yi(x), ah(x)(i = 1,,n;h = 1,,m) defined on an interval x1 x x2 with yi(x) continuous and i(x), ah(x) piecewise continuous. The arc is admissible if it satisfies the differential equations

˙yi = Pi(x,y,a) (i = 1,⋅⋅⋅ ,n)

on x1 x x2 and the end conditions

xs = Xs(b),yi(xs) = Yis(b) (s = 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 C0, which minimizes the integral

            ∫ x2
I(C ) = g(b)+    f(x,y,a)dx

where P(x,y,a) and f(x,y,a) are assumed to be class C′′ for (x,y,a) in an open set R while g(b), Xs(b), Y is(b) are of class C′′ on B. Under the added assumption that P(x,y,a) is Lipschitzian 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 (x,y,a) to lie in an open set.

Mathematical Subject Classification
Primary: 93.40
Secondary: 49.00
Received: 11 November 1964
Published: 1 May 1966
Edwin Haena Mookini