Pontryagin’s principle for Dieudonné-Rashevsky type problems with polyconvex integrands

with n, m > 2, Ω ⊂ R, m < p < ∞ and a compact set K ⊂ R with nonempty interior. In the case of a convex integrand f(s, ξ, · ) and a convex restriction set K, the global minimizers of (1.1) − (1.3) satisfy optimality conditions in the form of Pontryagin’s principle 01) even though the usual regularity condition for the equality operator (1.2) fails. 02) The question arises whether the Pontryagin principle and its proof can be extended to situations where the usual convexity of the data is replaced by generalized convexity notions. An answer to this question is of conceptual interest since the classical proof of the Pontryagin principle is based on an implicit convexification of the integrand as well as of the set of feasible controls. 03) Within the hierarchy of the generalized convexity notions, 04) polyconvexity is the closest one to usual convexity. In short, a polyconvex function arises as a composition of the vector of all minors of a matricial argument with a convex function. Appearing e. g. in problems from material science, 05) hydrodynamics 06) and mathematical image processing, 07) objectives with polyconvex integrands are of considerable practical importance. In the present paper, it will be shown that the proof of Pontryagin’s principle for the problem (1.1)− (1.3) can be maintained if the integrand f(s, ξ, v) is polyconvex with respect to v while the control restriction set K is still convex (Theorems 4.3., 4.4. and 4.11.). To the best of the author’s knowledge, a proof of optimality conditions, which makes explicit use of the polyconvex structure of the integrand, is still missing in the literature. The incorporation of polyconvex control constraints into the proof scheme, which turns out to be possible as well, will be achieved in a subsequent publication.