A pointwise constrained version of the Liapunov convexity theorem for single integrals

Abstract

Given any AC solution x : [a, b] → Rn to the convex ordinary differential inclusion x

(t) ∈ co{v1(t), . . . , vm(t)} a.e. on [a, b], (*) we aim at solving the associated nonconvex inclusion x

(t) ∈ {v1(t), . . . , vm(t)} a.e., x(a) = x(a), x(b) = x(b), (**) under an extra pointwise constraint (e.g. on the first coordinate): x1(t) ≤ x1(t) ∀t ∈ [a, b]. (***) While the unconstrained inclusion (**) had been solved already in 1940 by Liapunov, its constrained version, with (***), was solved in 1994 by Amar and Cellina in the scalar n = 1 case. In this paper we add an extra geometrical hypothesis which is necessary and sufficient, in the vector n > 1 case, for existence of solution to the constrained inclusion (**) and (***). We also present many examples and counterexamples to the 2 × 2 case.