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

Given any AC solution (x) over bar : [a, b] -> R-n to the convex ordinary differential inclusion x' (t) is an element of co{upsilon(1)(t) , ... , upsilon(m)(t)} a.e. on [a, b], (*) we aim at solving the associated nonconvex inclusion x'(t) is an element of {upsilon(1)(t) , ... , upsilon(m)(t)} a.e., x(a) = (x) over bar (a), x(b) = (x) over bar (b), (**) under an extra pointwise constraint (e.g. on the first coordinate): x(1)(t) = (x) over bar (1)(t) for all t is an element of [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 x 2 case.