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

Given any AC solution \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\overline{x} : [a,b] \rightarrow \mathbb{R}^{n}}$$\end{document} to the convex ordinary differential inclusion\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x^{\prime} ( t) \in co\{v^{1} ( t), \ldots, v^{m} ( t)\} \qquad a.e. on [ a,b], \qquad \qquad (^{*})$$\end{document}we aim at solving the associated nonconvex inclusion\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x^{\prime} ( t) \in \{v^{1} ( t), \ldots, v^{m} ( t)\} \qquad a.e.,x( a) = \overline{x} ( a), x( b) = \overline{x} ( b), \qquad \qquad (^{**})$$\end{document}under an extra pointwise constraint (e.g. on the first coordinate): \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_{1} ( t) \leq \overline{x}_{1} ( t) \qquad \forall t \in [ a,b]. \qquad \qquad \qquad (^{***})$$\end{document}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 it existence of solution to the constrained inclusion (**) and (***). We also present many examples and counterexamples to the 2 × 2 case.