Résumé en anglais

This paper discusses the following viability problem of a differential inclusion, x′(t) + Ax(t) ϵF(x(t))x(0) = x0ϵK∀t ⩾ 0, x(t) ϵK, where K is a compact subset of a Banach space X, − A is the infinitesimal generator of a compact differentiable semigroup of bounded linear operators on X, and F: K ⇉ X is upper semicontinuous with compact convex values. We show that a natural tanegential condition is necessary and sufficient for the existence of a global solution to this problem.