Lyapunov pairs for multi-valued semi-linear evolutions

We give here a characterization of a Lyapunov pair for a multi-valued semi-linear evolution equation on a Banach space by means of an appropriate lower contingent derivative. The contingent derivative introduced in this paper is related to a new concept of tangency introduced recently in [O. Cârj?, M. Necula, I.I. Vrabie, Necessary and sufficient conditions for viability for semilinear differential inclusions, Trans. Amer. Math. Soc. 361 (2009) 343-390]. As an application, we give a controllability result and a Lipschitz estimate of the corresponding minimum time function under a Petrov-like condition.