The existence of a mild solution of the abstract functional differential inclusion
$$P(\varphi)\begin{cases}u^\prime(t)+Au(t)-F(t,u_t)\ni 0, & t\in I=[0,T]\\ u(t)=\varphi(t),& t\in J=[-r,0]\end{cases}$$
is obtained by a Filippov technique. Here $A$ is an operator such that $A +\omega\mathcal I$ is m-accretive for some $\omega\in\mathbb R^+$, and the multimapping $F(t, .)$ is $h(t)$-Lipschitzian. The result is applied to a nonlinear functional control problem.