In the control system $ dot{x} = f(t,x(t)) + g(t,x(t))dot{u}, x(0) = ar{x}, t in [0,T], $ this paper shows that the map from u with $L^1(m)$-topology to $x_u$ with $L^1(mu)$-topology is Lipschitz continuous where f is $C^1$, $mu$ is the Stieltjes measure derived from the function g which is not smooth in the variable t and $x_u$ is the solution of the above system corresponding to u under the assumption that $dot{u}$ is bounded.