Suppose that we have two differential inclusions

$$\frac{dY^1}{dt}(t)\in b_1(Y^1,t)$$

with $Y^1(0)\in Y_0^1$ and

$$\frac{dY^2}{dt}(t)\in b_2(Y^2,t)$$

with $Y^2(0)\in Y_0^2$.

Can we then control $d(Y^1(t),Y^2(t))$ by the distance of $b_1$ and $b_2$ and the initial conditions? The distance here would be some appropriate set distance such as Kuratowski or Hausdorff distance or some other distance?