Subdifferentials of a minimal time function in normed spaces

In a general normed vector space, we study the minimal time function determined by a differential inclusion where the set-valued mapping involved has constant values of a bounded closed convex set U and by a closed target set S. We show that proximal and Fréchet subdifferentials of a minimal time function are representable by virtue of corresponding normal cones of sublevel sets of the function and level or suplevel sets of the support function of U. The known results in the literature require the set U to have the origin as an interior point or U be compact. (In particular, if the set U is the unit closed ball, the results obtained reduce to the subdifferential of the distance function defined by S.)