The relaxation of the optimal control problem with cost functional which is the supremum in time of some function h(t, x, z) is determined. The trajectory is convexified in the usual way but the cost functional is convexified in a nonobvious manner. Thus, if the original value function is V(t, z) = inf zeta is an element of Zh(s,xi(s),zeta(s))L(infinity)t,T, then the relaxed value function is [GRAPHICS] where the inner norm is the essential sup of h over z is an element of Z with respect to the measure mu(s). We prove that V and ($) over cap V coincide.