Relaxation in L∞ control

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. If the original value function is V(t,x)=inf_ζεz ||h(s,ξ(s),ζ(s))||L∞[t,T], then, the relaxed value function is Vˆ(t,x)=inf||||h(s,ξˆ(s),z)||L∞(z;μ(s)) ||L∞[t,T], μεzˆ[t,T] where, for each fixed s ε [t,T], the inner norm is the essential sup of h over zεZ with respect to the probability measure μ(s)