On the continuity of optimal controls and value functions with respect to initial conditions

For standard optimal control problems with cost-to-go function J(t,x,u)J(t,x,u), results are presented concerning the continuity in x∈Rnx∈Rn of globally infimizing controls uxux for J(t,x,u)J(t,x,u) and the continuity in (t,x)∈[0,T]×Rn(t,x)∈[0,T]×Rn of the value function V(t,x)=J(t,x,ux)V(t,x)=J(t,x,ux).