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 ) , results are presented concerning the continuity in x ∈ R n of globally infimizing controls u x for J ( t , x , u ) and the continuity in ( t , x ) ∈ [ 0 , T ] × R n of the value function V ( t , x ) = J ( t , x , u x ) .