An optimal control problem is studied, in which the state is requiredto remain in acompact set S. A control feedback law is constructed which, forgiven ε > 0, produces ε-optimal trajectories that satisfy thestate constraint universally with respect to all initial conditionsin S.The construction relies upon a constraint removal technique whichutilizes geometric properties of inner approximations of S and arelated trajectory tracking result.The control feedback is shown to possess a robustness property withrespect to state measurement error.