Distance Estimates for State Constrained Trajectories of Control Systems

I will discuss the validity of estimates on the distance of an arbitrary state trajectory of a control system from the set of all state trajectories which lie in a given state constraint set.
These estimates have wide-spread application in state constrained optimal control, including justifying the use of the Maximum Principle in normal form and establishing regularity properties of value functions. We focus on linear, $L^{ infty}$ and $W^{1,1}$ distance estimates which, of all the available estimates have, so far, been the most widely used.