Estimates on trajectories in a closed set with corners for $(t,x)$ dependent data

International audienceEstimates on the distance of a given process from the set of processes that satisfy a specified state constraint in terms of the state constraint violation are important analytical tools in state constrained optimal control theory; they have been employed to ensure the validity of the Maximum Principle in normal form, to establish regularity properties of the value function, to justify interpreting the value function as a unique solution of the Hamilton-Jacobi equation, and for other purposes. A range of estimates are required, which differ according the metrics used to measure the 'distance' and the modulus θ(h) of state constraint violation h in terms of which the estimates are expressed. Recent research has shown that simple linear estimates are valid when the state constraint set A has smooth boundary, but do not generalize to a setting in which the boundary of A has corners. Indeed, for a velocity set F which does not depend on (t,x) and for state constraints taking the form of the intersection of two closed spaces (the simplest case of a boundary with corners), the best distance estimates we can hope for, involving the W1,1, metric on state trajectories, is a super-linear estimate expressed in terms of the h∣∣log(h)∣∣ modulus. But, distance estimates involving the h∣∣log(h)∣∣ modulus are not in general valid when the velocity set F(.,x) is required merely to be continuous, while not even distance estimates involving the weaker, Hölder modulus hα (with α arbitrarily small) are in general valid, when F(.,x) is allowed to be discontinuous. This paper concerns the validity of distance estimates when the velocity set F(t,x) is (t,x)-dependent and satisfy standard hypotheses on the velocity set (linear growth, Lipschitz x-dependence and an inward pointing condition). Hypotheses are identified for the validity of distance estimates, involving both the h∣∣log(h)∣∣ and linear moduli, within the framework of control systems described by a controlled differential equation and state constraint sets having a functional inequality representation