In this section
@article{Bettiol:2013:10.3934/mcrf.2013.3.245,
author = {Bettiol, P and Vinter, RB},
doi = {10.3934/mcrf.2013.3.245},
journal = {Mathematical Control and Related Fields},
pages = {245--267},
title = {Estimates on trajectories in a closed set with corners for (t,x) dependent data},
url = {http://dx.doi.org/10.3934/mcrf.2013.3.245},
volume = {3},
year = {2013}
}
TY - JOUR
AB - Estimates 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 dif
AU - Bettiol,P
AU - Vinter,RB
DO - 10.3934/mcrf.2013.3.245
EP - 267
PY - 2013///
SN - 2156-8472
SP - 245
TI - Estimates on trajectories in a closed set with corners for (t,x) dependent data
T2 - Mathematical Control and Related Fields
UR - http://dx.doi.org/10.3934/mcrf.2013.3.245
UR - http://hdl.handle.net/10044/1/40476
VL - 3
ER -