ON TRAJECTORIES SATISFYING A STATE CONSTRAINT: W1,1 ESTIMATES AND COUNTEREXAMPLES

This paper concerns properties of solutions to a differential inclusion ("F-trajectories") satisfying a state constraint. Estimates on the W-1,W-1 distance of a given F-trajectory to the set of F-trajectories satisfying the constraint have an important role in state-constrained optimal control theory, regarding the derivation of nondegenerate necessary conditions, sensitivity analysis, characterization of the value function in terms of the Hamilton-Jacobi equation and other applications. According to some of the earlier literature, estimates, in which the W-1,W-1 distance is related linearly to the degree of constraint violation of the original F-trajectory, are valid for state constraints defined by a collection of one or more inequality constraint functionals. We show, by counterexample, that linear, W-1,W-1 estimates are not in general valid for multiple state constraints. We also identify cases involving several state constraints, where not even a weaker, linear L-infinity estimate holds. We further show that it is possible to justify linear, W-1,W-1 estimates by means of a modification of earlier constructive techniques, when there is only one state constraint. In a future companion paper [Estimates for trajectories confined to a cone in R-n, SIAM J. Control Optim., submitted] we develop weaker estimates for multiple state constraints and identify additional hypotheses under which W-1,W-1 estimates are valid in this broader setting.