On Trajectories Satisfying a State Constraint: $W^{1,1}$ Estimates and Counterexamples

Abstract

This paper concerns properties of solutions to a differential inclusion (“F-trajectories”) satisfying a state constraint. Estimates on the $W^{1,1}$ 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 $W^{1,1}$ 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, $W^{1,1}$ estimates are not in general valid for multiple state constraints. We also identify cases involving several state constraints, where not even a weaker, linear $L^{\infty}$ estimate holds. We further show that it is possible to justify linear, $W^{1,1}$ 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 $\mathbb{R}^n$, SIAM J. Control Optim., submitted] we develop weaker estimates for multiple state constraints and identify additional hypotheses under which $W^{1,1}$ estimates are valid in this broader setting.