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Existence of feasible approximating trajectories satisfying multiple state constraints

In many areas of state-constrained optimal control, an important role is played by distance estimates, i.e. estimates on the distance of a nominal state trajectory from the class of state trajectories that satisfy the state constraint, in terms of the state constraint violation. Recently, pathologies have been revealed for state constraint sets with corners, where linear distance estimates (which hold true when the state constraint is regular) are not valid. It is possible, however, to develop super-linear estimates. In earlier work, attention was restricted to constant dynamics, i.e. models for which the set of admissible velocities do not depend on t and x. In this paper, we investigate distance estimates for state constraint sets with corners when the dynamics are (t, x) dependent. The results are somewhat unexpected: a counter-example demonstrates that the constant dynamics estimates do not generalise in the obvious way, and suggests additional hypotheses that need to be imposed to arrive at the desired distance estimates.