Control For Energy and Sustainability

EPSRC Programme Grant

[BFV11] P.Bettiol, H.Frankowska and R.B.Vinter, L infinity estimates on trajectories confined to a closed subset, Journal of Differential Equations, 252, 2012, pp. 1912-1933., 2012


This paper concerns the validity of estimates on the distance of an arbitrary state trajectory from the set of state trajectories which lie in a given state constraint set. These so called distance estimates have wide-spread application in state constrained optimal control, including justifying the use of the Maximum Principle in normal form and establishing regularity properties of value functions. We focus on linear, L-infinity distance estimates which, of all the available estimates have, so far, been the most widely used. Such estimates are known to be valid for general, closed state constraint sets, provided the functions defining the dynamic constraint are Lipschitz continuous, with respect to the time and state variables. We ask whether linear, L-infinity distance estimates remain valid when the Lipschitz continuity hypothesis governing $t$-dependence of the data is relaxed. We show by counter-example that these distance estimates are not valid in general if the hypothesis of Lipschitz continuity is replaced by continuity. We also provide a new hypothesis, `absolute continuous from the left', for the validity of linear, L-infinity estimates. The new hypothesis is less restrictive than Lipschitz continuity and even allows discontinuous time dependence in certain cases. The new hypothesis allows non-Lipschitz timedependence at isolated points, governed, for example, by a fractional-power modulus of continuity. The relevance of distance estimates for state constrained control systems permitting fractional-power time dependence is illustrated by an example in engineering design, where we encounter an isolated, square root type singularity, concerning the $t$-dependence of the data.