Control For Energy and Sustainability

EPSRC Programme Grant

[BBV10] P.Bettiol, A.Bressan and R.B.Vinter, On trajectories satisfying a state constraint: W(1,1) Estimates and Counter-Examples, SIAM Journal on Control and Optimisation, vol 48, no 7, pp 4664-4679, 2010


This paper concerns properties of solutions to a differential inclusion (`trajectories') satisfying a state constraint. Estimates on the W(1,1) distance of a given F-trajectory to the set of trajectories satisfying the constraint have an important role in state-constrained optimal control theory, regarding the derivation of non-degenerate 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 counter-example, 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 infinity 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 companion paper we develop weaker estimates for multiple state constraints, and identify additional hypotheses under which W(1,1) estimates are valid in this broader setting.