Solutions to the Hamilton-Jacobi equation for state constrained Bolza problems with discontinuous time dependence

Abstract

This paper provides characterizations of the value function as the unique lower semicontinuous solution, appropriately defined, to the Hamilton-Jacobi equation, for optimal control problems of Bolza type with state constraints. Notable features of our analysis are that it covers problems in which the running cost is possibly non Lipschitz continuous w.r.t. the state variable and the dynamic constraint takes the form of a differential inclusion that is possibly discontinuous w.r.t. time. Alternative characterizations are given, in terms of lower Dini derivative type solutions and of proximal solutions to the Hamilton-Jacobi equation.Distance estimates, which permit us to approximate an arbitrary state trajectory by a state trajectory that strictly satisfies the state constraint, have a key role in the analysis. Earlier research, treating problems in which the running cost is absent or at least Lipschitz continuous w.r.t. the state, makes use of known distance estimates in which ‘approximate’ is interpreted in terms of the L∞ norm. L∞ distance estimates are inadequate, however, for problems in which the running cost may fail to be Lipschitz continuous. We show that, for this broader class of problems, the analysis can still be carried out, if it is based on stronger distance estimates involving the W1,1 norm. The new W1,1 distance estimates, developed here for this purpose, are of independent interest.Motivation for the study of problems with running costs that are not Lipschitz continuous is provided by a ‘growth versus consumption’ problem in neo-classical macro-economics. Our theory provides fresh insights into this much studied problem, by extending the class of initial data points for which a solution can be achieved.