On the solution map of control systems with multiple state constraints

Cost-to-go function is recognized to be an important tool of optimal control theory. For control systems under state constraints this function is, in general, discontinuous even when all data are smooth. Similarly to unconstrained optimal control problems, local Lipschitz continuity of the cost-to-go function can be deduced if feasible trajectories of constrained control system depend on initial conditions in a Lipschitz way. However, in general, feasible trajectories do not enjoy this property and an additional inward pointing condition has to be imposed. Given a control system, a state constraint with smooth boundary and a time interval [0, T], it is known that under standard assumptions on the dynamics (i.e. measurable in time, Lipschitz in the state variable and continuous in the control) an inward pointing condition implies that the sets of feasible trajectories defined on [0, T] are Lipschitz dependent on the initial states. This is a consequence of the so-called Neighboring Feasible Trajectories theorems (NFT). However, some recent counterexamples indicate that, if the state constraint is an intersection of two half spaces in ℝⁿ (which is a very simple case of multiple state constraints), surprisingly conclusions of NFT theorems might be no longer valid. We show here that for control systems under multiple state constraints, a relaxed inward pointing condition guarantees local Lipschitz dependence of feasible trajectories on the initial states taken from the interior of constraints. As an application, for the Mayer optimal control problem, we provide sufficient conditions for the local Lipschitz continuity of the cost-to-go function on the interior of state constraint.