(L ∞ + Bolza) control problems as dynamic differential games
Résumé
We consider a (L ∞ + Bolza) control problem, namely a problem where the payoff is the sum of a L ∞ functional and a classical Bolza functional (the latter being an integral plus an end-point functional). Owing to the ⟨L1,L∞⟩ duality, the (L ∞+Bolza) control problem is rephrased in terms of a static differential game, where a new variable k plays the role of maximizer (we regard 1−k as the available fuel for the maximizer). The relevant fact is that this static game is equivalent to the corresponding dynamic differential game, which allows the (upper) value function to verify a boundary value problem. This boundary value problem involves a Hamilton-Jacobi equation whose Hamiltonian is continuous. The fueled value function (t,x,k) --whose restriction to k = 0 coincides with the value function of the reference (L ∞ + Bolza) problem--is continuous and solves the established boundary value problem. Furthermore, is the unique viscosity solution in the class of (not necessarily continuous) bounded solutions.