(L ∞ + Bolza) control problems as dynamic differential games

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 W(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, W is the unique viscosity solution in the class of (not necessarily continuous) bounded solutions. ntial Equations and Applications NoDEA Look Inside Share Share this content on Facebook Share this content on Twitter Share this content on LinkedIn Other actions Export citations Register for Journal Updates About This Journal Reprints and Permissions