(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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\langle}L^1,L^\infty{\rangle}}$$\end{document} 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\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal W}(t,x,k)}$$\end{document} —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, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal W}}$$\end{document} is the unique viscosity solution in the class of (not necessarily continuous) bounded solutions.