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

Abstract

Abstract. 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 L 1 , 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.Mathematics Subject Classification. 49K35 minimax problems · 49N70 differential games · 49L25 viscosity solutions.

L1limit solutions for control systems

“…Let us observe that the dependence of g α on v is much more critical than the v-dependence of f, already in the case of u with bounded variation, in that a simultaneous jump of u and v would make the determination of the corresponding jump of x quite delicate (see e.g. [33,29,34]). The issue could be actually extended to more general equations of the form ẋ = Φ(t, x, u, v, u).…”

State constrained $L^\infty$ optimal control problems interpreted as differential games

“…In [9] a new approach has been developed for optimal control problems like (1)-( 2) in absence of state constraints (i.e. E = R n ).…”

“…(In [9] W is called fueled value function for, in some way, k might be regarded as the consumed fuel, and 1 − k as the available fuel. )…”

“…the maximum of a running cost along the trajectories. In absence of state constraints, a new approach has been suggested by a recent paper [9]. The main purpose of the present paper is to extend this approach and the related results to state constrained L ∞ optimal control problems.…”

Infinite Horizon Stochastic Optimal Control Problems with Running Maximum Cost

“…In the deteministic case, problems with supremum costs and finite time horizon have been extensively studied in the literature, we refer for instance to [8,9,2,10] where the value function is characterized as unique solution of a Hamilton-Jacobi-Bellman (HJB) equation with an obstacle. The value function is also analysed within the viability framework in [34,35], and linearization techniques for such problems are presented in [23].…”