Approximation of control problems involving ordinary and impulsive controls

Abstract

Abstract. In this paper we study an approximation scheme for a class of control problems involving an ordinary control v, an impulsive control u and its derivativeu. Adopting a space-time reparametrization of the problem which adds one variable to the state space we overcome some difficulties connected to the presence ofu. We construct an approximation scheme for that augmented system, prove that it converges to the value function of the augmented problem and establish an error estimates in L ∞ for this approximation. Moreover, a characterization of the limit of the discrete optimal controls is given showing that it converges (in a suitable sense) to an optimal control for the continuous problem.Résumé. Dans ce papier nousétudions un schéma d'approximation pour une classe de problèmes de contrôle où la dynamique contient un contrôle mesurable v, un contrôle impulsionnel u et sa dérivéė u. En utilisant une reparamétrisation espace-temps, du problème qui introduit une nouvelle variable d'état, nous arrivonsà résoudre les difficultés liéesà la présence deu. Nous proposons un schéma d'approximation pour le système augmenté, prouvons qu'il converge vers la fonction valeur du système augmenté et nous donnons une estimation a priori dans L ∞ pour cette approximation. Nous donnons aussi une caractérisation de la limite des contrôles discrets et nous montrons qu'elle converge (dans un sensà définir) vers un contrôle optimal du problème continu.

L1limit solutions for control systems

“…This fact is the common outcome of several investigations on the subject, which share a notion of solution here referred as graph completion solution (see e.g. [11,12,13,29,14,16,15,28,17,19]). In general graph completion solutions are set-valued at a countable subset of instants and here are referred as set-valued graph completion solutions, while their selection will be called simply graph completion solution.…”

“…We shall prove that ϕ 0 • σ(t) = t, on [a, b], which implies thatφ 0 = ϕ 0 . Indeed, observe that, due to (b), for any continuity point t ∈ [a, b] of σ one has (16) t =φ 0,k •σ k (t) →φ 0 (σ(t)).…”

Consistent Approximations to Impulsive Optimal Control Problem

“…Given η = (ξ 0 N , u N , Ω N ) ∈ S N , we can use the Euler's descretization to get the discrete dynamic below by the continuous dynamic given by (4). In this way, take N ∈ N , h = 1/N the step size and s k = kh, k = 0, ..., N .…”

“…By Arzelà-Ascoli's Theorem, there exist K ⊂ N and y : [0, 1] → R n such that y η N N (·) uniformly converge to y(·), N ∈ K, N → ∞. Now, we need to show that y(·) satisfies the system (4). For this, define…”

“…We can cite, for example, [1], [11], [15], [16], [20], [21], [23]. On the other hand, the literature about numerical methods for impulsive optimal control problems is rather scarce, [4]. In this paper, a space-time reparametrization of the impulsive problem is adopted, an approximation scheme for that augmented system is construct and it is proved that such approximation converge to the value function of the augmented problem.…”