Approximation of control problems involving ordinary and impulsive controls

Free access article

Issue		ESAIM: COCV Volume 4, 1999


Page(s)		159 - 176
DOI		10.1051/cocv:1999108

DOI: 10.1051/cocv:1999108

ESAIM: COCV, May 1999, Vol. 4, p. 159-176

Approximation of control problems involving ordinary and impulsive controls

Approximation de problèmes de contrôle avec des contrôles ordinaires et impulsionnels

Fabio Camilli
Università di Torino, Via Carlo Alberto 10, 10123 Torino, Italy; (Camilli@dm.unito.it)

Maurizio Falcone
Università di Roma "La Sapienza", P.le Aldo Moro 2, 00185 Roma, Italy; (Falcone@axcasp.caspur.it)

Received August 1, 1997. Revised August 6, 1998.

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 derivative $\dot u$ . Adopting a space-time reparametrization of the problem which adds one variable to the state space we overcome some difficulties connected to the presence of $\dot u$ .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 $L^\infty$ 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 nous étudions 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 dérivée $\dot u$ . En utilisant une reparamétrisation espace-temps, du problème qui introduit une nouvelle variable d'état, nous arrivons à résoudre les difficultés liées à la présence de $\dot u$ . 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 $L^\infty$ pour cette approximation. Nous donnons aussi une caractérisation de la limite des contrôles discrets et nous montrons qu'elle converge (dans un sens à définir) vers un contrôle optimal du problème continu.

Keywords and phrases: Impulsive control, approximation scheme, dynamic programming, viscosity solution

AMS Subject Classification: 65M12, 49N25; Secondary 49L20

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