Optimal control and differential games with measures

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.

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“…These two conditions correspond exactly to the boundary conditions added in [5] to the continuous equation.…”

Section: Discretization In Time and Variation Variablesmentioning

confidence: 69%

Approximation of control problems involving ordinary and impulsive controls

Camilli

Falcone

1999

ESAIM: COCV

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“…Differential games with switching strategies and switching costs for the case of finite horizon problems is discussed in [20] while the case of an infinite horizon with both control and competing disturbance but with a discount factor in the running cost is discussed in [21]. A more general formulation of the finite-horizon optimal control and differential game problems, where a general (not necessarily discrete) measure is allowed to enter both the dynamics and the running cost, is studied in [7]. These authors, under their various assumptions, were able to show that the value function is continuous and is the unique solution of the appropriate system of quasivariational inequalities.…”

Section: Introductionmentioning

confidence: 99%

Robust Optimal Switching Control for Nonlinear Systems

Ball¹,

Chudoung²,

Day³

2002

SIAM J. Control Optim.

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Abstract. We formulate a robust optimal control problem for a general nonlinear system with finitely many admissible control settings and with costs assigned to switching of controls. We formulate the problem both in an L 2 -gain/dissipative system framework and in a game-theoretic framework. We show that, under appropriate assumptions, a continuous switching-storage function is characterized as a viscosity supersolution of the appropriate system of quasivariational inequalities (the appropriate generalization of the Hamilton-Jacobi-Bellman-Isaacs equation for this context), and that the minimal such switching-storage function is equal to the continuous switching lower-value function for the game. Finally we show how a prototypical example with one-dimensional state space can be solved by a direct geometric construction.

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“…As considering a cumulative cost is not always the best method to qualify the performance of a controlled system with a unique scalar parameter, problems of this type have received considerable interest in recent publications (see e.g. [2][3][4][5][6][7][8][9][10]). Instead of being considered as an optimal control problem with a non-standard cost functional, this problem can also be studied through other different approaches, between them, the following two:…”

Section: Description Of the Problemmentioning

confidence: 99%

Minimax optimal control problems. Numerical analysis of the finite horizon case

Marco

González

1999

ESAIM: M2AN

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Abstract. In this paper we consider the numerical computation of the optimal cost function associated to the problem that consists in finding the minimum of the maximum of a scalar functional on a trajectory. We present an approximation method for the numerical solution which employs both discretization on time and on spatial variables. In this way, we obtain a fully discrete problem that has unique solution. We give an optimal estimate for the error between the approximated solution and the optimal cost function of the original problem. Also, numerical examples are presented.Résumé. Nousétudions ici la solution numérique d'une inéquation quasi-variationnelle associéeà la minimisation du maximum d'une fonctionnelle définie sur la trajectoire d'un système dynamique gouverné par uneéquation différentielle ordinaire. Nous faisons la présentation d'une méthode d'approximation en employant des discrétisations en espace et en temps. Nous obtenons des estimations optimales pour la vélocité de convergence des solutions approchées vers la fonction de coût optimal du problème originel.

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Optimal control and differential games with measures

Cited by 24 publications

References 26 publications

Approximation of control problems involving ordinary and impulsive controls

Approximation of control problems involving ordinary and impulsive controls

Robust Optimal Switching Control for Nonlinear Systems

Minimax optimal control problems. Numerical analysis of the finite horizon case

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