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

Abstract

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.

Penalization techniques in L ∞ optimization problems with unbounded horizon

“…The proof of the rate of convergence is a consequence of definition of U k ε and the results obtained in Di Marco and González (1999b).…”

Equivalent formulations of optimal control problems with maximum cost and applications

“…Let Π 0 = (x 0 , y 0 , y 0 ) and denote by S the set of absolutely continuous solutions of (7) with Π(0) = Π 0 ∈ D × R. We consider the Mayer problem…”

“…without boundary condition. Nevertheless, although necessary optimality conditions and numerical procedures have been formulated [2,6,7,8], there is no practical numerical tool to solve such problems as it exists for Mayer problems, to the best of our knowledge. The aim of the present work is to study different reformulations of this problem into Mayer form in higher dimension with possibly state or mixed constraint, for which existing numerical methods can be used.…”

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

“…L = 0, g = 0) has been extensively studied from various point of views, including dynamic programming, numerical approximations, and necessary conditions (see e.g. [4][5][6][7][8][9][10][11][14][15][16]19,20]). In particular, in [7] Barron and Ishii established a Hamilton-Jacobi equation by regarding a L ∞ problem as the limit of standard L p optimal control problems.…”