Minimax -: or feared value -: L1/L∞ control

This is the third paper, after Bernhard (Expected value, feared value and partial information optimal control, in G.J. Olsder (Ed.), New Trends in Dynamic Games and Applications, Annals of the International Society of Dynamic Games, vol. 3, Birkhauser, Boston, USA, 1995, pp. 95-100; A Separation Theorem for Expected Value and Feared Value Discrete Time Control, COCV, Vol. 1, pp. 191-206, SMAI, www.emath.fr/cocv, 1996.), where we attempt to develop and exploit the parallel between stochastic control and min-max control induced by the use of the max-plus algebra, using the concept of feared value as the parallel to expected value. The present paper builds on the main formula of Bernhard (1996), the results of which are a subset of those given here. Its new contribution is twofold. On the one hand we clarify the role of the integral part of the cost in that parallel. This leads to a more extensive theory of the so-called (imperfect information) minimax. L-infinity control problem than apparently available in the literature including a certainty equivalence theorem. On the other hand, we extend the parallel to the continuous time case as much as we can. In that direction, the present paper slightly extends the classical framework of the variational inequality of stopping time games, and gives a formal treatment of the partial information case. Altogether, this might be the first new results obtained with the tool of mathematical fear. (C) 2002 Elsevier Science B.V. All rights reserved.