Least regret control, virtual control and decomposition methods

Collège de France, 3 rue d'Ulm, 75231 Paris Cedex 05, France.

Received: 18 October 1999

Abstract

"Least regret control" consists in trying to find a control which "optimizes the situation" with the constraint of not making things too worse with respect to a known reference control, in presence of more or less significant perturbations. This notion was introduced in [7]. It is recalled on a simple example (an elliptic system, with distributed control and boundary perturbation) in Section 2. We show that the problem reduces to a standard optimal control problem for augmented state equations. On another hand, we have introduced in recent notes [9-12] the method of virtual control, aimed at the "decomposition of everything" (decomposition of the domain, of the operator, etc). An introduction to this method is presented, without a priori knowledge needed, in Sections 3 and 4, directly on the augmented state equations. For problems without control, or with "standard" control, numerical applications of the virtual control ideas have been given in the notes [9-12] and in the note [5]. One of the first systematic paper devoted to all kind of decomposition methods, including multicriteria, is a joint paper with A. Bensoussan and R. Temam, to whom this paper is dedicated, cf. [1].