Least regret control, virtual control anddecomposition methods

Abstract

"Least regret control" consists in trying tofind a control which "optimizes the situation" with the constraint of not making things tooworse with respect to a known reference control,in presence of more or less significantperturbations. This notion was introduced in [7].It is recalled on a simple example (an ellipticsystem, 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 recentnotes [9-12] the method ofvirtual control, aimed at the"decomposition of everything" (decomposition ofthe domain, of the operator, etc). Anintroduction to this method is presented, withouta 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 virtualcontrol ideas have been given in the notes[9-12] and in the note[5].One of the first systematic paper devoted to allkind of decomposition methods, including multicriteria, is a jointpaper with A. Bensoussan and R. Temam, towhom this paper is dedicated, cf. [1].