# Least regret control, virtual control and decomposition methods

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

- Volume: 34, Issue: 2, page 409-418
- ISSN: 0764-583X

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topLions, Jacques-Louis. "Least regret control, virtual control and decomposition methods." ESAIM: Mathematical Modelling and Numerical Analysis 34.2 (2010): 409-418. <http://eudml.org/doc/197478>.

@article{Lions2010,

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].
},

author = {Lions, Jacques-Louis},

journal = {ESAIM: Mathematical Modelling and Numerical Analysis},

keywords = {Control; least regret; domain decomposition; virtuel control.; least regret control; virtual control; second-order elliptic operator; quadratic functional; optimal control; algorithm},

language = {eng},

month = {3},

number = {2},

pages = {409-418},

publisher = {EDP Sciences},

title = {Least regret control, virtual control and decomposition methods},

url = {http://eudml.org/doc/197478},

volume = {34},

year = {2010},

}

TY - JOUR

AU - Lions, Jacques-Louis

TI - Least regret control, virtual control and decomposition methods

JO - ESAIM: Mathematical Modelling and Numerical Analysis

DA - 2010/3//

PB - EDP Sciences

VL - 34

IS - 2

SP - 409

EP - 418

AB -
"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].

LA - eng

KW - Control; least regret; domain decomposition; virtuel control.; least regret control; virtual control; second-order elliptic operator; quadratic functional; optimal control; algorithm

UR - http://eudml.org/doc/197478

ER -

## References

top- A. Bensoussan, J.L. Lions and R. Temam, Sur les méthodes de décomposition, de décentralisation et de coordination et applications, Méthodes Mathématiques de l'Informatique, J.L. Lions and G.I. Marchuk Eds, Dunod, Paris (1974) 133-257.
- I. Ekeland and R. Temam, Analyse convexe et problèmes variationnels, Dunod, Gauthier-Villars (1974).
- D. Gabay and J.L. Lions, Décisions stratégiques à moindres regrets. C.R. Acad. Sci. Paris319 (1994) 1049-1056.
- R. Glowinski and J.L. Lions, Exact and approximate controllability for distributed parameter systems. Acta Numer. (1994) 269-378 and (1995) 159-333.
- R. Glowinski, J.L. Lions and O.Pironneau, Decomposition of energy spaces and applications. C.R. Acad. Sci. Paris329 (1999) 445-452.
- J.L. Lions, Contrôle optimal des systèmes gouvernés par des équations aux dérivées partielles, Paris, Dunod, Gauthier-Villars (1968).
- J.L. Lions, Contrôle à moindres regrets des systèmes distribués. C.R. Acad. Sci. Paris315 (1992) 1253-1257.
- J.L. Lions and E.Magenes, Problèmes aux limites non homogènes et applications, Dunod, Paris (1968) Vol. 1 and 2.
- J.L. Lions and O. Pironneau, Algorithmes parallèles pour la solution de problèmes aux limites. C.R. Acad. Sci. Paris327 (1998) 947-952.
- J.L. Lions and O. Pironneau, Domain decomposition methods for CAD. C.R. Acad. Sci. Paris328 (1999) 73-80.
- J.L. Lions and O. Pironneau, Décomposition d'opérateurs, répliques et contrôle virtuel. C.R. Acad. Sci. Paris (to appear).
- J.L. Lions and O. Pironneau, Sur le contrôle parallèle de systèmes distribués. C.R. Acad. Sci. Paris327 (1998) 993-998.
- L.J. Savage, The foundations of Statistics, 2nd edition, Dover (1972).

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