Least regret control, virtual control and decomposition methods

Jacques-Louis Lions

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

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

Abstract

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"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].

How to cite

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Lions, 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

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

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