Optimal control for distributed systems subject to null-controllability. Application to discriminating sentinels

Ousseynou Nakoulima

ESAIM: Control, Optimisation and Calculus of Variations (2007)

  • Volume: 13, Issue: 4, page 623-638
  • ISSN: 1292-8119

Abstract

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We consider a distributed system in which the state q is governed by a parabolic equation and a pair of controls v = (h,k) where h and k play two different roles: the control k is of controllability type while h expresses that the state q does not move too far from a given state. Therefore, it is natural to introduce the control point of view. In fact, there are several ways to state and solve optimal control problems with a pair of controls h and k, in particular the Least Squares method with only one criteria for the pair (h,k) or the Pareto Optimal Control for multicriteria problems. We propose here to use the notion of Hierarchic Control. This notion assumes that we have two controls h, k where h will be the leader while k will be the follower. The main tool used to solve the null-controllability problem with constraints on the follower is an observability inequality of Carleman type which is “adapted” to the constraints. The obtained results are applied to the sentinels theory of Lions [Masson (1992)].

How to cite

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Nakoulima, Ousseynou. "Optimal control for distributed systems subject to null-controllability. Application to discriminating sentinels." ESAIM: Control, Optimisation and Calculus of Variations 13.4 (2007): 623-638. <http://eudml.org/doc/249999>.

@article{Nakoulima2007,
abstract = { We consider a distributed system in which the state q is governed by a parabolic equation and a pair of controls v = (h,k) where h and k play two different roles: the control k is of controllability type while h expresses that the state q does not move too far from a given state. Therefore, it is natural to introduce the control point of view. In fact, there are several ways to state and solve optimal control problems with a pair of controls h and k, in particular the Least Squares method with only one criteria for the pair (h,k) or the Pareto Optimal Control for multicriteria problems. We propose here to use the notion of Hierarchic Control. This notion assumes that we have two controls h, k where h will be the leader while k will be the follower. The main tool used to solve the null-controllability problem with constraints on the follower is an observability inequality of Carleman type which is “adapted” to the constraints. The obtained results are applied to the sentinels theory of Lions [Masson (1992)]. },
author = {Nakoulima, Ousseynou},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Heat equation; optimal control; controllability; Carleman inequalities; sentinels; Pareto optimal control},
language = {eng},
month = {9},
number = {4},
pages = {623-638},
publisher = {EDP Sciences},
title = {Optimal control for distributed systems subject to null-controllability. Application to discriminating sentinels},
url = {http://eudml.org/doc/249999},
volume = {13},
year = {2007},
}

TY - JOUR
AU - Nakoulima, Ousseynou
TI - Optimal control for distributed systems subject to null-controllability. Application to discriminating sentinels
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2007/9//
PB - EDP Sciences
VL - 13
IS - 4
SP - 623
EP - 638
AB - We consider a distributed system in which the state q is governed by a parabolic equation and a pair of controls v = (h,k) where h and k play two different roles: the control k is of controllability type while h expresses that the state q does not move too far from a given state. Therefore, it is natural to introduce the control point of view. In fact, there are several ways to state and solve optimal control problems with a pair of controls h and k, in particular the Least Squares method with only one criteria for the pair (h,k) or the Pareto Optimal Control for multicriteria problems. We propose here to use the notion of Hierarchic Control. This notion assumes that we have two controls h, k where h will be the leader while k will be the follower. The main tool used to solve the null-controllability problem with constraints on the follower is an observability inequality of Carleman type which is “adapted” to the constraints. The obtained results are applied to the sentinels theory of Lions [Masson (1992)].
LA - eng
KW - Heat equation; optimal control; controllability; Carleman inequalities; sentinels; Pareto optimal control
UR - http://eudml.org/doc/249999
ER -

References

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  1. V. Barbu, Exact controllability of the superlinear heat equation. Appl. Math. Optim.42 (2000) 73–89.  Zbl0964.93046
  2. T. Cazenave and A. Haraux, Introduction aux Problèmes d'Evolution Semi-Linéaires, Collection Mathématiques et Applications de la SMAI. Éditions Ellipses, Paris (1991).  
  3. R. Dorville, Sur le contrôle de quelques problèmes singuliers associés à l'équation de la chaleur. Ph.D. thesis, Université des Antilles et de la Guyane (2004).  
  4. R. Dorville, O. Nakoulima and A. Omrane, Low-regret control for singular distributed systems: The backwards heat ill-posed problem. Appl. Math. Lett.17 (2004) 549–552.  Zbl1065.49004
  5. A. Doubova, A. Osses and J.P. Puel, Exact controllability to trajectories for semilinear heat equations with discontinuous diffusion coefficients. ESAIM: COCV8 (2002) 621–661.  Zbl1092.93006
  6. C. Fabre, J.P. Puel and E. Zuazua, Approximate controllability of the semilinear heat equation. Proc. Royal Soc. Edinburg125A (1995) 31–61.  Zbl0818.93032
  7. E. Fernández-Cara, Nul controllability of the semilinear heat equation. ESAIM: COCV2 (1997) 87–103.  Zbl0897.93011
  8. E. Fernández-Cara and S. Guerrero, Global Carleman inequalities for parabolic systems and applications to controllability. SIAM J. Control Optim.45 (2006) 1395–1446.  Zbl1121.35017
  9. E. Fernández-Cara and E. Zuazua, The cost of approximate controllability for heat equations: the linear case. Adv. Differ. Equ.5 (2000) 465–514.  Zbl1007.93034
  10. A. Fursikov and O.Yu. Imanuvilov, Controllability of evolution equations, Lecture Notes. Research Institute of Mathematics, Seoul National University, Korea (1996).  
  11. O.Yu. Imanuvilov, Controllability of parabolic equations. Sbornik Math.186 (1995) 879–900.  
  12. G. Lebeau and L. Robbiano, Contrôle exacte de l'équation de la chaleur. Comm. Part. Diff. Eq.20 (1995) 335–356.  
  13. J.L. Lions, Contrôle optimal de systèmes gouvernés par des équations aux dérivées partielles. Dunod, Gauthier-Villars, Paris (1968).  Zbl0179.41801
  14. J.L. Lions, Sentinelles pour les systèmes distribués à données incomplètes. Masson, Paris (1992).  
  15. J.L. Lions and M. Magenes, Problèmes aux limites non homogènes et applications. Vols. 1 et 2, Dunod, Paris (1988).  Zbl0165.10801
  16. O. Nakoulima, Contrôlabilité à zéro avec contraintes sur le contrôle. C. R. Acad. Sci. Paris Ser. I Math.339 (2004) 405–410.  
  17. D.L. Russell, A unified boundary controllability theory for hyperbolic and parabolic partial differential equations. Stud. App. Math.52 (1973) 189–212.  Zbl0274.35041
  18. E. Zuazua, Exact boundary controllability for the semilinear wave equation. Non linear Partial Diff. Equ. Appl.10 (1989) 357–391.  
  19. E. Zuazua, Finite dimensional null controllability for the semilinear heat equation. J. Math. Pures Appl.76 (1997) 237–264.  Zbl0872.93014
  20. E. Zuazua, controllability of partial differential equations and its semi-discrete approximations. Discrete Continuous Dynam. Syst.8 (2002) 469–513.  Zbl1005.35019

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