Hostname: page-component-78c5997874-xbtfd Total loading time: 0 Render date: 2024-11-17T20:06:41.535Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Published online by Cambridge University Press:  05 September 2007

Ousseynou Nakoulima
Ousseynou Nakoulima*
Affiliation:
Université Antilles-Guyane, Département de Mathématiques et Informatique, 97159 Pointe-à-Pitre, Guadeloupe, France; [email protected]
Get access

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 controlpoint 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)].

Keywords

Heat equationoptimal controlcontrollabilityCarleman inequalitiessentinels
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 13 , Issue 4 , October 2007 , pp. 623 - 638
Copyright
© EDP Sciences, SMAI, 2007

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Barbu, V., Exact controllability of the superlinear heat equation. Appl. Math. Optim. 42 (2000) 73–89. CrossRef
T. Cazenave and A. Haraux, Introduction aux Problèmes d'Evolution Semi-Linéaires, Collection Mathématiques et Applications de la SMAI. Éditions Ellipses, Paris (1991).
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).
Dorville, R., Nakoulima, O. and Omrane, A., Low-regret control for singular distributed systems: The backwards heat ill-posed problem. Appl. Math. Lett. 17 (2004) 549–552. CrossRef
Doubova, A., Osses, A. and Puel, J.P., Exact controllability to trajectories for semilinear heat equations with discontinuous diffusion coefficients. ESAIM: COCV 8 (2002) 621–661. CrossRef
Fabre, C., Puel, J.P. and Zuazua, E., Approximate controllability of the semilinear heat equation. Proc. Royal Soc. Edinburg 125A (1995) 31–61. CrossRef
Fernández-Cara, E., Nul controllability of the semilinear heat equation. ESAIM: COCV 2 (1997) 87–103. CrossRef
Fernández-Cara, E. and Guerrero, S., Global Carleman inequalities for parabolic systems and applications to controllability. SIAM J. Control Optim. 45 (2006) 1395–1446. CrossRef
Fernández-Cara, E. and Zuazua, E., The cost of approximate controllability for heat equations: the linear case. Adv. Differ. Equ. 5 (2000) 465–514.
A. Fursikov and O.Yu. Imanuvilov, Controllability of evolution equations, Lecture Notes. Research Institute of Mathematics, Seoul National University, Korea (1996).
Imanuvilov, O.Yu., Controllability of parabolic equations. Sbornik Math. 186 (1995) 879–900.
Lebeau, G. and Robbiano, L., Contrôle exacte de l'équation de la chaleur. Comm. Part. Diff. Eq. 20 (1995) 335–356. CrossRef
J.L. Lions, Contrôle optimal de systèmes gouvernés par des équations aux dérivées partielles. Dunod, Gauthier-Villars, Paris (1968).
J.L. Lions, Sentinelles pour les systèmes distribués à données incomplètes. Masson, Paris (1992).
J.L. Lions and M. Magenes, Problèmes aux limites non homogènes et applications. Vols. 1 et 2, Dunod, Paris (1988).
Nakoulima, O., Contrôlabilité à zéro avec contraintes sur le contrôle. C. R. Acad. Sci. Paris Ser. I Math. 339 (2004) 405–410. CrossRef
Russell, D.L., A unified boundary controllability theory for hyperbolic and parabolic partial differential equations. Stud. App. Math. 52 (1973) 189–212. CrossRef
Zuazua, E., Exact boundary controllability for the semilinear wave equation. Non linear Partial Diff. Equ. Appl. 10 (1989) 357–391.
Zuazua, E., Finite dimensional null controllability for the semilinear heat equation. J. Math. Pures Appl. 76 (1997) 237–264. CrossRef
Zuazua, E., controllability of partial differential equations and its semi-discrete approximations. Discrete Continuous Dynam. Syst. 8 (2002) 469–513. CrossRef