Optimal control for distributed systems subject to null-controllability. Application to discriminating sentinels
ESAIM: Control, Optimisation and Calculus of Variations, Tome 13 (2007) no. 4, pp. 623-638.

We consider a distributed system in which the state $q$ is governed by a parabolic equation and a pair of controls $v=\left(h,k\right)$ 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 $\left(h,k\right)$ 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)].

DOI : https://doi.org/10.1051/cocv:2007038
Classification : 35K05,  35K15,  35K20,  49J20,  93B05
Mots clés : heat equation, optimal control, controllability, Carleman inequalities, sentinels
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author = {Nakoulima, Ousseynou},
title = {Optimal control for distributed systems subject to null-controllability. {Application} to discriminating sentinels},
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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, Tome 13 (2007) no. 4, pp. 623-638. doi : 10.1051/cocv:2007038. http://www.numdam.org/articles/10.1051/cocv:2007038/

[1] V. Barbu, Exact controllability of the superlinear heat equation. Appl. Math. Optim. 42 (2000) 73-89. | Zbl 0964.93046

[2] T. Cazenave and A. Haraux, Introduction aux Problèmes d'Evolution Semi-Linéaires, Collection Mathématiques et Applications de la SMAI. Éditions Ellipses, Paris (1991). | Zbl 0786.35070

[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. | Zbl 1065.49004

[5] A. Doubova, A. Osses and J.P. Puel, Exact controllability to trajectories for semilinear heat equations with discontinuous diffusion coefficients. ESAIM: COCV 8 (2002) 621-661. | Numdam | Zbl 1092.93006

[6] C. Fabre, J.P. Puel and E. Zuazua, Approximate controllability of the semilinear heat equation. Proc. Royal Soc. Edinburg 125A (1995) 31-61. | Zbl 0818.93032

[7] E. Fernández-Cara, Nul controllability of the semilinear heat equation. ESAIM: COCV 2 (1997) 87-103. | Numdam | Zbl 0897.93011

[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. | Zbl 1121.35017

[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. | Zbl 1007.93034

[10] A. Fursikov and O.Yu. Imanuvilov, Controllability of evolution equations, Lecture Notes. Research Institute of Mathematics, Seoul National University, Korea (1996). | MR 1406566 | Zbl 0862.49004

[11] O.Yu. Imanuvilov, Controllability of parabolic equations. Sbornik Math. 186 (1995) 879-900. | Zbl 0845.35040

[12] G. Lebeau and L. Robbiano, Contrôle exacte de l'équation de la chaleur. Comm. Part. Diff. Eq. 20 (1995) 335-356. | Zbl 0819.35071

[13] J.L. Lions, Contrôle optimal de systèmes gouvernés par des équations aux dérivées partielles. Dunod, Gauthier-Villars, Paris (1968). | MR 244606 | Zbl 0179.41801

[14] J.L. Lions, Sentinelles pour les systèmes distribués à données incomplètes. Masson, Paris (1992). | MR 1159093 | Zbl 0759.93043

[15] J.L. Lions and M. Magenes, Problèmes aux limites non homogènes et applications. Vols. 1 et 2, Dunod, Paris (1988). | Zbl 0165.10801

[16] O. Nakoulima, Contrôlabilité à zéro avec contraintes sur le contrôle. C. R. Acad. Sci. Paris Ser. I Math. 339 (2004) 405-410. | Zbl 1060.93015

[17] D.L. Russell, A unified boundary controllability theory for hyperbolic and parabolic partial differential equations. Stud. App. Math. 52 (1973) 189-212. | Zbl 0274.35041

[18] E. Zuazua, Exact boundary controllability for the semilinear wave equation. Non linear Partial Diff. Equ. Appl. 10 (1989) 357-391. | Zbl 0731.93011

[19] E. Zuazua, Finite dimensional null controllability for the semilinear heat equation. J. Math. Pures Appl. 76 (1997) 237-264. | Zbl 0872.93014

[20] E. Zuazua, controllability of partial differential equations and its semi-discrete approximations. Discrete Continuous Dynam. Syst. 8 (2002) 469-513. | Zbl 1005.35019

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