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

Nakoulima, Ousseynou

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 $v=(h,k)$ 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)].

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

[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

[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

[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

[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

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

[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

[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

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

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

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

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

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

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

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

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

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

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

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