Insensitizing control for linear and semi-linear heat equations with partially unknown domain

Lissy, Pierre; Privat, Yannick; Simporé, Yacouba

ESAIM: Control, Optimisation and Calculus of Variations, Tome 25 (2019), article no. 50.

Suite au passage du modèle économique de la revue en S20, le texte intégral des articles des années 2019 et 2020 est accessible uniquement sur le site de l'éditeur et est réservé aux abonnés.

We consider a semi-linear heat equation with Dirichlet boundary conditions and globally Lipschitz nonlinearity, posed on a bounded domain of ℝ$$ (N ∈ ℕ^*), assumed to be an unknown perturbation of a reference domain. We are interested in an insensitizing control problem, which consists in finding a distributed control such that some functional of the state is insensitive at the first order to the perturbations of the domain. Our first result consists of an approximate insensitization property on the semi-linear heat equation. It rests upon a linearization procedure together with the use of an appropriate fixed point theorem. For the linear case, an appropriate duality theory is developed, so that the problem can be seen as a consequence of well-known unique continuation theorems. Our second result is specific to the linear case. We show a property of exact insensitization for some families of deformation given by one or two parameters. Due to the nonlinearity of the intrinsic control problem, no duality theory is available, so that our proof relies on a geometrical approach and direct computations.

[1] O. Bodart and P. Demeestere, Sentinels for the identification of an unknown boundary. Math. Model. Methods Appl. Sci. 7 (1997) 871. | DOI | MR | Zbl

[2] O. Bodart and C. Fabre, Controls insensitizing the norm of the solution of a semi-linear heat equation. J. Math. Anal. Appl. 195 (1995) 658–683. | DOI | MR | Zbl

[3] O. Bodart, M. Gonzalez-Burgos and R. Pérez-Garcia, A local result on insensitizing control for a semi-linear heat equation with nonlinear boundary Fourier conditions. SIAM J. Control Optim. 43 (2004) 955–969. | DOI | MR | Zbl

[4] N. Carreño, S. Guerrero and M. Gueye, Insensitizing controls with two vanishing components for the three-dimensional Boussinesq system. ESAIM: COCV 21 (2015) 73–100. | Numdam | MR | Zbl

[5] N. Carreño and M. Gueye, Insensitizing controls with one vanishing component for the Navier–Stokes system. J. Math. Pures Appl. 101 (2014) 27–53. | DOI | MR | Zbl

[6] N. Carreño, Insensitizing controls for the Boussinesq system with no control on the temperature equation. Adv. Differ. Equ. 22 (2017) 235–258. | MR | Zbl

[7] M. Delfour and J.P. Zolésio, Advances in design and control, in Shapes and geometries. Analysis, differential calculus, and optimization. SIAM, Philadelphia, PA (2001). | MR | Zbl

[8] L. De Teresa, Insensitizing Controls for a semi-linear heat equation. Commun. Part. Differ. Equ. 25 (2000) 39–72. | DOI | MR | Zbl

[9] L. De Teresa and O. Kavian, Unique continuation principle for systems of parabolic equations. ESAIM: COCV 16 (2010) 247–274. | Numdam | MR | Zbl

[10] L. De Teresa and E. Zuazua, Identification of the class of initial data for the insensitizing control of the heat equation. Commun. Pure Appl. Anal. 8 (2009) 457–471. | DOI | MR | Zbl

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

[12] S. Guerrero, Null controllability of some systems of two parabolic equations with one control force. SIAM J. Control Optim. 46 (2007) 379–394. | DOI | MR | Zbl

[13] S. Guerrero, Controllability of systems of Stokes equations with one control force: existence of insensitizing controls. Ann. Inst. Henri Poincaré Anal. Non Linéaire 24 (2007) 1029–1054. | DOI | Numdam | MR | Zbl

[14] M. Gueye, Insensitizing controls for the Navier–Stokes equations. Ann. Inst. Henri Poincaré Anal. Non Linéaire 30 (2013) 825–844. | DOI | Numdam | MR | Zbl

[15] A. Henrot and M. Pierre, Variation et optimisation de formes. In Vol. 48. Springer-Verlag, Berlin, Heidelberg (2005). | DOI | MR | Zbl

[16] J.-L. Lions, Remarques préliminaires sur le contrôle des systèmes a données incomplètes. in Actas del Congreso de Ecuaciones Diferenciales y Aplicaciones (CEDYA), Universidad de Malaga (1989) 43–54. | Zbl

[17] J.-L. Lions, Contrôlabilité exacte, perturbations et stabilisation des systèmes distribués. Tome 1, Controlabilitè exacte, Collection R.M.A 8, Paris, Masson (1988). | MR | Zbl

[18] X. Liu, Insensitizing controls for a class of quasilinear parabolic equations. J. Differ. Equ. 253 (2012) 1287–1316. | DOI | MR | Zbl

[19] F. Méhats, Y. Privat and M. Sigalotti, On the controllability of quantum transport in an electronic nanostructure. SIAM J. Appl. Math. 74 (2014) 1870–1894. | DOI | MR | Zbl

[20] S. Micu, J.H. Ortega and L. De Teresa, An example of ε-insensitizing controls for the heat equation with no intersecting observation and control regions. Appl. Math. Lett. 8 (2004) 927–932. | DOI | MR | Zbl

[21] Y. Privat and M. Sigalotti, The squares of the Laplacian–Dirichlet eigenfunctions are generically linearly independent. ESAIM: COCV 16 (2010) 794–805. | Numdam | MR | Zbl

[22] J.-C. Saut and B. Scheurer, Unique continuation for some evolution equations. J. Differ. Equ. 66 (1987) 118–139. | DOI | MR | Zbl

[23] Y. Simporé, O. Traoré and O. Nakoulima, Insensitizing control with constraints on the control for the semi-linear heat equation. Nonlinear Stud. 20 (2013) 203–216. | MR | Zbl

[24] Y. Simporé and O. Traoré, Insensitizing control with constraints on the control of the semi-linear heat equation. J. Nonlinear Evol. Equ. Appl. 1 (2017) 1–12. | Zbl

[25] J. Sokołowski and J.-P. Zolésio, Shape sensitivity analysis, in Introduction to shape optimization. Vol. 16 of Springer Series in Computational Mathematics. Springer-Verlag, Berlin (1992). | MR | Zbl

[26] M. Tucsnak and G. Weiss, Observation and control for operator semigroups. Birkhäuser, Basel (2009). | DOI | MR | Zbl