March  2019, 9(1): 117-158. doi: 10.3934/mcrf.2019007

Insensitizing controls for a semilinear parabolic equation: A numerical approach

1. 

Institut de Mathématiques de Toulouse & Institut universitaire de France, UMR 5219, Université de Toulouse, CNRS, UPS IMT, F-31062 Toulouse Cedex 9, France

2. 

Depto. de Control Automático, CINVESTAV-IPN, Apartado Postal 14-740, 07000 México, D.F., México

3. 

DeustoTech, University of Deusto, 48007 Bilbao, Basque Country, Spain

4. 

Facultad de Ingeniería, Universidad de Deusto, Avda Universidades 24, 48007 Bilbao, Basque Country, Spain

5. 

Instituto de Matemáticas, Universidad Nacional Autónoma de México, Circuito Exterior, C.U. 04510 México D.F., México

* Corresponding author: F. Boyer

Received  May 2017 Revised  November 2017 Published  September 2018

In this paper, we study the insensitizing control problem in the discrete setting of finite-differences. We prove the existence of a control that insensitizes the norm of the observed solution of a 1-D semi discrete parabolic equation. We derive a (relaxed) observability estimate that yields a controllability result for the cascade system arising in the insensitizing control formulation. Moreover, we deal with the problem of computing numerical approximations of insensitizing controls for the heat equation by using the Hilbert Uniqueness Method (HUM). We present various numerical illustrations.

Citation: Franck Boyer, Víctor Hernández-Santamaría, Luz De Teresa. Insensitizing controls for a semilinear parabolic equation: A numerical approach. Mathematical Control & Related Fields, 2019, 9 (1) : 117-158. doi: 10.3934/mcrf.2019007
References:
[1]

F. Ammar-KhojdaA. BenabdallahM. González-Burgos and L. de Teresa, New phenomena for the null controllability of parabolic systems: Minimal time and geometrical dependence, J. Math. Anal. Appl., 444 (2016), 1071-1113.  doi: 10.1016/j.jmaa.2016.06.058.  Google Scholar

[2]

O. Bodart and C. Fabre, Controls insensitizing the norm of the solutions of a semilinear heat equation, J. Math. Anal. and App., 195 (1995), 658-683.  doi: 10.1006/jmaa.1995.1382.  Google Scholar

[3]

O. BodartM. González-Burgos and R. Pérez-García, Existence of insensitizing controls for a semilinear heat equation with a superlinear nonlinearity, Communications in Partial Differential Equations, 29 (2004), 1017-1050.  doi: 10.1081/PDE-200033749.  Google Scholar

[4]

O. BodartM. González-Burgos and R. Pérez-García, Insensitizing controls for a heat equation with a nonlinear term involving the state and the gradient, Nonlinear Anal., 57 (2004), 687-711.  doi: 10.1016/j.na.2004.03.012.  Google Scholar

[5]

F. Boyer, On the penalised HUM approach and its applications to the numerical approximation of null-controls for parabolic problems, ESAIM Proceedings, 41 (2013), 15-58.  doi: 10.1051/proc/201341002.  Google Scholar

[6]

F. BoyerF. Hubert and J. Le Rousseau, Discrete Carleman estimates for elliptic operators and uniform controllability of semi-discretized parabolic equations, J. Math Pures Appl., 93 (2010), 240-276.  doi: 10.1016/j.matpur.2009.11.003.  Google Scholar

[7]

F. BoyerF. Hubert and J. Le Rousseau, Discrete Carleman estimates for elliptic operators in arbitrary dimension and applications, SIAM J. Control Optim., 48 (2010), 5357-5397.  doi: 10.1137/100784278.  Google Scholar

[8]

F. BoyerF. Hubert and J. Le Rousseau, Uniform null-controllability for space/time-discretized parabolic equations, Numer. Math., 118 (2011), 601-661.  doi: 10.1007/s00211-011-0368-1.  Google Scholar

[9]

F. Boyer and J. Le Rousseau, Carleman estimates for semi-discrete parabolic operators and application to the controllability of semi-linear and semi-discrete parabolic equations, Ann. I. H. Poincaré-AN, 31 (2014), 1035-1078.  doi: 10.1016/j.anihpc.2013.07.011.  Google Scholar

[10]

N. CarreñoM. Gueye and S. Guerrero, Insensitizing control with two vanishing components for the three-dimensional Boussinesq system, ESAIM Control Optim. Calc. Var., 21 (2015), 73-100.  doi: 10.1051/cocv/2014020.  Google Scholar

[11]

I. Ekeland and R. Temam, Convex Analysis and Variational Problems, North-Holland, 1976.  Google Scholar

[12]

C. FabreJ. P. Puel and E. Zuazua, Approximate controllability of the semilinear heat equation, Proc. Royal Soc. Edinburgh, 125 (1995), 31-61.  doi: 10.1017/S0308210500030742.  Google Scholar

[13]

E. Fernández-Cara and E. Zuazua, Null and approximate controllability for weakly blowing up semilinear heat equations, Ann. I. H. Poincaré-AN, 17 (2000), 583-616.  doi: 10.1016/S0294-1449(00)00117-7.  Google Scholar

[14]

A. Fursikov and O. Yu. Imanuvilov, Controllability of Evolution Equations, Lecture Notes, Research Institute of Mathematics, Seoul National University, Korea, 1996.  Google Scholar

[15]

R. Glowinski and J.-L. Lions, Exact and approximate controllability for distributed parameter systems, Acta Numer., (1994), 269-378.  doi: 10.1017/S0962492900002452.  Google Scholar

[16]

R. Glowinski, J.-L. Lions and J. He, Exact and Approximate Controllability for Distributed Parameter Systems, Encyclopedia of Mathematics and its Applications, vol. 117, Cambridge University Press, Cambridge, 2008. doi: 10.1017/CBO9780511721595.  Google Scholar

[17]

M. González-Burgos and L. de Teresa, Controllability results for cascade systems of $m $ coupled parabolic PDEs by one control force, Portugal. Math., 67 (2010), 91-113.  doi: 10.4171/PM/1859.  Google Scholar

[18]

S. Guerrero, Controllability of systems of Stokes equations with one control force: existence of insensitizing controls, Ann. I. H. Poincaré-AN, 24 (2007), 1029-1054.  doi: 10.1016/j.anihpc.2006.11.001.  Google Scholar

[19]

S. Guerrero, Null controllability of some systems of two parabolic equations with one control force, SIAM J. Control Optim., 46 (2007), 379-394.  doi: 10.1137/060653135.  Google Scholar

[20]

M. Gueye, Insensitizing controls for the Navier-Stokes equations, Ann. I. H. Poincaré-AN, 30 (2013), 825-844.  doi: 10.1016/j.anihpc.2012.09.005.  Google Scholar

[21]

O. Kavian and L. de Teresa, Unique continuation principle for systems of parabolic equations, ESAIM Control Optim. Calc. Var., 16 (2010), 247-274.  doi: 10.1051/cocv/2008077.  Google Scholar

[22]

S. Labbé and E. Trélat, Uniform controllability of semidiscrete approximations of parabolic control systems, Syst. Control Lett., 55 (2006), 597-609.  doi: 10.1016/j.sysconle.2006.01.004.  Google Scholar

[23]

J.-L. Lions, Quelques notions dans l'analyse et le contrôle de systèmes à données incomplètes, Proceedings of the XIth Congress on Differential Equations and Applications/First Congress on Applied Mathematics, University of Málaga, 1990, 43-54.  Google Scholar

[24]

L. de Teresa, Insensitizing controls for a semilinear heat equation, Comm. Partial Differential Equations, 25 (2000), 39-72.  doi: 10.1080/03605300008821507.  Google Scholar

[25]

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: 10.3934/cpaa.2009.8.457.  Google Scholar

[26]

E. Zuazua, Control and numerical approximation of the wave and heat equation, International Congress of Mathematicians, Madrid, Spain, 3 (2006), 1389-1417.  Google Scholar

show all references

References:
[1]

F. Ammar-KhojdaA. BenabdallahM. González-Burgos and L. de Teresa, New phenomena for the null controllability of parabolic systems: Minimal time and geometrical dependence, J. Math. Anal. Appl., 444 (2016), 1071-1113.  doi: 10.1016/j.jmaa.2016.06.058.  Google Scholar

[2]

O. Bodart and C. Fabre, Controls insensitizing the norm of the solutions of a semilinear heat equation, J. Math. Anal. and App., 195 (1995), 658-683.  doi: 10.1006/jmaa.1995.1382.  Google Scholar

[3]

O. BodartM. González-Burgos and R. Pérez-García, Existence of insensitizing controls for a semilinear heat equation with a superlinear nonlinearity, Communications in Partial Differential Equations, 29 (2004), 1017-1050.  doi: 10.1081/PDE-200033749.  Google Scholar

[4]

O. BodartM. González-Burgos and R. Pérez-García, Insensitizing controls for a heat equation with a nonlinear term involving the state and the gradient, Nonlinear Anal., 57 (2004), 687-711.  doi: 10.1016/j.na.2004.03.012.  Google Scholar

[5]

F. Boyer, On the penalised HUM approach and its applications to the numerical approximation of null-controls for parabolic problems, ESAIM Proceedings, 41 (2013), 15-58.  doi: 10.1051/proc/201341002.  Google Scholar

[6]

F. BoyerF. Hubert and J. Le Rousseau, Discrete Carleman estimates for elliptic operators and uniform controllability of semi-discretized parabolic equations, J. Math Pures Appl., 93 (2010), 240-276.  doi: 10.1016/j.matpur.2009.11.003.  Google Scholar

[7]

F. BoyerF. Hubert and J. Le Rousseau, Discrete Carleman estimates for elliptic operators in arbitrary dimension and applications, SIAM J. Control Optim., 48 (2010), 5357-5397.  doi: 10.1137/100784278.  Google Scholar

[8]

F. BoyerF. Hubert and J. Le Rousseau, Uniform null-controllability for space/time-discretized parabolic equations, Numer. Math., 118 (2011), 601-661.  doi: 10.1007/s00211-011-0368-1.  Google Scholar

[9]

F. Boyer and J. Le Rousseau, Carleman estimates for semi-discrete parabolic operators and application to the controllability of semi-linear and semi-discrete parabolic equations, Ann. I. H. Poincaré-AN, 31 (2014), 1035-1078.  doi: 10.1016/j.anihpc.2013.07.011.  Google Scholar

[10]

N. CarreñoM. Gueye and S. Guerrero, Insensitizing control with two vanishing components for the three-dimensional Boussinesq system, ESAIM Control Optim. Calc. Var., 21 (2015), 73-100.  doi: 10.1051/cocv/2014020.  Google Scholar

[11]

I. Ekeland and R. Temam, Convex Analysis and Variational Problems, North-Holland, 1976.  Google Scholar

[12]

C. FabreJ. P. Puel and E. Zuazua, Approximate controllability of the semilinear heat equation, Proc. Royal Soc. Edinburgh, 125 (1995), 31-61.  doi: 10.1017/S0308210500030742.  Google Scholar

[13]

E. Fernández-Cara and E. Zuazua, Null and approximate controllability for weakly blowing up semilinear heat equations, Ann. I. H. Poincaré-AN, 17 (2000), 583-616.  doi: 10.1016/S0294-1449(00)00117-7.  Google Scholar

[14]

A. Fursikov and O. Yu. Imanuvilov, Controllability of Evolution Equations, Lecture Notes, Research Institute of Mathematics, Seoul National University, Korea, 1996.  Google Scholar

[15]

R. Glowinski and J.-L. Lions, Exact and approximate controllability for distributed parameter systems, Acta Numer., (1994), 269-378.  doi: 10.1017/S0962492900002452.  Google Scholar

[16]

R. Glowinski, J.-L. Lions and J. He, Exact and Approximate Controllability for Distributed Parameter Systems, Encyclopedia of Mathematics and its Applications, vol. 117, Cambridge University Press, Cambridge, 2008. doi: 10.1017/CBO9780511721595.  Google Scholar

[17]

M. González-Burgos and L. de Teresa, Controllability results for cascade systems of $m $ coupled parabolic PDEs by one control force, Portugal. Math., 67 (2010), 91-113.  doi: 10.4171/PM/1859.  Google Scholar

[18]

S. Guerrero, Controllability of systems of Stokes equations with one control force: existence of insensitizing controls, Ann. I. H. Poincaré-AN, 24 (2007), 1029-1054.  doi: 10.1016/j.anihpc.2006.11.001.  Google Scholar

[19]

S. Guerrero, Null controllability of some systems of two parabolic equations with one control force, SIAM J. Control Optim., 46 (2007), 379-394.  doi: 10.1137/060653135.  Google Scholar

[20]

M. Gueye, Insensitizing controls for the Navier-Stokes equations, Ann. I. H. Poincaré-AN, 30 (2013), 825-844.  doi: 10.1016/j.anihpc.2012.09.005.  Google Scholar

[21]

O. Kavian and L. de Teresa, Unique continuation principle for systems of parabolic equations, ESAIM Control Optim. Calc. Var., 16 (2010), 247-274.  doi: 10.1051/cocv/2008077.  Google Scholar

[22]

S. Labbé and E. Trélat, Uniform controllability of semidiscrete approximations of parabolic control systems, Syst. Control Lett., 55 (2006), 597-609.  doi: 10.1016/j.sysconle.2006.01.004.  Google Scholar

[23]

J.-L. Lions, Quelques notions dans l'analyse et le contrôle de systèmes à données incomplètes, Proceedings of the XIth Congress on Differential Equations and Applications/First Congress on Applied Mathematics, University of Málaga, 1990, 43-54.  Google Scholar

[24]

L. de Teresa, Insensitizing controls for a semilinear heat equation, Comm. Partial Differential Equations, 25 (2000), 39-72.  doi: 10.1080/03605300008821507.  Google Scholar

[25]

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: 10.3934/cpaa.2009.8.457.  Google Scholar

[26]

E. Zuazua, Control and numerical approximation of the wave and heat equation, International Congress of Mathematicians, Madrid, Spain, 3 (2006), 1389-1417.  Google Scholar

Figure 1.  $f(y) = -0.1 \sin(y)$, $y_0 = 0$, $\xi(x, t) = {\bf{1}}_{[0.4, 1]}(t)$. Uncontrolled solution
Figure 2.  $f(y) = -0.1 \sin(y)$, $y_0 = 0$, $\xi(x, t) = {\bf{1}}_{[0.4, 1]}(t)$. Controlled solution
Figure 3.  Convergence properties of the method for insensitizing problem
Figure 4.  Value of $\Psi(y)$ for different parameters $\tau$ and initial perturbations $w_0$
Figure 5.  $y_0(x) = {\bf{1}}_{(0.2, 0.7)}(x)$, $\xi = 0$, $f = 0$, $\omega = (0.3, 0.8)$. Same legend as in Figure 3
Figure 6.  The case where $\mathcal{O} = \Omega$ with $\xi = 0$, $\omega = (0, 0.5)$. Same legend as in Figure 3
Figure 7.  Different values of $\mathcal{M}$ in the source term. Same legend as in Figure 3
Figure 8.  The case where $\omega \cap \mathcal{O} = \emptyset$. Same legend as in Figure 3
Figure 9.  Convergence properties for the quadratic case. Same legend as in Figure 3.
Figure 10.  $f(y) = -y^2$, $y_0 = 0$, $\xi(x, t) = 8\times {\bf{1}}_{[0.2, 1]}(t)$. Time evolution
Figure 11.  $f(y) = -y^2$, $y_0 = 0$, $\xi(x, t) = 8\times {\bf{1}}_{[0.2, 1]}(t)$. Controlled solution
Figure 12.  Simultaneous insensitizing and null-control
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