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doi: 10.3934/mcrf.2022001
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## Constructive exact control of semilinear 1D heat equations

 Université Clermont Auvergne, CNRS, LMBP, F-63000 Clermont-Ferrand, France

* Corresponding author: Arnaud Münch

Received  March 2021 Revised  September 2021 Early access January 2022

Fund Project: The second author is supported by the french government research program "Investissements d'Avenir" through the IDEX-ISITE initiative 16-IDEX-0001 (CAP 20-25)

The exact distributed controllability of the semilinear heat equation $\partial_{t}y-\Delta y + g(y) = f \,1_{\omega}$ posed over multi-dimensional and bounded domains, assuming that $g\in C^1(\mathbb{R})$ satisfies the growth condition $\limsup_{r\to \infty} g(r)/ (\vert r\vert \ln^{3/2}\vert r\vert) = 0$ has been obtained by Fernández-Cara and Zuazua in 2000. The proof based on a non constructive fixed point arguments makes use of precise estimates of the observability constant for a linearized heat equation. In the one dimensional setting, assuming that $g^\prime$ does not grow faster than $\beta \ln^{3/2}\vert r\vert$ at infinity for $\beta>0$ small enough and that $g^\prime$ is uniformly Hölder continuous on $\mathbb{R}$ with exponent $p\in [0,1]$, we design a constructive proof yielding an explicit sequence converging to a controlled solution for the semilinear equation, at least with order $1+p$ after a finite number of iterations.

Citation: Jérôme Lemoine, Arnaud Münch. Constructive exact control of semilinear 1D heat equations. Mathematical Control and Related Fields, doi: 10.3934/mcrf.2022001
##### References:
 [1] M. Badra, S. Ervedoza and S. Guerrero, Local controllability to trajectories for non-homogeneous incompressible Navier-Stokes equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 33 (2016), 529-574.  doi: 10.1016/j.anihpc.2014.11.006. [2] V. Barbu, Exact controllability of the superlinear heat equation, Appl. Math. Optim., 42 (2000), 73-89.  doi: 10.1007/s002450010004. [3] A. Bottois, J. Lemoine and A. Münch, Constructive exact control of semilinear multi-dimensional wave equations, Preprint. arXiv: 2101.06446. [4] F. Boyer, On the penalised HUM approach and its applications to the numerical approximation of null-controls for parabolic problems, In CANUM 2012, 41e Congrès National d'Analyse Numérique, volume 41 of ESAIM Proc., 15–58. EDP Sci., Les Ulis, 2013. doi: 10.1051/proc/201341002. [5] F. Boyer and J. Le Rousseau, Carleman estimates for semi-discrete parabolic operators and application to the controllability of semi-linear semi-discrete parabolic equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 31 (2014), 1035-1078.  doi: 10.1016/j.anihpc.2013.07.011. [6] M. O. Bristeau, O. Pironneau, R. Glowinski, J. Periaux and P. Perrier, On the numerical solution of nonlinear problems in fluid dynamics by least squares and finite element methods. I. Least square formulations and conjugate gradient, Comput. Methods Appl. Mech. Engrg., 17 (1979), 619-657.  doi: 10.1016/0045-7825(79)90048-3. [7] T. Cazenave and A. Haraux, An Introduction to Semilinear Evolution Equations, volume 13 of Oxford Lecture Series in Mathematics and its Applications, The Clarendon Press, Oxford University Press, New York, 1998. Translated from the 1990 French original by Yvan Martel and revised by the authors. [8] J.-M. Coron, Control and Nonlinearity, volume 136 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 2007. doi: 10.1090/surv/136. [9] J.-M. Coron and E. Trélat, Global steady-state controllability of one-dimensional semilinear heat equations, SIAM J. Control Optim., 43 (2004), 549-569.  doi: 10.1137/S036301290342471X. [10] P. Deuflhard, Newton Methods for Nonlinear Problems, volume 35 of Springer Series in Computational Mathematics, Springer, Heidelberg, 2011. Affine invariance and adaptive algorithms, First softcover printing of the 2006 corrected printing. doi: 10.1007/978-3-642-23899-4. [11] A. Doubova, E. Fernández-Cara, M. González-Burgos and E. Zuazua, On the controllability of parabolic systems with a nonlinear term involving the state and the gradient, SIAM J. Control Optim., 41 (2002), 798-819.  doi: 10.1137/S0363012901386465. [12] T. Duyckaerts, X. Zhang and E. Zuazua, On the optimality of the observability inequalities for parabolic and hyperbolic systems with potentials, Ann. Inst. H. Poincaré Anal. Non Linéaire, 25 (2008), 1-41.  doi: 10.1016/j.anihpc.2006.07.005. [13] S. Ervedoza, J. Lemoine and A. Münch, Exact controllability of semilinear heat equations through a constructive approach, Preprint, hal-03350534. [14] E. Fernández-Cara and A. Münch, Numerical null controllability of semi-linear 1-D heat equations: Fixed point, least squares and Newton methods, Math. Control Relat. Fields, 2 (2012), 217-246.  doi: 10.3934/mcrf.2012.2.217. [15] E. Fernández-Cara and A. Münch, Strong convergent approximations of null controls for the 1D heat equation, SeMA J., 61 (2013), 49-78.  doi: 10.1007/s40324-013-0001-6. [16] E. Fernández-Cara, A. Münch and D. A. Souza, On the numerical controllability of the two-dimensional heat, Stokes and Navier-Stokes equations, J. Sci. Comput., 70 (2017), 819-858.  doi: 10.1007/s10915-016-0266-x. [17] E. Fernández-Cara and E. Zuazua, The cost of approximate controllability for heat equations: The linear case, Adv. Differential Equations, 5 (2000), 465-514. [18] E. Fernández-Cara and E. Zuazua, Null and approximate controllability for weakly blowing up semilinear heat equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 17 (2000), 583-616.  doi: 10.1016/s0294-1449(00)00117-7. [19] A. V. Fursikov and O. Yu. Imanuvilov, Controllability of Evolution Equations, volume 34 of Lecture Notes Series, Seoul National University, Research Institute of Mathematics, Global Analysis Research Center, Seoul, 1996. [20] R. Glowinski, J.-L. Lions and J. He, Exact and Approximate Controllability for Distributed Parameter Systems, volume 117 of Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, 2008. A numerical approach. doi: 10.1017/CBO9780511721595. [21] A. A. Lacey, Global blow-up of a nonlinear heat equation, Proc. Roy. Soc. Edinburgh Sect. A, 104 (1986), 161-167.  doi: 10.1017/S0308210500019120. [22] K. Le Balc'h, Global null-controllability and nonnegative-controllability of slightly superlinear heat equations, J. Math. Pures Appl., 135 (2020), 103-139.  doi: 10.1016/j.matpur.2019.10.009. [23] J. Lemoine, I. Marín-Gayte and A. Münch, Approximation of null controls for semilinear heat equations using a least-squares approach, ESAIM Control Optim. Calc. Var., 27 (2021), Paper No. 63, 28 pp. doi: 10.1051/cocv/2021062. [24] J. Lemoine and A. Münch, A fully space-time least-squares method for the unsteady Navier-Stokes system, J. Math. Fluid Mech., 23 (2021), Paper No. 102, 38 pp. doi: 10.1007/s00021-021-00627-6. [25] J. Lemoine and A. Münch, Resolution of the implicit Euler scheme for the Navier-Stokes equation through a least-squares method, Numer. Math., 147 (2021), 349-391.  doi: 10.1007/s00211-021-01171-1. [26] J. Lemoine, A. Münch and P. Pedregal, Analysis of continuous $H^{-1}$-least-squares approaches for the steady Navier-Stokes system, Appl. Math. Optim., 83 (2021), 461-488.  doi: 10.1007/s00245-019-09554-5. [27] A. Münch, A least-squares formulation for the approximation of controls for the Stokes system, Math. Control Signals Systems, 27 (2015), 49-75.  doi: 10.1007/s00498-014-0134-x. [28] A. Münch and P. Pedregal, Numerical null controllability of the heat equation through a least squares and variational approach, European J. Appl. Math., 25 (2014), 277-306.  doi: 10.1017/S0956792514000023. [29] A. Münch and D. A. Souza, A mixed formulation for the direct approximation of $L^2$-weighted controls for the linear heat equation, Adv. Comput. Math., 42 (2016), 85-125.  doi: 10.1007/s10444-015-9412-5. [30] A. Münch and E. Trélat, Constructive exact control of semilinear 1d wave equations by a least-squares approach, Preprint. arXiv: 2011.08462, To appear in SICON. [31] P. Saramito, A damped Newton algorithm for computing viscoplastic fluid flows, J. Non-Newton. Fluid Mech., 238 (2016), 6-15.  doi: 10.1016/j.jnnfm.2016.05.007. [32] E. Zuazua, Exact controllability for semilinear wave equations in one space dimension, Ann. Inst. H. Poincaré Anal. Non Linéaire, 10 (1993), 109-129.  doi: 10.1016/s0294-1449(16)30221-9.

show all references

##### References:
 [1] M. Badra, S. Ervedoza and S. Guerrero, Local controllability to trajectories for non-homogeneous incompressible Navier-Stokes equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 33 (2016), 529-574.  doi: 10.1016/j.anihpc.2014.11.006. [2] V. Barbu, Exact controllability of the superlinear heat equation, Appl. Math. Optim., 42 (2000), 73-89.  doi: 10.1007/s002450010004. [3] A. Bottois, J. Lemoine and A. Münch, Constructive exact control of semilinear multi-dimensional wave equations, Preprint. arXiv: 2101.06446. [4] F. Boyer, On the penalised HUM approach and its applications to the numerical approximation of null-controls for parabolic problems, In CANUM 2012, 41e Congrès National d'Analyse Numérique, volume 41 of ESAIM Proc., 15–58. EDP Sci., Les Ulis, 2013. doi: 10.1051/proc/201341002. [5] F. Boyer and J. Le Rousseau, Carleman estimates for semi-discrete parabolic operators and application to the controllability of semi-linear semi-discrete parabolic equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 31 (2014), 1035-1078.  doi: 10.1016/j.anihpc.2013.07.011. [6] M. O. Bristeau, O. Pironneau, R. Glowinski, J. Periaux and P. Perrier, On the numerical solution of nonlinear problems in fluid dynamics by least squares and finite element methods. I. Least square formulations and conjugate gradient, Comput. Methods Appl. Mech. Engrg., 17 (1979), 619-657.  doi: 10.1016/0045-7825(79)90048-3. [7] T. Cazenave and A. Haraux, An Introduction to Semilinear Evolution Equations, volume 13 of Oxford Lecture Series in Mathematics and its Applications, The Clarendon Press, Oxford University Press, New York, 1998. Translated from the 1990 French original by Yvan Martel and revised by the authors. [8] J.-M. Coron, Control and Nonlinearity, volume 136 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 2007. doi: 10.1090/surv/136. [9] J.-M. Coron and E. Trélat, Global steady-state controllability of one-dimensional semilinear heat equations, SIAM J. Control Optim., 43 (2004), 549-569.  doi: 10.1137/S036301290342471X. [10] P. Deuflhard, Newton Methods for Nonlinear Problems, volume 35 of Springer Series in Computational Mathematics, Springer, Heidelberg, 2011. Affine invariance and adaptive algorithms, First softcover printing of the 2006 corrected printing. doi: 10.1007/978-3-642-23899-4. [11] A. Doubova, E. Fernández-Cara, M. González-Burgos and E. Zuazua, On the controllability of parabolic systems with a nonlinear term involving the state and the gradient, SIAM J. Control Optim., 41 (2002), 798-819.  doi: 10.1137/S0363012901386465. [12] T. Duyckaerts, X. Zhang and E. Zuazua, On the optimality of the observability inequalities for parabolic and hyperbolic systems with potentials, Ann. Inst. H. Poincaré Anal. Non Linéaire, 25 (2008), 1-41.  doi: 10.1016/j.anihpc.2006.07.005. [13] S. Ervedoza, J. Lemoine and A. Münch, Exact controllability of semilinear heat equations through a constructive approach, Preprint, hal-03350534. [14] E. Fernández-Cara and A. Münch, Numerical null controllability of semi-linear 1-D heat equations: Fixed point, least squares and Newton methods, Math. Control Relat. Fields, 2 (2012), 217-246.  doi: 10.3934/mcrf.2012.2.217. [15] E. Fernández-Cara and A. Münch, Strong convergent approximations of null controls for the 1D heat equation, SeMA J., 61 (2013), 49-78.  doi: 10.1007/s40324-013-0001-6. [16] E. Fernández-Cara, A. Münch and D. A. Souza, On the numerical controllability of the two-dimensional heat, Stokes and Navier-Stokes equations, J. Sci. Comput., 70 (2017), 819-858.  doi: 10.1007/s10915-016-0266-x. [17] E. Fernández-Cara and E. Zuazua, The cost of approximate controllability for heat equations: The linear case, Adv. Differential Equations, 5 (2000), 465-514. [18] E. Fernández-Cara and E. Zuazua, Null and approximate controllability for weakly blowing up semilinear heat equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 17 (2000), 583-616.  doi: 10.1016/s0294-1449(00)00117-7. [19] A. V. Fursikov and O. Yu. Imanuvilov, Controllability of Evolution Equations, volume 34 of Lecture Notes Series, Seoul National University, Research Institute of Mathematics, Global Analysis Research Center, Seoul, 1996. [20] R. Glowinski, J.-L. Lions and J. He, Exact and Approximate Controllability for Distributed Parameter Systems, volume 117 of Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, 2008. A numerical approach. doi: 10.1017/CBO9780511721595. [21] A. A. Lacey, Global blow-up of a nonlinear heat equation, Proc. Roy. Soc. Edinburgh Sect. A, 104 (1986), 161-167.  doi: 10.1017/S0308210500019120. [22] K. Le Balc'h, Global null-controllability and nonnegative-controllability of slightly superlinear heat equations, J. Math. Pures Appl., 135 (2020), 103-139.  doi: 10.1016/j.matpur.2019.10.009. [23] J. Lemoine, I. Marín-Gayte and A. Münch, Approximation of null controls for semilinear heat equations using a least-squares approach, ESAIM Control Optim. Calc. Var., 27 (2021), Paper No. 63, 28 pp. doi: 10.1051/cocv/2021062. [24] J. Lemoine and A. Münch, A fully space-time least-squares method for the unsteady Navier-Stokes system, J. Math. Fluid Mech., 23 (2021), Paper No. 102, 38 pp. doi: 10.1007/s00021-021-00627-6. [25] J. Lemoine and A. Münch, Resolution of the implicit Euler scheme for the Navier-Stokes equation through a least-squares method, Numer. Math., 147 (2021), 349-391.  doi: 10.1007/s00211-021-01171-1. [26] J. Lemoine, A. Münch and P. Pedregal, Analysis of continuous $H^{-1}$-least-squares approaches for the steady Navier-Stokes system, Appl. Math. Optim., 83 (2021), 461-488.  doi: 10.1007/s00245-019-09554-5. [27] A. Münch, A least-squares formulation for the approximation of controls for the Stokes system, Math. Control Signals Systems, 27 (2015), 49-75.  doi: 10.1007/s00498-014-0134-x. [28] A. Münch and P. Pedregal, Numerical null controllability of the heat equation through a least squares and variational approach, European J. Appl. Math., 25 (2014), 277-306.  doi: 10.1017/S0956792514000023. [29] A. Münch and D. A. Souza, A mixed formulation for the direct approximation of $L^2$-weighted controls for the linear heat equation, Adv. Comput. Math., 42 (2016), 85-125.  doi: 10.1007/s10444-015-9412-5. [30] A. Münch and E. Trélat, Constructive exact control of semilinear 1d wave equations by a least-squares approach, Preprint. arXiv: 2011.08462, To appear in SICON. [31] P. Saramito, A damped Newton algorithm for computing viscoplastic fluid flows, J. Non-Newton. Fluid Mech., 238 (2016), 6-15.  doi: 10.1016/j.jnnfm.2016.05.007. [32] E. Zuazua, Exact controllability for semilinear wave equations in one space dimension, Ann. Inst. H. Poincaré Anal. Non Linéaire, 10 (1993), 109-129.  doi: 10.1016/s0294-1449(16)30221-9.
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