June  2022, 12(2): 327-341. doi: 10.3934/mcrf.2021024

Controllability under positive constraints for quasilinear parabolic PDEs

DeustoTech, Fundación Deusto, Av. Universidades, 24, 48007, Bilbao, Basque Country, Spain, Instituto de Matemática e Estatística, Universidade Federal Fluminense, R. Prof. Marcos Waldemar de Freitas, s/n, 24210-201, Niterói, RJ, Brazil

* Corresponding author: Miguel R. Nuñez-Chávez

Received  October 2019 Revised  March 2021 Published  June 2022 Early access  April 2021

This paper deals with the analysis of the internal controllability with constraint of positive kind of a quasilinear parabolic PDE. We prove two results about this PDE: First, we prove a global steady state constrained controllability result. For this purpose, we employ the called "stair-case method". And second, we prove a global trajectory constrained controllability result. For this purpose, we employ the well-known "stabilization property" in $ L^2 $ norms. Furthermore, for both results an important argument is needed: the exact local controllability to trajectories. Then we prove the positivity of the minimal controllability time using arguments of comparison principle. Some additional comments and open problems concerning other systems are presented.

Citation: Miguel R. Nuñez-Chávez. Controllability under positive constraints for quasilinear parabolic PDEs. Mathematical Control and Related Fields, 2022, 12 (2) : 327-341. doi: 10.3934/mcrf.2021024
References:
[1]

V. M. Alekseev, V. M. Tikhomorov and S. V. Formin, Optimal Control, Consultants Bureau, New York, 1987. doi: 10.1007/978-1-4615-7551-1.

[2]

M. Beceanu, Local exact controllability of the diffusion equation in one dimension, Abstr. Appl. Anal., 2003 (2003), 793-811.  doi: 10.1155/S1085337503303033.

[3]

H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer New York, 2010. Available from: https://books.google.es/books?id=GAA2XqOIIGoC.

[4]

R. Du, Null controllability for a class of degenerate parabolic equations with the gradient terms, J. Evol. Equ., 19 (2019), 585-613.  doi: 10.1007/s00028-019-00487-8.

[5]

O. Yu. Émanuvilov, Controllability of parabolic equations, Mat. Sb., 186 (1995), 879-900.  doi: 10.1070/SM1995v186n06ABEH000047.

[6]

L. C. Evans, Partial Differential Equations, Graduate studies in mathematics, American Mathematical Society, 2010. Available from: https://books.google.es/books?id=Xnu0o_EJrCQC. doi: 10.1090/gsm/019.

[7]

E. Fernández-CaraD. Nina-HuamánM. R. Nuñez-Chávez and F. B. Vieira, On the theoretical and numerical control of a one-dimensional nonlinear parabolic partial differential equation, J. Optim. Theory Appl.s, 175 (2017), 652-682.  doi: 10.1007/s10957-017-1190-4.

[8]

E. Fernández-Cara and E. Zuazua, The cost of approximate controllability for heat equations: The linear case, Adv. Differential Equations, 5 (2000), 465-514. 

[9]

A. V. Fursikov and O. Yu. Imanuvilov, Controllability of Evolution Equations, Lecture Notes Series, vol. 34, Seoul National University, Research Institute of Mathematics, Global Analysis Research Center, Seoul, 1996.

[10]

O. Yu. Imanuvilov and M. Yamamoto, Carleman inequalities for parabolic equations in Sobolev Spaces of negative order and exact controllability for semilinear parabolic equations, Publ. Res. Inst. Math. Sci., 39 (2003), 227-274.  doi: 10.2977/prims/1145476103.

[11]

O. A. Ladyzhenskaya and N. N. Ural'ceva, Linear and Quasilinear Elliptic Equations, Translations by Scripta Technica, Inc, Academy Press, New York and London, 1968.

[12]

O. A. Ladyžhenskaya, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type, Translations of Mathematical Monographs, vol. 23, AMS, Providence, RI, 1968. Available from: https://books.google.es/books?id=dolUcRSDPgkC.

[13]

K. Le Balc'h, Controllability of a $4 \times 4$ quadratic reaction-diffusion system, J. Differential Equations, 266 (2019), 3100-3188.  doi: 10.1016/j.jde.2018.08.046.

[14]

G. Lebeau and L. Robbiano, Contrôle exact de léquation de la chaleur, Communications in Partial Differential Equations, 20 (1995), 335-356. 

[15]

G. M. Lieberman, Second Order Parabolic Differential Equations, World Scientific, 1996. Available from: https://books.google.es/books?id=s9Guiwylm3cC. doi: 10.1142/3302.

[16]

J.-L. Lions, Controlablité exacte des syst$\grave{e}$mes distribués: Remarques sur la théorie générale et les applications, Analysis and Optimization System, Springer, (1986), 3–14. doi: 10.1007/BFb0007542.

[17]

J. L. Lions and E. Magenes, Problemes aux Limites non Homogenes et Applications, Grundlehren der mathematischen Wissenschaften, Springer Berlin Heidelberg, vol 1, 1968.

[18]

X. Liu and X. Zhang, Local controllability of multidimensional quasi-linear parabolic equations, SIAM J. Control Optim., 50 (2012), 2046-2064.  doi: 10.1137/110851808.

[19]

J. LohéacE. Trélat and E. Zuazua, Minimal controllability time for the heat equation under unilateral state or control constraints, Math. Models Methods Appl. Sci., 27 (2017), 1587-1644.  doi: 10.1142/S0218202517500270.

[20]

D. Pighin and E. Zuazua, Controllability under positive constraints of semilinear heat equations, Math. Control Relat. Fields, 8 (2018), 935-964.  doi: 10.3934/mcrf.2018041.

[21]

C. PoucholE. Trélat and E. Zuazua, Phase portrait control for 1d monostable and bistable reaction-diffusion equations, Nonlinearity, 32 (2019), 884-909.  doi: 10.1088/1361-6544/aaf07e.

[22]

M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations, Springer, New York, 2012. Available from: https://books.google.es/books?id=JUXhBwAAQBAJ.

[23]

E. J. P. G. Schmidt, Boundary control for the heat equation with steady-state targets, J. Differential Equations, 78 (1989), 89-121.  doi: 10.1016/0022-0396(89)90077-6.

[24]

E. Zeidler, Nonlinear Functional Analysis and its Applications I: Fixed-Point Theorems, Springer-Verlag, New York, Berlin, Heidelberg, Tokio, 1986.

show all references

References:
[1]

V. M. Alekseev, V. M. Tikhomorov and S. V. Formin, Optimal Control, Consultants Bureau, New York, 1987. doi: 10.1007/978-1-4615-7551-1.

[2]

M. Beceanu, Local exact controllability of the diffusion equation in one dimension, Abstr. Appl. Anal., 2003 (2003), 793-811.  doi: 10.1155/S1085337503303033.

[3]

H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer New York, 2010. Available from: https://books.google.es/books?id=GAA2XqOIIGoC.

[4]

R. Du, Null controllability for a class of degenerate parabolic equations with the gradient terms, J. Evol. Equ., 19 (2019), 585-613.  doi: 10.1007/s00028-019-00487-8.

[5]

O. Yu. Émanuvilov, Controllability of parabolic equations, Mat. Sb., 186 (1995), 879-900.  doi: 10.1070/SM1995v186n06ABEH000047.

[6]

L. C. Evans, Partial Differential Equations, Graduate studies in mathematics, American Mathematical Society, 2010. Available from: https://books.google.es/books?id=Xnu0o_EJrCQC. doi: 10.1090/gsm/019.

[7]

E. Fernández-CaraD. Nina-HuamánM. R. Nuñez-Chávez and F. B. Vieira, On the theoretical and numerical control of a one-dimensional nonlinear parabolic partial differential equation, J. Optim. Theory Appl.s, 175 (2017), 652-682.  doi: 10.1007/s10957-017-1190-4.

[8]

E. Fernández-Cara and E. Zuazua, The cost of approximate controllability for heat equations: The linear case, Adv. Differential Equations, 5 (2000), 465-514. 

[9]

A. V. Fursikov and O. Yu. Imanuvilov, Controllability of Evolution Equations, Lecture Notes Series, vol. 34, Seoul National University, Research Institute of Mathematics, Global Analysis Research Center, Seoul, 1996.

[10]

O. Yu. Imanuvilov and M. Yamamoto, Carleman inequalities for parabolic equations in Sobolev Spaces of negative order and exact controllability for semilinear parabolic equations, Publ. Res. Inst. Math. Sci., 39 (2003), 227-274.  doi: 10.2977/prims/1145476103.

[11]

O. A. Ladyzhenskaya and N. N. Ural'ceva, Linear and Quasilinear Elliptic Equations, Translations by Scripta Technica, Inc, Academy Press, New York and London, 1968.

[12]

O. A. Ladyžhenskaya, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type, Translations of Mathematical Monographs, vol. 23, AMS, Providence, RI, 1968. Available from: https://books.google.es/books?id=dolUcRSDPgkC.

[13]

K. Le Balc'h, Controllability of a $4 \times 4$ quadratic reaction-diffusion system, J. Differential Equations, 266 (2019), 3100-3188.  doi: 10.1016/j.jde.2018.08.046.

[14]

G. Lebeau and L. Robbiano, Contrôle exact de léquation de la chaleur, Communications in Partial Differential Equations, 20 (1995), 335-356. 

[15]

G. M. Lieberman, Second Order Parabolic Differential Equations, World Scientific, 1996. Available from: https://books.google.es/books?id=s9Guiwylm3cC. doi: 10.1142/3302.

[16]

J.-L. Lions, Controlablité exacte des syst$\grave{e}$mes distribués: Remarques sur la théorie générale et les applications, Analysis and Optimization System, Springer, (1986), 3–14. doi: 10.1007/BFb0007542.

[17]

J. L. Lions and E. Magenes, Problemes aux Limites non Homogenes et Applications, Grundlehren der mathematischen Wissenschaften, Springer Berlin Heidelberg, vol 1, 1968.

[18]

X. Liu and X. Zhang, Local controllability of multidimensional quasi-linear parabolic equations, SIAM J. Control Optim., 50 (2012), 2046-2064.  doi: 10.1137/110851808.

[19]

J. LohéacE. Trélat and E. Zuazua, Minimal controllability time for the heat equation under unilateral state or control constraints, Math. Models Methods Appl. Sci., 27 (2017), 1587-1644.  doi: 10.1142/S0218202517500270.

[20]

D. Pighin and E. Zuazua, Controllability under positive constraints of semilinear heat equations, Math. Control Relat. Fields, 8 (2018), 935-964.  doi: 10.3934/mcrf.2018041.

[21]

C. PoucholE. Trélat and E. Zuazua, Phase portrait control for 1d monostable and bistable reaction-diffusion equations, Nonlinearity, 32 (2019), 884-909.  doi: 10.1088/1361-6544/aaf07e.

[22]

M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations, Springer, New York, 2012. Available from: https://books.google.es/books?id=JUXhBwAAQBAJ.

[23]

E. J. P. G. Schmidt, Boundary control for the heat equation with steady-state targets, J. Differential Equations, 78 (1989), 89-121.  doi: 10.1016/0022-0396(89)90077-6.

[24]

E. Zeidler, Nonlinear Functional Analysis and its Applications I: Fixed-Point Theorems, Springer-Verlag, New York, Berlin, Heidelberg, Tokio, 1986.

Figure 1.  The stair-case method. Trajectory of state-control in blue. Path of steady states in red
Figure 2.  Trajectory of state-control in blue. Target trajectory in red
Figure 3.  Initial datum $ \varphi^T $ for the adjoint system
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