# American Institute of Mathematical Sciences

September  2018, 8(3&4): 935-964. doi: 10.3934/mcrf.2018041

## Controllability under positivity constraints of semilinear heat equations

 1 Departamento de Matemáticas, Universidad Autónoma de Madrid, 28049 Madrid, Spain 2 DeustoTech, Fundación Deusto, Avda. Universidades, 24, 48007, Bilbao, Basque Country, Spain 3 Facultad de Ingeniería, Universidad de Deusto, Avda. Universidades, 24, 48007, Bilbao, Basque Country, Spain 4 Laboratoire Jacques-Louis Lions, UPMC Univ. Paris 06, CNRSUMR 7598, Sorbonne Universités, F-75005, Paris, France

* Corresponding author: Dario Pighin

Dedicated to Professor Jiongmin Yong on the occasion of his 60th birthday

Received  November 2017 Revised  April 2018 Published  September 2018

Fund Project: This work was partially supported by the Advanced Grant DYCON (Dynamic Control) of the European Research Council Executive Agency, FA9550-15-1-0027 of AFOSR, FA9550-14-1-0214 of the EOARD-AFOSR, the MTM2014-52347 and MTM2017 Grants of the MINECO (Spain) and ICON of the French ANR

In many practical applications of control theory some constraints on the state and/or on the control need to be imposed.

In this paper, we prove controllability results for semilinear parabolic equations under positivity constraints on the control, when the time horizon is long enough. As we shall see, in fact, the minimal controllability time turns out to be strictly positive.

More precisely, we prove a global steady state constrained controllability result for a semilinear parabolic equation with $C^1$ nonlinearity, without sign or globally Lipschitz assumptions on the nonlinear term. Then, under suitable dissipativity assumptions on the system, we extend the result to any initial datum and any target trajectory.

We conclude with some numerical simulations that confirm the theoretical results that provide further information of the sparse structure of constrained controls in minimal time.

Citation: Dario Pighin, Enrique Zuazua. Controllability under positivity constraints of semilinear heat equations. Mathematical Control & Related Fields, 2018, 8 (3&4) : 935-964. doi: 10.3934/mcrf.2018041
##### References:
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Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer New York, 2011, URL https://books.google.es/books?id=GAA2XqOIIGoC. doi: 10.1007/978-0-387-70914-7. Google Scholar [6] W. Chan and B. Z. Guo, Optimal birth control of population dynamics. ⅱ. problems with free final time, phase constraints, and mini-max costs, Journal of Mathematical Analysis and Applications, 146 (1990), 523-539. doi: 10.1016/0022-247X(90)90322-7. Google Scholar [7] R. M. Colombo, G. Guerra, M. Herty and V. Schleper, Optimal control in networks of pipes and canals, SIAM Journal on Control and Optimization, 48 (2009), 2032-2050. doi: 10.1137/080716372. Google Scholar [8] J.-M. Coron and E. Trélat, Global steady-state controllability of one-dimensional semilinear heat equations, SIAM journal on control and optimization, 43 (2004), 549-569. doi: 10.1137/S036301290342471X. Google Scholar [9] J. Coron, Control and Nonlinearity, Mathematical surveys and monographs, American Mathematical Society, 2007, URL https://books.google.es/books?id=aEKv1bpcrKQC. doi: 10.1090/surv/136. Google Scholar [10] J. I. Diaz, Sur la contrôlabilité approchée des inéquations variationelles et dutres problèmes paraboliques non linéaires, CR Acad. Sci. Paris, 312 (1991), 519-522. Google Scholar [11] O. Y. Emanuilov, Controllability of parabolic equations, Sbornik: Mathematics, 186 (1995), 879-900. doi: 10.1070/SM1995v186n06ABEH000047. Google Scholar [12] L. Evans, Partial Differential Equations, Graduate studies in mathematics, American Mathematical Society, 2010, URL https://books.google.es/books?id=Xnu0o_EJrCQC. doi: 10.1090/gsm/019. Google Scholar [13] H. O. Fattorini and D. L. Russell, Exact controllability theorems for linear parabolic equations in one space dimension, Archive for Rational Mechanics and Analysis, 43 (1971), 272-292. doi: 10.1007/BF00250466. Google Scholar [14] E. Fernández-Cara and E. Zuazua, Null and approximate controllability for weakly blowing up semilinear heat equations, Annales de l'Institut Henri Poincare (C) Non Linear Analysis, 17 (2000), 583-616, URL http://www.sciencedirect.com/science/article/pii/S0294144900001177. doi: 10.1016/S0294-1449(00)00117-7. Google Scholar [15] R. Fourer, D. M. Gay and B. W. Kernighan, A modeling language for mathematical programming, Management Science, 36 (1990), 519-554. URL https://orfe.princeton.edu/~rvdb/307/textbook/AMPLbook.pdf. doi: 10.1287/mnsc.36.5.519. Google Scholar [16] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Classics in Mathematics. Springer-Verlag, Berlin, 2001. Google Scholar [17] P. Grisvard, Elliptic Problems in Nonsmooth Domains, SIAM, 2011. doi: 10.1137/1.9781611972030.ch1. Google Scholar [18] J. Henry, Etude de la contrôlabilité de certaines équations paraboliques non linéaires, These, Paris.Google Scholar [19] O. Y. Imanuvilov and M. Yamamoto, Carleman estimate for a parabolic equation in a sobolev space of negative order and their applications, Control of Nonlinear Distributed Parameter Systems, 218 (2011), 113-137. Google Scholar [20] O. Ladyzhenskaia, V. Solonnikov and N. Ural'tseva, Linear and Quasi-linear Equations of Parabolic Type, American Mathematical Society, translations of mathematical monographs, American Mathematical Society, 1988, URL https://books.google.es/books?id=dolUcRSDPgkC.Google Scholar [21] G. Lebeau and L. Robbiano, Contrôle exact de léquation de la chaleur, Communications in Partial Differential Equations, 20 (1995), 335-356. doi: 10.1080/03605309508821097. Google Scholar [22] X. Li and J. Yong, Systems & Control: Foundations & Applications. Birkhäuser Boston, Inc., Boston, MA, 1995, URL https://books.google.es/books?id=ryfUBwAAQBAJ. doi: 10.1007/978-1-4612-4260-4. Google Scholar [23] G. Lieberman, Second Order Parabolic Differential Equations, World Scientific Publishing Co., Inc., River Edge, NJ, 1996, URL https://books.google.es/books?id=s9Guiwylm3cC. doi: 10.1142/3302. Google Scholar [24] J.-L. Lions, Controlabilité exacte des systèmes distribués: remarques sur la théorie générale et les applications, Springer, 83 (1986), 3-14. doi: 10.1007/BFb0007542. Google Scholar [25] J. Lions and E. Magenes, Problmes aux Limites Non Homognes et Applications, no. v. 1 in Grundlehren der mathematischen Wissenschaften, Springer Berlin Heidelberg, 1968. Google Scholar [26] J. Lohéac, E. Trélat and E. Zuazua, Minimal controllability time for the heat equation under unilateral state or control constraints, Mathematical Models and Methods in Applied Sciences, 27 (2017), 1587-1644, URL http://www.worldscientific.com/doi/abs/10.1142/S0218202517500270. doi: 10.1142/S0218202517500270. Google Scholar [27] S. Mitter and J. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Grundlehren der mathematischen Wissenschaften, Springer Berlin Heidelberg, 1971, URL https://books.google.it/books?id=KDlhRQAACAAJ. Google Scholar [28] A. Porretta, Local existence and uniqueness of weak solutions for non-linear parabolic equations with superlinear growth and unbounded initial data, Advances in Differential Equations, 6 (2001), 73-128. Google Scholar [29] M. Protter and H. Weinberger, Maximum Principles in Differential Equations, Springer-Verlag, New York, 1984, URL https://books.google.es/books?id=JUXhBwAAQBAJ. doi: 10.1007/978-1-4612-5282-5. Google Scholar [30] J. Simon, Compact sets in the space $L^p(0, T; B)$, Annali di Matematica Pura ed Applicata, 146 (1986), 65-96, URL https://doi.org/10.1007/BF01762360. doi: 10.1007/BF01762360. Google Scholar [31] A. Wächter and L. T. Biegler, On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming, Mathematical Programming, 106 (2006), 25-57. doi: 10.1007/s10107-004-0559-y. Google Scholar [32] Z. Wu, J. Yin and C. Wang, Elliptic & Parabolic Equations, World Scientific, 2006, URL https://books.google.es/books?id=DnCH1_1YffYC. doi: 10.1142/6238. Google Scholar [33] E. Zuazua, Controllability and observability of partial differential equations: Some results and open problems, Handbook of Differential Equations: Evolutionary Equations, 3 (2007), 527-621. doi: 10.1016/S1874-5717(07)80010-7. Google Scholar

show all references

##### References:
 [1] F. Ammar-Khodja, S. Micu and A. Münch, Controllability of a string submitted to unilateral constraint, in Annales de l'Institut Henri Poincare (C) Non Linear Analysis, Elsevier, 27 (2010), 1097-1119. doi: 10.1016/j.anihpc.2010.02.003. Google Scholar [2] S. Aniţa and D. Tataru, Null controllability for the dissipative semilinear heat equation, Applied Mathematics & Optimization, 46 (2002), 97-105. doi: 10.1007/s00245-002-0746-2. Google Scholar [3] V. Barbu, Optimal Control of Variational Inequalities, Research notes in mathematics, Pitman Advanced Pub. Program, 1984, URL https://books.google.es/books?id=PRKoAAAAIAAJ. Google Scholar [4] V. Barbu, Analysis and Control of Nonlinear Infinite Dimensional Systems, Mathematics in Science and Engineering, Academic Press, Inc., Boston, MA, 1993, URL https://books.google.es/books?id=IaqpPMvArqEC. Google Scholar [5] H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer New York, 2011, URL https://books.google.es/books?id=GAA2XqOIIGoC. doi: 10.1007/978-0-387-70914-7. Google Scholar [6] W. Chan and B. Z. Guo, Optimal birth control of population dynamics. ⅱ. problems with free final time, phase constraints, and mini-max costs, Journal of Mathematical Analysis and Applications, 146 (1990), 523-539. doi: 10.1016/0022-247X(90)90322-7. Google Scholar [7] R. M. Colombo, G. Guerra, M. Herty and V. Schleper, Optimal control in networks of pipes and canals, SIAM Journal on Control and Optimization, 48 (2009), 2032-2050. doi: 10.1137/080716372. Google Scholar [8] J.-M. Coron and E. Trélat, Global steady-state controllability of one-dimensional semilinear heat equations, SIAM journal on control and optimization, 43 (2004), 549-569. doi: 10.1137/S036301290342471X. Google Scholar [9] J. Coron, Control and Nonlinearity, Mathematical surveys and monographs, American Mathematical Society, 2007, URL https://books.google.es/books?id=aEKv1bpcrKQC. doi: 10.1090/surv/136. Google Scholar [10] J. I. Diaz, Sur la contrôlabilité approchée des inéquations variationelles et dutres problèmes paraboliques non linéaires, CR Acad. Sci. Paris, 312 (1991), 519-522. Google Scholar [11] O. Y. Emanuilov, Controllability of parabolic equations, Sbornik: Mathematics, 186 (1995), 879-900. doi: 10.1070/SM1995v186n06ABEH000047. Google Scholar [12] L. Evans, Partial Differential Equations, Graduate studies in mathematics, American Mathematical Society, 2010, URL https://books.google.es/books?id=Xnu0o_EJrCQC. doi: 10.1090/gsm/019. Google Scholar [13] H. O. Fattorini and D. L. Russell, Exact controllability theorems for linear parabolic equations in one space dimension, Archive for Rational Mechanics and Analysis, 43 (1971), 272-292. doi: 10.1007/BF00250466. Google Scholar [14] E. Fernández-Cara and E. Zuazua, Null and approximate controllability for weakly blowing up semilinear heat equations, Annales de l'Institut Henri Poincare (C) Non Linear Analysis, 17 (2000), 583-616, URL http://www.sciencedirect.com/science/article/pii/S0294144900001177. doi: 10.1016/S0294-1449(00)00117-7. Google Scholar [15] R. Fourer, D. M. Gay and B. W. Kernighan, A modeling language for mathematical programming, Management Science, 36 (1990), 519-554. URL https://orfe.princeton.edu/~rvdb/307/textbook/AMPLbook.pdf. doi: 10.1287/mnsc.36.5.519. Google Scholar [16] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Classics in Mathematics. Springer-Verlag, Berlin, 2001. Google Scholar [17] P. Grisvard, Elliptic Problems in Nonsmooth Domains, SIAM, 2011. doi: 10.1137/1.9781611972030.ch1. Google Scholar [18] J. Henry, Etude de la contrôlabilité de certaines équations paraboliques non linéaires, These, Paris.Google Scholar [19] O. Y. Imanuvilov and M. Yamamoto, Carleman estimate for a parabolic equation in a sobolev space of negative order and their applications, Control of Nonlinear Distributed Parameter Systems, 218 (2011), 113-137. Google Scholar [20] O. Ladyzhenskaia, V. Solonnikov and N. Ural'tseva, Linear and Quasi-linear Equations of Parabolic Type, American Mathematical Society, translations of mathematical monographs, American Mathematical Society, 1988, URL https://books.google.es/books?id=dolUcRSDPgkC.Google Scholar [21] G. Lebeau and L. Robbiano, Contrôle exact de léquation de la chaleur, Communications in Partial Differential Equations, 20 (1995), 335-356. doi: 10.1080/03605309508821097. Google Scholar [22] X. Li and J. Yong, Systems & Control: Foundations & Applications. Birkhäuser Boston, Inc., Boston, MA, 1995, URL https://books.google.es/books?id=ryfUBwAAQBAJ. doi: 10.1007/978-1-4612-4260-4. Google Scholar [23] G. Lieberman, Second Order Parabolic Differential Equations, World Scientific Publishing Co., Inc., River Edge, NJ, 1996, URL https://books.google.es/books?id=s9Guiwylm3cC. doi: 10.1142/3302. Google Scholar [24] J.-L. Lions, Controlabilité exacte des systèmes distribués: remarques sur la théorie générale et les applications, Springer, 83 (1986), 3-14. doi: 10.1007/BFb0007542. Google Scholar [25] J. Lions and E. Magenes, Problmes aux Limites Non Homognes et Applications, no. v. 1 in Grundlehren der mathematischen Wissenschaften, Springer Berlin Heidelberg, 1968. Google Scholar [26] J. Lohéac, E. Trélat and E. Zuazua, Minimal controllability time for the heat equation under unilateral state or control constraints, Mathematical Models and Methods in Applied Sciences, 27 (2017), 1587-1644, URL http://www.worldscientific.com/doi/abs/10.1142/S0218202517500270. doi: 10.1142/S0218202517500270. Google Scholar [27] S. Mitter and J. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Grundlehren der mathematischen Wissenschaften, Springer Berlin Heidelberg, 1971, URL https://books.google.it/books?id=KDlhRQAACAAJ. Google Scholar [28] A. Porretta, Local existence and uniqueness of weak solutions for non-linear parabolic equations with superlinear growth and unbounded initial data, Advances in Differential Equations, 6 (2001), 73-128. Google Scholar [29] M. Protter and H. Weinberger, Maximum Principles in Differential Equations, Springer-Verlag, New York, 1984, URL https://books.google.es/books?id=JUXhBwAAQBAJ. doi: 10.1007/978-1-4612-5282-5. Google Scholar [30] J. Simon, Compact sets in the space $L^p(0, T; B)$, Annali di Matematica Pura ed Applicata, 146 (1986), 65-96, URL https://doi.org/10.1007/BF01762360. doi: 10.1007/BF01762360. Google Scholar [31] A. Wächter and L. T. Biegler, On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming, Mathematical Programming, 106 (2006), 25-57. doi: 10.1007/s10107-004-0559-y. Google Scholar [32] Z. Wu, J. Yin and C. Wang, Elliptic & Parabolic Equations, World Scientific, 2006, URL https://books.google.es/books?id=DnCH1_1YffYC. doi: 10.1142/6238. Google Scholar [33] E. Zuazua, Controllability and observability of partial differential equations: Some results and open problems, Handbook of Differential Equations: Evolutionary Equations, 3 (2007), 527-621. doi: 10.1016/S1874-5717(07)80010-7. Google Scholar
Stepwise procedure
Illustration of the proof of Theorem 1.3 in two steps: Stabilization + Control
Final data for the adjoint system
Evolution of the adjoint heat equation with final datum $\varphi_T$
graph of the control in the minimal time
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