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Hierarchical exact controllability of semilinear parabolic equations with distributed and boundary controls

F. D. Araruna

Departamento de Matemática, Universidade Federal da Paraíba, 58051-900, João Pessoa — PB, Brazil

E-mail Address: fagner@mat.ufpb.br

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E. Fernández-Cara

Departamento EDAN and IMUS, University of Sevilla, Aptdo. 1160, 41080 Sevilla, Spain

E-mail Address: cara@us.es

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L. C. da Silva

Departamento de Matemática, Universidade Federal da Paraíba, 58051-900, João Pessoa — PB, Brazil

E-mail Address: luciano.cipri@yahoo.com.br

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https://doi.org/10.1142/S0219199719500342Cited by:15 (Source: Crossref)

Abstract

We present some exact controllability results for parabolic equations in the context of hierarchic control using Stackelberg–Nash strategies. We analyze two cases: in the first one, the main control (the leader) acts in the interior of the domain and the secondary controls (the followers) act on small parts of the boundary; in the second one, we consider a leader acting on the boundary while the followers are of the distributed kind. In both cases, for each leader, an associated Nash equilibrium pair is found; then, we obtain a leader that leads the system exactly to a prescribed (but arbitrary) trajectory. We consider linear and semilinear problems.

Keywords:

AMSC: 35K50, 90C29, 93C20

References

1. F. D. Araruna, E. Fernández-Cara, S. Guerrero and M. C. Santos , New results on the Stackelberg–Nash exact controllability of linear parabolic equations, Syst. Control Lett. 104 (2017) 78–85. Crossref, Web of Science, Google Scholar
2. F. D. Araruna, E. Fernández-Cara and M. C. Santos , Stackelberg–Nash exact controllabilty for linear and semilinear parabolic equations, ESAIM, Control Optimisation Calc. Var. 21 (2015) 835–856. Crossref, Web of Science, Google Scholar
3. F. D. Araruna, E. Fernández-Cara and L. C. da Silva , Hierarchic control for the wave equation, J. Optim. Theory Appl. 178(1) (2018) 264–288. Crossref, Web of Science, Google Scholar
4. J. I. Díaz and J.-L. Lions , On the approximate controllability of Stackelberg–Nash strategies, in Ocean Circulation and Pollution Control: A Mathematical and Numerical Investigations (Springer, Berlin, 2004), pp. 17–27. Crossref, Google Scholar
5. C. Fabre, J.-P. Puel and E. Zuazua , Approximate controllability of the semilinear heat equation, Proc. R. Soc. Edinb. A 125(1) (1995) 31–61. Crossref, Web of Science, Google Scholar
6. E. Fernández-Cara and S. Guerrero , Global Carleman inequalities for parabolic systems and applications to controllability, SIAM J. Control Optim. 45(4) (2006) 1395–1446. Crossref, Web of Science, Google Scholar
7. E. Fernández-Cara, S. Guerrero, O. Y. Imanuvilov and J.-P. Puel , Local exact controllability of the Navier–Stokes system, J. Math. Pures Appl. 83(2004) 1501–1542. Crossref, Web of Science, Google Scholar
8. A. V. Fursikov and O. Y. Imanuvilov , Controllability of Evolution Equations, Lectures Notes Series, Vol. 34 (Seoul National University, Research Institute of Mathematics, Global Analysis Research Center, Seoul, 1996). Google Scholar
9. F. Guillén-González, F. P. Marques-Lopes and M. A. Rojas-Medar , On the approximate controllability of Stackelberg–Nash strategies for Stokes equation, Proc. Amer. Math. Soc. 141(5) (2013) 1759–1773. Crossref, Web of Science, Google Scholar
10. O. Y. Imanuilov , Controllability of parabolic equations, Mat. Sb. 186 (1995) 109–132. Web of Science, Google Scholar
11. O. Y. Imanuvilov, J.-P. Puel and M. Yamamoto , Carleman estimates for parabolic equations with nonhomogeneous boundary conditions, Chin. Ann. Math. Ser. B 30(4) (2009) 333–378. Crossref, Web of Science, Google Scholar
12. O. Y. Imanuvilov and M. Yamamoto , Carleman estimate for a parabolic equation in a Sobolev space of negative order and its applications, in Control of Nonlinear Distributed Parameter Systems, Lecture Notes in Pure and Appl. Math., Vol. 218 (Dekker, New York, 2001), pp. 113–137. Crossref, Google Scholar
13. J.-L. Lions , Hierarchic control, Proc. Indian Acad. Sci. Math. Sci. 104(4) (1994) 295–304. Crossref, Web of Science, Google Scholar
14. J.-L. Lions and E. Magenes , Non-Homogeneous Boundary Value Problems and Applications II (Springer-Verlag, Berlin, Heidelberg, New York, 1972). Google Scholar