Abstract
In this paper, we study Stackelberg-Nash strategies to control a system of two coupled parabolic equations. We assume that we act in the system by means of a hierarchy of controls. First, a leader (vectorial) control achieve their objectives, and then other controls, named followers, react optimally to the leader action. We prove an observability inequality for an extended system, which yields the Stackelberg-Nash optimization. Then, we remove the action of one of the components of the leader control. In this way, we control a system of various equations by acting only on the first component.
Dedicated to Prof. Enrique Fernández-Cara on the occasion of his 60th birthday.
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References
Ammar-Khojda, F., Benabdallah, A., Dupaix, C., Kostin, I.: Null controllability of some systems of parabolic type by one control force. ESAIM Control Optim. Calc. Var. 11(3), 426–448 (2005)
Ammar-Khojda, F., Benabdallah, A., Dupaix C., González-Burgos, M.: A generalization of the Kalman rank condition for time-dependent coupled linear parabolic systems. Differ. Equ. Appl. 1(3), 427–457 (2009)
Ammar-Khojda, F., Benabdallah, A., González-Burgos, M., de Teresa, L.: Recent results on the controllability of linear coupled parabolic problems: a survey. Math. Control Relat. Fields 1(3), 267–306 (2011)
Araruna, F.D., de Menezes, S.D.B., Rojas-Medar, M.A.: On the approximate controllability of Stackelberg-Nash strategies for linearized micropolar fluids. Appl. Math. Optim. 70(3), 373–393 (2014)
Araruna, F.D., Fernández-Cara E., Santos, M.C.: Stackelberg-Nash exact controllability for linear and semilinear parabolic equations. ESAIM Control Optim. Calc. Var. 21(3), 835–856 (2015)
Araruna, F. D., Fernández-Cara, E., Guerrero, S., Santos, M. C. New results on the Stackelberg-Nash exact control of linear parabolic equations. Systems Control Lett. 104, 78–85 (2017).
Calsarava, B.M.R., Carreño, N., Cerpa, E.: Insensitizing controls for a phase field system. Nonlinear Anal. 143, 120–137 (2016)
Chakrabarty, S.P., Hanson, F.B.: Optimal control of drug delivery to brain tumors for a distributed parameters model. In: Proceedings of the American Control Conference, pp. 973–978 (2005)
Corrias, L., Perthame, B., Zaag, H.: Global solutions of some chemotaxis and angiogenesis systems in high space dimensions. Milan J. Math. 72, 1–28 (2004)
Díaz, J.I.: On the von Neumann problem and the approximate controllability of Stackelberg-Nash strategies for some environmental problems. Rev. Real Acad. Cienc. Exact. Fís. Natur. 96(3), 343–356 (2002)
de Teresa, L.: Insensitizing controls for a semilinear heat equation. Commun. Partial Differ. Equ. 25(1–2), 39–72 (2000)
Fernández-Cara, E., Guerrero, S.: Global Carleman inequalities for parabolic systems and applications to controllability. SIAM J. Control Optim. 45(4), 1395–1446 (2006)
Fernández-Cara, E., Guerrero, S., Imanuvilov, O.Y., Puel, J.P.: Local exact controllability of the Navier-Stokes system. J. Math. Pures Appl. 83(12), 1501–1542 (2004)
Fursikov, A., Imanuvilov, O.Y.: Controllability of Evolution Equations. Lecture Notes, Research Institute of Mathematics. Seoul National University, Seoul (1996)
González-Burgos, M., de Teresa, L.: Controllability results for cascade systems of m coupled parabolic PDEs by one control force. Port. Math. 67(1), 91–113 (2010)
Guillén-González, F., Marques-Lopes, F., Rojas-Medar, M.: On the approximate controllability of Stackelberg-Nash strategies for Stokes equations. Proc. Am. Math. Soc. 141(5), 1759–1773 (2013)
Hernández-Santamaría, V., de Teresa, L., Poznyak, A.: Hierarchic control for a coupled parabolic system. Port. Math. 73(2), 115–137 (2016)
Imanuvilov, O.Y., Yamamoto, M.: Carleman inequalities for parabolic equations in Sobolev spaces of negative order and exact controllability for semilinear parabolic equations. Publ. Res. Inst. Math. Sci. 39(2), 227–274 (2003)
Limaco, F., Clark, H.R., Medeiros, L.A.: Remarks on hierarchic control. J. Math. Anal. Appl. 359(1), 368–383 (2009)
Lions, J.-L.: Hierarchic control. Indian Acad. Sci. Proc. Math. Sci. 104(1), 295–304 (1994)
Lions, J.-L.: Some remarks on Stackelberg’s optimization. Math. Models Methods Appl. Sci. 4(4), 477–487 (1994)
Nagai, T., Senba, T., Suzuki, T.: Chemotactic collapse in a parabolic system of mathematical biology. Hiroshima Math. J. 30(3), 463–497 (2000)
Nash, J.F.: Non-cooperative games. Ann. Math. 54(2), 286–295 (1951)
von Stackelberg, H.: Marktform und Gleichgewicht. Springer, Vienna (1934)
Zabczyk, J.: Mathematical Control Theory: An Introduction. Birkhäuser, Boston (1992)
Acknowledgements
This work was partially supported by CONACyT and UNAM-DGAPA-PAPIIT IN102116.
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Hernández-Santamaría, V., de Teresa, L. (2018). Some Remarks on the Hierarchic Control for Coupled Parabolic PDEs. In: Doubova, A., González-Burgos, M., Guillén-González, F., Marín Beltrán, M. (eds) Recent Advances in PDEs: Analysis, Numerics and Control. SEMA SIMAI Springer Series, vol 17. Springer, Cham. https://doi.org/10.1007/978-3-319-97613-6_7
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