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Hierarchical Controllability for a Nonlinear Parabolic Equation in One Dimension

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Abstract

This paper deals with the hierarchical control of a nonlinear parabolic equation in one dimension. The novelty in this work is the appearance of the spatial derivative of the solution instead of considering only the solution in the quasilinear term (nonlinearity), here lies the difficulty of approaching said equation. We use Stackelberg–Nash strategies. As usual, we consider one control called leader and two controls called followers. To each leader, we associate a Nash equilibrium corresponding to a bi-objective optimal control problem; then, we look for a leader that solves null and trajectory controllability problems. First, we study the linear problem and then, we use the results obtained in the linear case to conclude the nonlinear problem by applying the Right Inverse Function Theorem. Some comments will be put at the end.

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References

  1. Alabau-Boussouira, F., Cannarsa, P., Fragnelli, G.: Carleman estimates for degenerate parabolic operators with applications to null controllability. J. Evol. Equ. 6(2), 161–204 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  2. Alekseev, V.M., Tikhomirov, V.M., Fomin, S.V.: Optimal Control. Contemporary Soviet Mathematics, Consultants Bureau, New York (1987)

    Book  MATH  Google Scholar 

  3. Araruna, F., Fernández-Cara, E., Santos, M.: Stackelberg-Nash Controllability for linear and semilinear parabolic equations,. ESAIM Control Optim. Calc. Var. 21(3), 835–856 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  4. Araruna, F., Fernández-Cara, E., Guerrero, S., Santos, M.: New results on the Stackelberg Nash exact control of linear parabolic equations. Syst. Control Lett. 104, 78–85 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  5. Araruna, F., Fernández-Cara, E., Da Silva, L.: Hierarchical exact controllability of semilinear parabolic equations with distributed and boundary controls. Commun. Contemp. Math. 22(7), 1950034-1–1950034-41 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  6. Cannarsa, P., Fragnelli, G.: Null controllability of semilinear degenerate parabolic equations in bounded domains. Electron. J. Differ. Equ. 2006(136), 1–20 (2006)

    MathSciNet  MATH  Google Scholar 

  7. Cherniha, R., King, J., Kovalenko, S.: Lie symmetry properties of nonlinear reaction-diffusion equations with gradient-dependent diffusivity. Commun. Nonlinear Sci. Numer. Simul. 36, 98–108 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  8. De Carvalho, P., Fernández-Cara, E.: Numerical Stackelberg-Nash control for the heat equation. SIAM J. Sci. Comput. 5(42), A2678–A2700 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  9. Díaz J.I., Lions J.L.: On the Approximate Controllability of Stackelberg-Nash Strategies Ocean Circulation and Pollution Control - A Mathematical and Numerical Investigation, pp. 17–27. Springer, Berlin (2004)

  10. Fursikov, A.V., Imanuvilov, O.Y.: Controllability of Evolution Equations. Research Institute of Mathematics, Lecture Note Series. Seoul National University (1996)

  11. Gawinecki, J., Kacprzyk, P., Bar-Yoseph, P.: Initial-boundary value problem for some coupled nonlinear parabolic system of partial differential equations appearing in thermodiffusion in solid body. J. Anal. Appl. 19(1), 121–130 (2000)

    MathSciNet  MATH  Google Scholar 

  12. Ghisi, M., Gobbino, M.: A class of local classical solutions for the one-dimensional Perona-Malik equation. Trans. Am. Math. Soc. 361(12), 6429–6446 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  13. Glowinski, R., Ramos, A., Periaux, J.: Nash equilibria for the multiobjective control of linear partial differential equations. J. Optim. Theory Appl. 112(3), 457–498 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  14. Guillén-González, F., Marques-Lopes, F., Rojas-Medar, M.: On the approximate controllability of Stackelberg-Nash strategies for Stokes equation. Proc. Am. Math. Soc. 141(5), 1759–1773 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  15. Hernández, V., De Teresa, L., Poznyak, A.: Hierarchic control for a coupled parabolic system. Port. Math. 73(2), 115–137 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  16. Höllig, K.: Existence of infinitely many solutions for a forward backward heat equation. Trans. Am. Math. Soc. 278(1), 299–316 (1983)

    Article  MathSciNet  Google Scholar 

  17. 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. Kyoto Univ. 39, 227–274 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  18. Kawohl, B., Kutev, N.: Maximum and comparison principle for one-dimensional anisotropic diffusion. Math. Ann. 311, 107–123 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  19. Límaco, J., Clark, H., Medeiros, L.: Remarks on hierarchic control. J. Math. Anal. Appl. 359(1), 368–383 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  20. Lions, J.L.: Hierarchic control. Proc. Indian Acad. Sci. (Math. Sci.) 104(1), 295–304 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  21. Lions, J.L.: Some remarks on Stackelberg’s optimization. Math. Models Methods Appl. Sci. 4(4), 477–487 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  22. Liu, X., Zhang, X.: Local Controllability of Multidimensional Quasi-Linear Parabolic Equations. SIAM J. Control. Optim. 50(4), 2046–2064 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  23. Loheac, J., Trelat, E., Zuazua, E.: Minimal controllability time for the heat equation under unilateral state or control constraints. Math. Models Methods Appl. Sci. 27, 1587–1644 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  24. Nash, J.: Non-cooperative games. Ann. Math. 54(2), 286–295 (1951)

    Article  MathSciNet  MATH  Google Scholar 

  25. Nina-Huaman, D., Límaco, J.: Stackelberg-Nash Controllability for N-Dimensional Nonlinear Parabolic Partial Differential Equations. Appl. Math. Optim. 84, 1401–1452 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  26. Nuñez-Chávez, M.R.: Controllability under positive constraints for quasilinear parabolic PDEs. Math. Control Relat. Fields 12(2), 327–341 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  27. Opanesenko, S., Boyko, V., Popovych, R.: Enhanced group classification of nonlinear diffusion-reaction equations with gradient-dependent diffusivity. J. Math. Anal. Appl. 484, 123739 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  28. Pareto, V.: Cours d’ économie politique. Rouge, Laussane (1896)

    Google Scholar 

  29. Perona, P., Malik, J.: Scale-space and edge detection using anisotropic diffusion. IEEE Trans. Pattern Anal. Mach. Intell. 12(7), 629–639 (1990)

    Article  Google Scholar 

  30. Rincon M., Límaco J., Liu I-S.: A nonlinear heat equation with temperature-dependent parameters, Math. Phys. Electron. J. 12 (2006)

  31. Stackelberg, H.: Marktform und Gleichgewicht. Springer, Berlin (1934)

    MATH  Google Scholar 

  32. Taheri, S., Tang, Q., Zhang, K.: Young measure solutions and instability of the one-dimensional Perona-Malik equation. J. Math. Anal. Appl. 308, 467–490 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  33. Wei, G.: Generalized Perona-Malik equation for image restoration. Signal Process. Lett. 6(7), 165–167 (1999)

    Article  Google Scholar 

  34. Wei, G.: Generalized reaction-diffusion equations. Chem. Phys. Lett. 303, 531–536 (1999)

    Article  Google Scholar 

  35. Zhang, K.: On existence of weak solutions for one-dimensional forward-backward diffusion equations. J. Differ. Equ. 220, 322–353 (2006)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Departamento de Matemática, UFMT, Cuiabá, MT, Brazil

    Miguel R. Nuñez-Chávez

  2. IME, UFF, Rio de Janeiro, RJ, Brazil

    Juan B. Límaco Ferrel

Authors
  1. Miguel R. Nuñez-Chávez
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  2. Juan B. Límaco Ferrel
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Correspondence to Juan B. Límaco Ferrel.

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Communicated by Stefan Ulbrich.

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Nuñez-Chávez, M.R., Ferrel, J.B.L. Hierarchical Controllability for a Nonlinear Parabolic Equation in One Dimension. J Optim Theory Appl 198, 1–48 (2023). https://doi.org/10.1007/s10957-023-02217-0

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