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Optimal Design of the Time-Dependent Support of Bang–Bang Type Controls for the Approximate Controllability of the Heat Equation

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Abstract

We consider the nonlinear optimal shape design problem, which consists in minimizing the amplitude of bang–bang type controls for the approximate controllability of a linear heat equation with a bounded potential. The design variable is the time-dependent support of the control. Precisely, we look for the best space–time shape and location of the support of the control among those, which have the same Lebesgue measure. Since the admissibility set for the problem is not convex, we first obtain a well-posed relaxation of the original problem and then use it to derive a descent method for the numerical resolution of the problem. Numerical experiments in 2D suggest that, even for a regular initial datum, a true relaxation phenomenon occurs in this context. Also, we implement a simple algorithm for computing a quasi-optimal domain for the original problem from the optimal solution of its associated relaxed one.

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References

  1. Frecker, M.I.: Recent advances in optimization of smart structures actuators. J. Intell. Mater. Syst. Struct. 14, 207–216 (2003)

    Article  Google Scholar 

  2. Morris, K.: Linear-quadratic optimal actuator location. IEEE Trans. Autom. Control 56(1), 113–124 (2011)

    Article  Google Scholar 

  3. Henrot, A., Maillot, H.: Optimization of the shape and location of the actuators in an internal control problem. Boll. Unione Mat. It. Sez. B Artic. Ric. Mat. 8(4), 737–757 (2001)

    MathSciNet  Google Scholar 

  4. Hébrard, P., Henrot, A.: Optimal shape and position of the actuators for the stabilization of a string. Syst. Control Lett. 48(3–4), 199–209 (2003)

    Article  MATH  Google Scholar 

  5. Hébrard, P., Henrot, A.: A spillover phenomenon in the optimal location of actuators. SIAM J. Control Optim. 44(1), 349–366 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  6. Münch, A.: Optimal location of the support of the control for the 1-D wave equation: numerical investigations. Comput. Optim. Appl. 42(3), 443–470 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  7. Periago, F.: Optimal shape and position of the support of the internal exact control of a string. Syst. Control Lett. 58(2), 136–140 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  8. Privat, Y., Trélat, E., Zuazua, E.: Optimal location of controllers for the one-dimensional wave equation. Ann. Inst. Henri Poincare Anal. Non Lineaire (2012). doi:10.1016/j.anihpc.2012.11.005

    Google Scholar 

  9. Münch, A.: Optimal design of the support of the control for the 2-D wave equation: a numerical method. Int. J. Numer. Anal. Model. 5(2), 331–351 (2008)

    MATH  MathSciNet  Google Scholar 

  10. Fabré, C., Puel, J.P., Zuazua, E.: Approximate controllability of the semilinear heat equation. Proc. R. Soc. Edinb., Sect. A 125(1), 31–61 (1995)

    Article  MATH  Google Scholar 

  11. Münch, A., Zuazua, E.: Numerical approximation of null controls for the heat equation: ill-posedness and remedies. Inverse Probl. 26(8), 085018 (2010)

    Article  Google Scholar 

  12. Münch, A., Periago, F.: Numerical approximation of optimal bang–bang controls for the heat equation: an optimal design approach. Syst. Control Lett. 62, 643–655 (2013)

    Article  MATH  Google Scholar 

  13. Boyer, F., Hubert, F., Le Rousseau, J.: Uniform controllability properties for space/time-discretized parabolic equations. Numer. Math. 118, 601–661 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  14. Micu, S., Zuazua, E.: Regularity issues for the null-controllability of the linear 1-d heat equation. Syst. Control Lett. 60, 406–413 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  15. Carthel, C., Glowinski, R., Lions, J.L.: On exact and approximate boundary controllabilities for the heat equation: a numerical approach. J. Optim. Theory Appl. 82(3), 429–484 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  16. Münch, A., Periago, F.: Optimal distribution of the internal null control for the one-dimensional heat equation. J. Differ. Equ. 250(1), 95–111 (2011)

    Article  MATH  Google Scholar 

  17. Angement, S.: The zero set of a solution of a parabolic equation. J. Reine Angew. Math. 399, 79–96 (1988)

    Google Scholar 

  18. Allaire, G., Münch, A., Periago, F.: Long time behavior of a two-phase optimal design for the heat equation. SIAM J. Control Optim. 48(8), 5333–5356 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  19. Henrot, A., Pierre, M.: Variation et Optimization de Formes. Mathématiques & Applications. Springer, Berlin (2005)

    MATH  Google Scholar 

  20. Münch, A., Pedregal, P., Periago, F.: Optimal design of the damping set for the stabilization of the wave equation. J. Differ. Equ. 231, 331–358 (2006)

    Article  MATH  Google Scholar 

  21. Fernández-Cara, E., Münch, A.: Numerical null controllability of semi-linear 1D heat equations: fixed point, least squares and Newton methods. Math. Control Relat. Fields 3(2), 217–246 (2012)

    Google Scholar 

  22. Castro, C., Zuazua, E.: Unique continuation and control for the heat equation from an oscillating lower dimensional manifold. SIAM J. Control Optim. 43(4), 1400–1434 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  23. Castro, C.: Exact controllability of the 1-d wave equation from a moving interior point. ESAIM Control Optim. Calc. Var. 19(1), 301–316 (2013)

    Article  MATH  MathSciNet  Google Scholar 

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Acknowledgements

Work supported by projects MTM2010-19739 from Ministerio de Educación y Ciencia (Spain) and 08720/PI/08 from Fundación Séneca (Agencia de Ciencia y Tecnología de la Región de Murcia (Spain). II PCTRM 2007-10).

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Authors and Affiliations

  1. Departamento de Matemática Aplicada y Estadística, ETSI Industriales, Universidad Politécnica de Cartagena, 30202, Cartagena, Spain

    Francisco Periago

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  1. Francisco Periago
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Correspondence to Francisco Periago.

Additional information

Communicated by Emmanuel Trélat.

Dedicated to the memory of Professor Miguel A. Sanz Alix.

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Periago, F. Optimal Design of the Time-Dependent Support of Bang–Bang Type Controls for the Approximate Controllability of the Heat Equation. J Optim Theory Appl 161, 951–968 (2014). https://doi.org/10.1007/s10957-013-0447-9

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  • DOI: https://doi.org/10.1007/s10957-013-0447-9

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