Skip to main content

Optimal Control of a Parabolic Equation with Dynamic Boundary Condition

Abstract

We investigate a control problem for the heat equation. The goal is to find an optimal heat transfer coefficient in the dynamic boundary condition such that a desired temperature distribution at the boundary is adhered. To this end we consider a function space setting in which the heat flux across the boundary is forced to be an L p function with respect to the surface measure, which in turn implies higher regularity for the time derivative of temperature. We show that the corresponding elliptic operator generates a strongly continuous semigroup of contractions and apply the concept of maximal parabolic regularity. This allows to show the existence of an optimal control and the derivation of necessary and sufficient optimality conditions.

This is a preview of subscription content, access via your institution.

Fig. 1

References

  1. Alibert, J.-J., Raymond, J.-P.: Boundary control of semilinear elliptic equations with discontinuous leading coefficients and unbounded controls. Numer. Funct. Anal. Optim. 3–4, 235–250 (1997)

    MathSciNet  Article  Google Scholar 

  2. Amann, H.: Linear and Quasilinear Parabolic Problems. Birkhäuser, Basel (1995)

    MATH  Book  Google Scholar 

  3. Amann, H.: Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems. In: Schmeisser, H.-J., et al. (eds.) Function Spaces, Differential Operators and Nonlinear Analysis. Teubner-Texte Math., vol. 133, pp. 9–126. Teubner, Stuttgart (1993)

    Google Scholar 

  4. Amann, H., Escher, J.: Strongly continuous dual semigroups. Ann. Mat. Pura Appl. 171, 41–62 (1996)

    MathSciNet  MATH  Article  Google Scholar 

  5. Arendt, W., Chill, R., Fornaro, S., Poupaud, C.: L p-Maximal regularity for nonautonomous evolution equations. J. Differ. Equ. 237(1), 1–26 (2007)

    MathSciNet  MATH  Article  Google Scholar 

  6. Arrieta, J.M., Quittner, P., Rodríguez-Bernal, A.: Parabolic problems with nonlinear dynamical boundary conditions and singular initial data. Differ. Integral Equ. 14(12), 1487–1510 (2001)

    MATH  Google Scholar 

  7. Bandle, C., von Below, J., Reichel, W.: Parabolic problems with dynamical boundary conditions: eigenvalue expansions and blow up. Atti Accad. Naz. Lincei, Rend. Lincei, Mat. Appl. 17(1), 35–67 (2006)

    MathSciNet  MATH  Article  Google Scholar 

  8. Bank, R.E., Rose, D.J., Fichtner, W.: Numerical methods for semiconductor device simulation. SIAM J. Sci. Stat. Comput. 4(3), 416–435 (1983)

    MathSciNet  MATH  Article  Google Scholar 

  9. von Below, J., De Coster, C.: A qualitative theory for parabolic problems under dynamical boundary conditions. J. Inequal. Appl. 5(5), 467–486 (2000)

    MathSciNet  MATH  Google Scholar 

  10. Berezanskij, Y.M.: Selfadjoint Operators in Spaces of Functions of Infinitely Many Variables. Translations of Mathematical Monographs, vol. 63. Am. Math. Soc., Providence (1986)

    MATH  Google Scholar 

  11. Bonnans, J.F.: Second-order analysis for control constrained optimal control problems of semillinear elliptic equations. Appl. Math. Optim. 38, 303–325 (1998)

    MathSciNet  MATH  Article  Google Scholar 

  12. Casas, E.: Boundary control of semilinear elliptic equations with pointwise state constraints. SIAM J. Control Optim. 31, 993–1006 (1993)

    MathSciNet  MATH  Article  Google Scholar 

  13. Casas, E., de los Reyes, J., Tröltzsch, F.: Sufficient second-order optimality conditions for semilinear control problems with pointwise state constraints. SIAM J. Optim. 19, 616–643 (2008)

    MathSciNet  MATH  Article  Google Scholar 

  14. Casas, E., Tröltzsch, F., Unger, A.: Second order sufficient optimality conditions for some state-constrained control problems of semilinear elliptic equations. SIAM J. Control Optim. 38, 1369–1391 (2000)

    MathSciNet  MATH  Article  Google Scholar 

  15. Cialdea, A., Maz’ya, V.: Criterion for the L p-dissipativity of second order differential operators with complex coefficients. J. Math. Pures Appl. 84(8), 1067–1100 (2005)

    MathSciNet  MATH  Google Scholar 

  16. Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. Studies in Mathematics and Its Applications, vol. 4. North Holland, Amsterdam (1978)

    MATH  Book  Google Scholar 

  17. Crank, J.: The Mathematics of Diffusion. Clarendon Press, Oxford (1975)

    Google Scholar 

  18. Denk, R., Hieber, M., Prüss, J.: \({\mathcal{R}}\)-boundedness, Fourier multipliers and problems of elliptic and parabolic type. Mem. Am. Math. Soc. 788 (2003)

  19. de Los Reyes, J.C., Merino, P., Rehberg, J., Tröltzsch, F.: Optimality conditions for state-constrained PDE control problems with time-dependent controls. Control Cybern. 37(1), 5–38 (2008)

    MATH  Google Scholar 

  20. de Simon, L.: Un’applicazione della teoria degli integrali singolari allo studio delle equazione differenziali lineari astratte del primo ordine. Rend. Semin. Mat. Univ. Padova 34, 205–223 (1964)

    MATH  Google Scholar 

  21. Dore, G.: L p regularity for abstract differential equations. In: Komatsu, H. (ed.) Proceedings of the International Conference in Memory of Professor Kosaku Yosida held at RIMS on Functional Analysis and Related Topics, Kyoto University, Japan, July 29–August 2, 1991. Lect. Notes Math., vol. 1540, pp. 25–38. Springer, Berlin (1993)

    Google Scholar 

  22. Duderstadt, F., Hömberg, D., Khludnev, A.M.: A mathematical model for impulse resistance welding. Math. Methods Appl. Sci. 26, 717–737 (2003)

    MathSciNet  MATH  Article  Google Scholar 

  23. Dunford, N., Schwartz, J.T., Bade, W.G., Bartle, R.G.: Linear Operators. I. General Theory. Pure and Applied Mathematics, vol. 6. Interscience, New York (1958)

    MATH  Google Scholar 

  24. Escher, J.: Quasilinear parabolic systems with dynamical boundary conditions. Commun. Partial Differ. Equ. 18(7–8), 1309–1364 (1993)

    MathSciNet  MATH  Article  Google Scholar 

  25. Evans, L.C., Gariepy, R.F.: Measure Theory and Fine Properties of Functions. Studies in Advanced Mathematics. CRC Press, Boca Raton (1992)

    MATH  Google Scholar 

  26. Favini, A., Goldstein, G.R., Goldstein, J.A., Romanelli, S.: The heat equation with generalized Wentzell boundary condition. J. Evol. Equ. 2(1), 1–19 (2002)

    MathSciNet  MATH  Article  Google Scholar 

  27. Fuhrmann, J., Langmach, H.: Stability and existence of solutions of time-implicit finite volume schemes for viscous nonlinear conservation laws. Appl. Numer. Math. 37(1–2), 201–230 (2001)

    MathSciNet  MATH  Article  Google Scholar 

  28. Gärtner, K.: Existence of bounded discrete steady-state solutions of the van Roosbroeck system on boundary conforming Delaunay grids. SIAM J. Sci. Comput. 31(2), 1347–1362 (2009)

    MATH  Article  Google Scholar 

  29. Gajewski, H.: Analysis und Numerik von Ladungstransport in Halbleitern. Mitt. Ges. Angew. Math. Mech. 16(1), 35–57 (1993)

    MathSciNet  Google Scholar 

  30. Gajewski, H., Gröger, K., Zacharias, K.: Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen. Akademie Verlag, Berlin (1974)

    MATH  Google Scholar 

  31. Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Grundlehren der Mathematischen Wissenschaften, vol. 224. Springer, Berlin (1983)

    MATH  Book  Google Scholar 

  32. Goldberg, H., Tröltzsch, F.: Second order sufficient optimality conditions for a class of nonlinear parabolic boundary control problems. SIAM J. Control Optim. 31(4), 1007–1025 (1993)

    MathSciNet  MATH  Article  Google Scholar 

  33. Goldstein, G.R.: Derivation and physical interpretation of general boundary conditions. Adv. Differ. Equ. 11(4), 457–480 (2006)

    MATH  Google Scholar 

  34. Griepentrog, J.A., Kaiser, H.-C., Rehberg, J.: Heat kernel and resolvent properties for second order elliptic differential operators with general boundary conditions on L p. Adv. Math. Sci. Appl. 11, 87–112 (2001)

    MathSciNet  MATH  Google Scholar 

  35. Griepentrog, J.A., Recke, L.: Linear elliptic boundary value problems with non-smooth data: normal solvability on Sobolev-Campanato spaces. Math. Nachr. 225, 39–74 (2001)

    MathSciNet  MATH  Article  Google Scholar 

  36. Griepentrog, J.A.: Linear elliptic boundary value problems with non-smooth data: Campanato spaces of functionals. Math. Nachr. 243, 19–42 (2002)

    MathSciNet  MATH  Article  Google Scholar 

  37. Gröger, K.: Private communication (1998)

  38. Gröger, K.: A W 1,p-estimate for solutions to mixed boundary value problems for second order elliptic differential equations. Math. Ann. 283, 679–687 (1989)

    MathSciNet  MATH  Article  Google Scholar 

  39. Haller-Dintelmann, R., Rehberg, J.: Coercivity for elliptic operators and positivity of solutions on Lipschitz domains. Arch. Math. 95, 457–468 (2010)

    MathSciNet  MATH  Article  Google Scholar 

  40. Haller-Dintelmann, R., Rehberg, J.: Maximal parabolic regularity for divergence operators including mixed boundary conditions. J. Differ. Equ. 247(5), 1354–1396 (2009)

    MathSciNet  MATH  Article  Google Scholar 

  41. Haller-Dintelmann, R., Meyer, C., Rehberg, J., Schiela, A.: Hölder continuity and optimal control for nonsmooth elliptic problems. Appl. Math. Optim. 60(3), 397–428 (2009)

    MathSciNet  MATH  Article  Google Scholar 

  42. Hille, E., Phillips, R.S.: Functional analysis and semi-groups. American Mathematical Society Colloquium Publications, vol. 31, pp. 1001–1009. American Mathematical Society, Providence (1957)

    Google Scholar 

  43. Hintermann, T.: Evolution equations with dynamic boundary conditions. Proc. R. Soc. Edinb., Sect. A, Math. 113(1–2), 43–60 (1989)

    MathSciNet  MATH  Article  Google Scholar 

  44. Kato, T.: Perturbation Theory for Linear Operators. Classics in Mathematics. Springer, Berlin (1980). Reprint of the corr. print. of the 2nd edn.

    MATH  Google Scholar 

  45. Lamberton, D.: Equations d’évolution linéaires associées à des semi-groupes de contractions dans les espaces L p. J. Funct. Anal. 72, 252–262 (1987)

    MathSciNet  MATH  Article  Google Scholar 

  46. Langer, R.E.: A problem in diffusion or in the flow of heat for a solid in contact with a fluid. Tohoku Math. J. 35, 260–275 (1932)

    Google Scholar 

  47. Lions, J.L.: Optimal control of systems governed by partial differential equations. In: Die Grundlehren der mathematischen Wissenschaften, vol. 170. Springer, Berlin (1971)

    Google Scholar 

  48. Lunardi, A.: Analytic Semigroups and Optimal Regularity in Parabolic Equations. Birkhäuser, Basel (1995)

    Book  Google Scholar 

  49. Marcus, M., Mizel, V.: Every superposition operator mapping one Sobolev space into another is continuous. J. Funct. Anal. 33, 217–229 (1979)

    MathSciNet  MATH  Article  Google Scholar 

  50. Ouhabaz, E.: Analysis of Heat Equations on Domains. London Mathematical Society Monographs Series, vol. 31. Princeton University Press, Princeton (2005)

    MATH  Google Scholar 

  51. Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer, Berlin (1983)

    MATH  Book  Google Scholar 

  52. Raymond, J.P., Tröltzsch, F.: Second order sufficient optimality conditions for nonlinear parabolic control problems with state constraints. Discrete Contin. Dyn. Syst. 6, 431–450 (2000)

    MATH  Article  Google Scholar 

  53. Reed, M., Simon, B.: Methods of Modern Mathematical Physics. II. Fourier Analysis, Self-Adjointness. Academic Press, New York (1975)

    MATH  Google Scholar 

  54. Stein, E.M.: Singular Integrals and Differentiability Properties of Functions. Princeton University Press, Princeton (1970)

    MATH  Google Scholar 

  55. Triebel, H.: Interpolation Theory, Function Spaces, Differential Operators. North-Holland, Amsterdam (1978)

    Google Scholar 

  56. Triebel, H.: On spaces of \(B^{s}_{\infty,q}\) and C s type. Math. Nachr. 85, 75–90 (1978)

    MathSciNet  MATH  Article  Google Scholar 

  57. Tröltzsch, F.: Optimal Control of Partial Differential Equations: Theory, Methods and Applications. Am. Math. Soc., Providence (2010)

    MATH  Google Scholar 

  58. Vogt, H., Voigt, J.: Wentzell boundary conditions in the context of Dirichlet forms. Adv. Differ. Equ. 8(7), 821–842 (2003)

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

We wish to thank our colleagues K. Gröger (Berlin), H. Amann (Zürich) and H. Vogt (Clausthal) for valuable discussions on the subject of the paper. Moreover, we thank the referees for the carefully reading of the paper.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to J. Rehberg.

Additional information

Den Therapeuten der Parkklinik Wandlitz in Dankbarkeit gewidmet.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Hömberg, D., Krumbiegel, K. & Rehberg, J. Optimal Control of a Parabolic Equation with Dynamic Boundary Condition. Appl Math Optim 67, 3–31 (2013). https://doi.org/10.1007/s00245-012-9178-9

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00245-012-9178-9

Keywords

  • Parabolic equation
  • Mixed boundary condition
  • Maximal parabolic L p-regularity
  • Optimal control
  • Sufficient optimality conditions