Applied Mathematics & Optimization

, Volume 75, Issue 2, pp 151–173 | Cite as

Optimal Wentzell Boundary Control of Parabolic Equations

Article
  • 169 Downloads

Abstract

This paper deals with a class of optimal control problems governed by an initial-boundary value problem of a parabolic equation. The case of semi-linear boundary control is studied where the control is applied to the system via the Wentzell boundary condition. The differentiability of the state variable with respect to the control is established and hence a necessary condition is derived for the optimal solution in the case of both unconstrained and constrained problems. The condition is also sufficient for the unconstrained convex problems. A second order condition is also derived.

Keywords

Optimality condition Parabolic equation Boundary control Wentzell boundary condition 

Mathematics Subject Classification

49K20 35K20 

References

  1. 1.
    Apushkinskaya, D.E., Nazarov, A.I.: Hlder estimates of solutions to initial-boundary value problems for parabolic equations of nondivergent form with Wentzel boundary condition. Am. Math. Soc. Transl. 2(64), 1–13 (1995)CrossRefMATHGoogle Scholar
  2. 2.
    Apushkinskaya, D.E., Nazarov, A.I.: The nonstationary Venttsel problem with quadratic growth with respect to the gradient. J. Math. Sci. 80(6), 2197–2207 (1996)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Apushkinskaya, D.E., Nazarov, A.I.: A survey of results on nonlinear Venttsel problems. Appl. Math. 45(1), 69–80 (2000)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Arena, O., Littman, W.: Boundary control of two PDE’s separated by interface conditions. J. Syst. Sci. Complex. 23(3), 431–437 (2010)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Ben Tal, A., Zowe J.: A unified theory of first and second order conditions for extremum problems in topological vector spaces. Math. Progr. Study 19, 39-76 (1999)Google Scholar
  6. 6.
    Bonnans, J., Casas, E.: Contrôle de systèmes elliptiques semilineaires comportant des contraintes sur l’ètat. In: Brezis, H., lions, J. (eds.) Nonlinear Partial Differential Equations and Their Applications, Collège de France Seminar, vol. 8, pp. 69–86. Londonman Scientific & Technical, New York (1988)Google Scholar
  7. 7.
    Carslaw, H.S., Jaeger, J.C.: Conduction of Heat in Solids. Clarendon Press, Oxford (1959)MATHGoogle Scholar
  8. 8.
    Casas, E., Tröltzsch, F.: Second-order necessary optimality conditions for some state-constrained control problems of semilinear elliptic equations. Appl. Math. Optim. 39, 211–227 (1999)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Casas, E., Tröltzsch, F.: Second-order sufficient optimality conditions for some state-constrained control problems of semilinear elliptic equations. SIAM J. Control Optim. 38(5), 1369–1391 (2000)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Favini, A., Goldestein, R.G., Romanelli, S.: The heat equation with generalized wentzell boundary condition. J. Evol. Equ. 2, 1–19 (2002)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Favini, A., Goldstein, G.R., Goldstein, J.A., Obrecht, E., Romanelli, S.: Elliptic operators with general Wentzell boundary conditions, analytic semigroups and the angle concavity theorem. Math. Nachr. 283(4), 504–521 (2010)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Friedman, A.: Partial Differential Equations of Parabolic Type. Dover Publications, New York (1992)Google Scholar
  13. 13.
    Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of the Second Order, 2nd edn. Springer (1975)Google Scholar
  14. 14.
    Krumbiegel, K., Rehberg, J.: Second order sufficient optimality conditions for parabolic optimal control problems with pointwise state constraints. SIAM J. Control Optim. 51(1), 304–331 (2013)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Lions J.-L.: Optimal Control of Systems Governed by Partial Differential Equations Springer (1971)Google Scholar
  16. 16.
    Littman, W., Taylor, S.: The heat and Schrdinger equations: boundary control with one shot, Control methods in PDE-dynamical systems, 293305, Contemp. Math., vol. 426.Amer. Math. Soc., Providence (2007)Google Scholar
  17. 17.
    Luo, Y.: The heat conduction in a medium enclosed by a thin shell of higher diffusivity. Proc. 5th Colloquium on Diff. Equations, Bulgaria (1994)Google Scholar
  18. 18.
    Luo, Y.: Quasilinear second order elliptic equations with elliptic Venttsel boundary conditions. Nonlinear Anal. 16, 761–769 (1991)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Luo, Y.: Necessary optimality conditions for some control problems of elliptic equations with Venttsel boundary conditions. Appl. Math. Optim. 61, 337–351 (2010)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Luo, Y., Trudinger, N.S.: Linear second order elliptic equations with Venttsel boundary conditions. Proc. R. Soc. Edinb. 118A, 193–207 (1991)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Nocedal, J., Wright S.: Numerical Optimization, Springer Series in Operations Research and Financial Engineering, 2nd edn. Springer (2006)Google Scholar
  22. 22.
    Raymond, J.-P., Tröltzsch, F.-J.: Second order sufficient optimality conditions for nonlinear parabolic optimal control problems with state constraints. Discrete Contin. Dyn Syst. 6(2), 431–450 (2000)CrossRefMATHGoogle Scholar
  23. 23.
    Robinson, S.M.: First order conditions for general nonlinear optimization. SIAM J. Appl. Math. 30(4), 597–607 (1976)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Venttsel’, A.D.: On boundary conditions for multidimensional diffusion processes. Theor. Probab. Appl. 4, 164–177 (1959)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Zowe, L., Kurcyusz, S.: Regularity and stability for the mathematical programming problem in Babach spaces. Appl. Math. Optim. 5, 49–62 (1979)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.School of Mathematical and Geospatial SciencesRMIT UniversityMelbourneAustralia

Personalised recommendations