Applied Mathematics and Optimization

, Volume 16, Issue 1, pp 147–168 | Cite as

The regulator problem for parabolic equations with dirichlet boundary control

Part I: Riccati's Feedback Synthesis and Regularity of Optimal Solution
  • I. Lasiecka
  • R. Triggiani


This paper considers the regulator problem for a parabolic equation (generally unstable), defined on an open, bounded domain Ω, with control functionu acting in the Dirichlet boundary condition: minimize the quadratic functional which penalizes theL2(0, ∞; L2(Ω))-norm of the solutiony and theL2(0, ∞; L2(Γ))-norm of the Dirichlet controlu. The paper is divided in two parts. Part I derives, in a constructive way, the algebraic Riccati equation satisfied by the candidate Riccati operator solution (unique in our case) and, moreover, studies the regularity properties of the optimal pairu0, y0. Part II studies a Galerkin approximation of the regulator problem. It shows first the uniform analyticity and the uniform exponential stability of the underlying discrete (approximating) semigroups. Then it establishes the desired convergence properties, in particular, pointwise Riccati operators convergence and, as a final goal, convergence of the original dynamics acted upon by the discrete feedbacks.


Parabolic Equation Dirichlet Boundary Dirichlet Boundary Condition Convergence Property Exponential Stability 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Anselone PM (1971) Collectively Compact Operator Approximation Theory. Prentice-Hall, Englewood Cliffs, NJGoogle Scholar
  2. 2.
    Balakrishnan AV (1981) Applied Functional Analysis, 2nd ed. Springer-Verlag, New YorkGoogle Scholar
  3. 3.
    Banks TH, Kunish K (1984) The linear regulator problem for parabolic systems. SIAM J Control Optim 22:684–699Google Scholar
  4. 4.
    Bramble J, Schatz A (1970) Rayleigh-Ritz-Galerkin methods for Dirichlet's problem using subspaces without boundary conditions. Comm Pure Appl Math 23:653–657Google Scholar
  5. 5.
    Datko R (1970) Extending a theorem of Liapunov to Hilbert spaces, J Math Anal Appl 32:610–616Google Scholar
  6. 6.
    Dunford N, Schwartz J (1963) Linear Operators, II. Wiley-Interscience, New YorkGoogle Scholar
  7. 7.
    Fattorini HO (1983) The Cauchy problem. In Encylcopedia of Mathematics and Its Applications, vol. 18. Addison-Wesley, Reading, MAGoogle Scholar
  8. 8.
    Flandoli F (1984) Riccati equation arising in a boundary control problem with distributed parameters. SIAM J Control Optim 22:76–86Google Scholar
  9. 9.
    Flandoli F (1985) Algebraic Riccati equation arising in boundary control problems. PreprintGoogle Scholar
  10. 10.
    Friedman A (1976) Partial Differential Equations (reprint). R. Knegar Huntington, New YorkGoogle Scholar
  11. 11.
    Fujiwara D (1967) Concrete characterization of the domains of fractional powers of some elliptic differential operators of the second order. Proc. Japan Acad 43:82–86Google Scholar
  12. 12.
    Kato T (1966) Perturbation Theory of Linear Operators. Springer-Verlag, New YorkGoogle Scholar
  13. 13.
    Kondratiev VA (1967) Boundary problems for elliptic equations in domains with conical or angular points. Trans Moscow Math Soc 16Google Scholar
  14. 14.
    Lasiecka I (1980) Unified theory for abstract parabolic boundary problems; a semigroup approach. Appl Math Optimiz 6:287–333Google Scholar
  15. 15.
    Lasiecka I (1984) Convergence estimates for semidiscrete approximations of non-selfadjoint parabolic equations. SIAM J Numer Anal 21:894–909Google Scholar
  16. 16.
    Lasiecka I (1986) Galerkin approximations of abstract parabolic boundary value problems with rough boundary data,L p-theory. Math Comput 47:55–75Google Scholar
  17. 17.
    Lasiecka I, Lions JL, Triggiani R (1986) Non-homogeneous boundary value problems for second-order hyperbolic operators. J Math Pures Appl 65:149–192Google Scholar
  18. 18.
    Lasiecka I, Manitius A (1985) Differentiability and convergence rates of approximating semigroups for retarded functional differential equations. SIAM J Numer Anal (to appear)Google Scholar
  19. 19.
    Lasiecka I, Triggiani R (1983) Dirichlet boundary control problem for parabolic equations with quadratic cost: analyticity and Riccati's feedback synthesis. SIAM J Control Optim 21:41–68Google Scholar
  20. 20.
    Lasiecka I, Triggiani R (1983) Stabilization and structural assignement of Dirichlet boundary feedback parabolic equations. SIAM J Control Optim 21:766–803Google Scholar
  21. 21.
    Lasiecka I, Triggiani R (1983) Feedback semigroups and cosine operators for boundary feedback parabolic and hyperbolic equations. J Differential Equations 47:245–272Google Scholar
  22. 22.
    Lasiecka I, Triggiani R (1987) Riccati equations for hyperbolic partial differential equations withL 2(0, T; L2(Γ))—Dirichlet boundary terms. SIAM J Control Optim (to appear)Google Scholar
  23. 23.
    Lions JL, Magenes E (1972) Non-homogeneous Boundary Value Problems, I, II. Springer-Verlag, New YorkGoogle Scholar
  24. 24.
    Necas J (1967) Les Methodes Direct en Theorie des Equations Elliptiques. Masson et Cie (Édit.) Paris-VieGoogle Scholar
  25. 25.
    Nitsche J (1971) Über ein Variationsprincip der Lösung von Dirichlet Problem bei Verwendung von Teilräumen, die keinen Randbegingungen unterworfen sind. Abh Math Sem Univ Hamburg 36:9–15Google Scholar
  26. 26.
    Pazy A (1974) Semigroups of operators and applications to partial differential equations. Lecture Notes. Department of Mathematics, University of MarylandGoogle Scholar
  27. 27.
    Sadosky C (1979) Interpolation of operators and singular integrals. In Pure and Applied Mathematics. Marcel Dekker, New YorkGoogle Scholar
  28. 28.
    Sorine M (1981) Une resultat d'existence et unicité pour l'equation de Riccati stationnaire. Rapport INRIA No 55.Google Scholar
  29. 29.
    Triebel H (1978) Interpolation Theory, Function Spaces, Differential Operators. VEB Deutscher Verlag der Wissenshaften, Berlin (licenced edition of North-Holland Publishing Company, 1978)Google Scholar
  30. 30.
    Triggiani R (1980) Boundary feedback stabilizability of parabolic equations. Appl Math Optim 6:201–220Google Scholar
  31. 31.
    Zabczyk J (1978) On decomposition of generators. SIAM J Control Optim 16:523–534Google Scholar

Copyright information

© Springer-Verlag New York Inc. 1987

Authors and Affiliations

  • I. Lasiecka
    • 1
  • R. Triggiani
    • 1
  1. 1.Department of Applied MathematicsUniversity of VirginiaCharlottesvilleUSA

Personalised recommendations