Advertisement

Optimality Conditions for Boundary Control Problems of Parabolic Type

  • Piermarco Cannarsa
  • Maria Elisabetta Tessitore
Conference paper
Part of the ISNM International Series of Numerical Mathematics book series (ISNM, volume 118)

Abstract

This paper is devoted to the study of finite horizon optimal control problems with boundary control. We prove a sufficient condition for optimality of trajectory—control pairs, using a non—smooth analysis approach. We formulate this condition in terms of an Hamiltonian system for which we show an existence and uniqueness result.

1991 Mathematics Subject Classification

49K20 49K27 

Key words and phrases

Optimality conditions boundary control parabolic equations Neumann boundary conditions analytic semigroups backward uniqueness 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    V. Barbu, Boundary control problemswith convex cost criterion, SIAM J. Control andOptim., 18, 227–254, 1980.MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    V. Barbu and G. Da Prato,Hamilton-Jacobi equations in Hilbert spaces, Pitman, Boston,1982.Google Scholar
  3. 3.
    A. Bensoussan, G. Da Prato, M. C.Delfour and S. K. Mitter, Representation and control ofinfinite dimensional systems, Birkhäuser,Boston, 1992.Google Scholar
  4. 4.
    P. Cannarsa, H. Frankowska, Valuefunction and optimality conditions for semilinear controlproblems, Applied Math. Optim., 26, 139–169, 1992.MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    P. Cannarsa and F. Gozzi, On thesmoothness of the value function along optimal trajectories,IFIP Workshop on Boundary Control and Boundary Variation (to appear in LectureNotesin Control and Information Sciences, Springer-Verlag).Google Scholar
  6. 6.
    P. Cannarsa, F. Gozzi and H. M.Soner, A dynamic programming approach to nonlinearboundary control problems of parabolic type, J. Funct. Anal. (to appear).Google Scholar
  7. 7.
    P. Cannarsa and M. E. Tessitore,Cauchy problem for the dynamic programming equation ofboundary control, Proceedings IFIP Workshop on Boundary Control and BoundaryVariation(to appear).Google Scholar
  8. 8.
    F. Clarke and R. B. Vinter, Therelationship between the maximum principle and dynamicprogramming, SIAM J. Control Optim., 25, 1291–1311, 1989.MathSciNetCrossRefGoogle Scholar
  9. 9.
    H. O. Fattorini, Boundary control system, SIAM J.Control, 6, 349–385, 1968.MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    H. O. Fattorini, A unified theory of necessary conditions fornonlinear nonconvex controlsystems, Appl. Math. Opt., 15, 141– 185, 1987.MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    A. Faviniand A. Venni, On a two-point problem for a system of abstract differentialequations, Numer. Funct. Anal. and Optimiz., 2(4), 301–322, 1980.CrossRefGoogle Scholar
  12. 12.
    H. Goldberg and F. Tröltzsch, Second-order sufficient optimality conditionsfor a class ofnonlinear parabolic boundary control problems, SIAM J. Control and Optim., vol.31, 4,1007–1025, 1993.CrossRefGoogle Scholar
  13. 13.
    V. Iftode, Hamilton-Jacobiequations and boundary convex control problems, Rev. RoumaineMath. Pures Appl., 34, (1989), 2, 117–127.MathSciNetzbMATHGoogle Scholar
  14. 14.
    G. E. Ladas and V.Lakshmikantham, Differential equations in abstract spaces, AcademicPress, New York and London, 1972.Google Scholar
  15. 15.
    I. Lasiecka and R. Triggiani,Differential and algebraic equations with application to boundary controlproblems:continuous theory and approximation theory, Lecture Notes in Controland Information Sciences, Springer-Verlag,164, 1991.Google Scholar
  16. 16.
    J. L.Lions, Optimal control of systems described by partial differential equations, Springer-Verlag, Wiesbaden, 1972.Google Scholar
  17. 17.
    J. L. Lions and B. Malgrange, Surl’unicité retrograde dans les problèmesmixtes paraboliques,Math. Scand. 8, 277–286, 1960MathSciNetGoogle Scholar
  18. 18.
    A. Pazy, Semigroups of linearoperators and applications to partial differential equations,Springer-Verlag, New York-Heidelberg- Berlin, 1983.CrossRefGoogle Scholar
  19. 19.
    D. Preiss, Differentiability ofLipschitz functions on Banach spaces, J. Funct. Anal., 91,312–345, 1990.MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    F. Tröltzsch, On the semigroup approach for the optimal control of semilinear parabolicequations including distributed and boundary control, Zeitschrift für Analysisund ihre Anwendungen Bd., 8, 431–443, 1989.zbMATHGoogle Scholar

Copyright information

© Springer Basel AG 1994

Authors and Affiliations

  • Piermarco Cannarsa
    • 1
  • Maria Elisabetta Tessitore
    • 2
  1. 1.Dipartimento di MatematicaUniversitá di Roma “Tor Vergata”RomaItaly
  2. 2.Dipartimento di MatematicaUniversitá di Roma “La Sapienza”RomaItaly

Personalised recommendations