Invariance of the Hamiltonian in Control Problems for Semilinear Parabolic Distributed Parameter Systems

  • H. O. Fattorini
Conference paper
Part of the ISNM International Series of Numerical Mathematics book series (ISNM, volume 118)


We consider a general optimal control problem for a distributed parameter described by a semilinear parabolic equation. We show that the Hamiltonian is invariant over optimally controlled trajectories, and that it vanishes when the terminal time is free. The method is that of approximation of the original control system by a sequence of “smoothed” control systems

1991 Mathematics Subject Classification

93E20 93E25 

Key words and phrases

Lagrange multiplier rule Kuhn-Tucker conditions maximum principle optimal control 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    L. DE SIMON, Un’applicazionedella teoria degli integrali singolari allo studio delle equazioni differenzialiastratte del primo ordine, Rend Sem.Mat. Univ. Padova 34 (1964), 205–22.zbMATHGoogle Scholar
  2. 2.
    G. DORE and A. VENNI, On theclosedness of the sum of two closed operators, Math Z. 196 (1987),189–201.MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    H. O.FATTORINI, Constancy of the Hamiltonian in infinite dimensional systems, inControl and Estimation of Distributed Parameter Systems, (F. Kappel, K.Kunisch, W. Schappacher, Eds.), Int. Series Numerical Math., vol. 91, Birkhäuser, Basel, 1989, pp. 123–133.Google Scholar
  4. 4.
    H. O. FATTORINI, Relaxedcontrols in infinite dimensional control systems, in Estimation and Controlof Distributed Parameter Systems, (W. Desch, F. Kappel, K. Kunisch, Eds.), International Series of Numerical Mathematics, vol.100, Birkhäuser, Basel, 1991, pp. 115– 128.Google Scholar
  5. 5.
    H. O. FATTORINI, Optimal controlproblems for distributed parameter systems governed by semilinear parabolic equations in L1 and L蜴spaces, in Optimal Control of PartialDifferential Equations, (K.-H. Hoffmann, W. Krabs, Eds.), Springer LectureNotes in Control and Information Sciences, vol. 149, 1991, pp. 68–80.CrossRefGoogle Scholar
  6. 6.
    H. O. FATTORINI, Optimal controlproblems in Banach spaces, Appl. Math. Optim. 28 (1993), 225–257.MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    H. O. FATTORINI, Relaxation in semilinear infinite dimensional control systems, in L. MarkusFestschrift, (K. D. Elworthy, W. N.Everitt, E. B. Lee, Eds.), Marcel Dekker, New York, 1993, pp. 505–522.Google Scholar
  8. 8.
    H. O. FATTORINI and H. FRANKOWSKA, Necessaryconditions for infinite dimensional control problems, Math. Control,Signals and Systems 4 (1991), 41–67.MathSciNetzbMATHCrossRefGoogle Scholar
  9. H. O. FATTORINI and T. MURPHY, Optimalproblems for nonlinear parabolic boundary control systems, to appear inSIAM J. Control & Optimization.Google Scholar
  10. 10.
    H. FRANKOWSKA, Some inversemapping theorems, Ann Inst. Henri Poincaré 7 (1990), 183–234.MathSciNetzbMATHGoogle Scholar
  11. 11.
    D. HENRY, “Geometric Theory of Semilinear ParabolicEquations”, Springer, Berlin 1981.Google Scholar
  12. 12.
    E. HILLE and R. S. PHILLIPS,“Functional Analysis and Semi-Groups”, AMS Colloquium Pubs. vol 31,Providence, 1957.Google Scholar

Copyright information

© Springer Basel AG 1994

Authors and Affiliations

  • H. O. Fattorini
    • 1
  1. 1.Department of MathematicsUniversity of CaliforniaLos AngelesUSA

Personalised recommendations