Annali di Matematica Pura ed Applicata

, Volume 154, Issue 1, pp 259–279 | Cite as

On the optimal control and relaxation of nonlinear infinite dimensional systems

  • Nikolaos S. Papageorgiou


In this paper we study optimal control problems for infinite dimensional systems governed by a semilinear evolution equation. First under appropriate convexity and growth conditions, we establish the existence of optimal pairs. Then we drop the convexity hypothesis and we pass to a larger system known as the « relaxed system ». We show that this system has a solution and the value of the relaxed optimization problem is equal to the value of the original one. Next we restrict our attention to linear systems and establish two « bang-bang » type theorems. Finally we present some examples from systems governed by partial differential equations.


Differential Equation Growth Condition Linear System Partial Differential Equation Control Problem 
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]
    T. Angell,On the optimal control of systems governed by nonlinear Volterra equations, J. Optim. Th. Appl.,19 (1976), pp. 29–45.Google Scholar
  2. [2]
    T. Angell,Existence of optimal control without convexity and a bang-bang theorem for linear Volterra equations, J. Optim. Th. Appl.,19 (1976), pp. 63–79.Google Scholar
  3. [3]
    E. Balder,A general denseness result for relaxed control theory, Bull. Austr. Math. Soc.,30 (1984), pp. 463–475.Google Scholar
  4. [4]
    E. Balder,Lower closure for orientor fields by lower semicontinuity of outer integral functionals, Ann. Mat. Pura Appl.,139 (1985), pp. 349–360.Google Scholar
  5. [5]
    V. Barbu,Boundary control problems with convex cost criterion, SIAM J. Control Optim.,18 (1980), pp. 227–243.Google Scholar
  6. [6]
    M. Benamara,Sections mesurables extrémales d'une multiapplication, C. R. Acad. Sci. Paris,278 (1974), pp. 1249–1252.Google Scholar
  7. [7]
    H.Brezis,New results concerning monotone operators and nonlinear semigroups, Proc. RIMS « Analysis of Nonlinear Problems », Kyoto University (1975), pp. 2–27.Google Scholar
  8. [8]
    C.Castaing - P.Clauzure,Semicontinuité inférieure des fonctionelles intégrales, Seminaire Anal. Convexe, Montpellier (1981), exposé no. 15.Google Scholar
  9. [9]
    C. Castaing -M. Valadier,Convex Analysis and Measurable Multifunctions, Lecture Notes in Math., Vol.580, Springer, Berlin (1977).Google Scholar
  10. [10]
    L. Cesari,Existence theorems for weak and usual optimal solutions in Lagrange problems with unilateral constraints, I and II, Trans. Amer. Math. Soc.,124 (1966), pp. 369–412.Google Scholar
  11. [11]
    L. Cesari,Closure theorems for orientor fields and weak convergence, Arch. Rat. Mech. Anal.,55 (1974), pp. 332–356.Google Scholar
  12. [12]
    L. Cesari,Convexity and property (Q) in optimal control theory, SIAM J. Control,12 (1974), pp. 705–720.Google Scholar
  13. [13]
    L. Cesari,Optimization, Theory and Applications, Springer, New York (1983).Google Scholar
  14. [14]
    L. Cesari -R. LaPalm -T. Nishiura,Remarks on some existence theorems for optimal control, J. Optim. Th. Appl.,3 (1969), pp. 296–305.Google Scholar
  15. [15]
    C. Dellacherie,Ensembles analytiques: Théorèmes de séparation et applications, Sém. de probabilités, IX, Univ. Strasbourg, Lecture Notes in Math., Vol.465, Springer, Berlin, New York (1975).Google Scholar
  16. [16]
    C. Dellacherie -P. A. Meyer,Probabilities and Potential, North Holland, Amsterdam (1978).Google Scholar
  17. [17]
    J. Diestel -J. J. Uhl,Vector Measures, Math. Surveys, Vol.15, A.M.S., Providence, R.I. (1977).Google Scholar
  18. [18]
    N. Dunford -J. Schwartz,Linear Operators I, Wiley, New York (1958).Google Scholar
  19. [19]
    G. Edgar,Measurability in a Banach space, Indiana J. Math.,28 (1979), pp. 559–579.Google Scholar
  20. [20]
    H. Fattorini,A remark on the Bang-Bang principle for linear control systems in infinite dimensional spaces, SIAM J. Control,6 (1968), pp. 109–113.Google Scholar
  21. [21]
    A. Filippov,On certain question in the theory of optimal control, SIAM J. Control,1 (1962), pp. 76–84.Google Scholar
  22. [22]
    A. Ghouila-Houri,Sur la généralisation de la notion de la commande d'un système guidable, Rev. d'Inform. et de Resch. Oper., no. 4 (1967), pp. 7–32.Google Scholar
  23. [23]
    H.Hermes - J.LaSalle,Functional Analysis and Time Optimal Control, Academic Press (1969).Google Scholar
  24. [24]
    F. Hiai -H. Umegaki,Integrals, conditional expectations and martingales of multivalued mappings, J. Multiv. Anal.,7 (1977), pp. 149–182.Google Scholar
  25. [25]
    A.Jawhar,Mesures de transition et applications, Sém. Anal. Convexe, Montpellier (1984), exposé no. 13.Google Scholar
  26. [26]
    D.Kandilakis - N. S.Papageorgiou,On the properties of the Aumann integral with applications to differential inclusions and control systems, Czech. Math. J., to appear.Google Scholar
  27. [27]
    N. Pavel,Differential Equations, Flow Invariance and Applications, Research Notes in Math., Vol.113, Pitman, Boston (1984).Google Scholar
  28. [28]
    N. S. Papageorgiou,On multivalued evolution equations and differential inclusions in Banach spaces, Comm. Math. Univ. Sancti Pauli,36 (1987), pp. 21–39.Google Scholar
  29. [29]
    N. S. Papageorgiou,Representation of set valued operators, Trans. Amer. Math. Soc.,292 (1985), pp. 557–572.Google Scholar
  30. [30]
    N. S. Papageorgiou,On the theory of Banach space valued multifunctions. Part 1:Integration and conditional expectation, J. Mult. Anal.,17 (1985), pp. 185–206.Google Scholar
  31. [31]
    M. F. Saint-Beuve,On the extension of Von Neumann-Aumann's theorem, J. Funct. Anal.,17 (1974), pp. 112–129.Google Scholar
  32. [32]
    M. F.Saint-Beuve,Une extension des théorèmes de Novikav et d'Arsenin, Sém. Anal. Convàxe, Montpellier (1981), exposé no. 18.Google Scholar
  33. [33]
    D. Wagner,Survey of measurable selection theorems, SIAM J. Control and Optim.,15 (1977), pp. 859–903.Google Scholar
  34. [34]
    J. Warga,Relaxed variational problems, J. Math. Anal. Appl.,4 (1962), pp. 111–128.Google Scholar
  35. [35]
    J. Warga,Optimal Control of Differential and Functional Equations, Academic Press, New York (1972).Google Scholar
  36. [36]
    L. C. Young,Lectures on the Calculus of Variations and Optimal Control Theory, Saunders, Philadelphia (1969).Google Scholar

Copyright information

© Fondazione Annali di Matematica Pura ed Applicata 1989

Authors and Affiliations

  • Nikolaos S. Papageorgiou
    • 1
  1. 1.1015 Department of MathematicsUniversity of CaliforniaDavisUSA

Personalised recommendations