Optimization and relaxation of nonlinear elliptic control systems

  • Nikolaos C. Kourogenis
  • Nikolaos S. Papageorgiou


In this paper we study the existence and relaxation theory for distributed parameter systems driven by nonlinear elliptic differential equations. The existence theory is developed for systems with a priori feedback (closed loop systems), while the main results of the relaxation results are proved for open loop systems. We start with some existence theorems for nonlinear elliptic inclusions which are then used to establish the existence of admissible states for the optimal control problem. Then with the help of a convexity-type condition, we formulate and prove the main existence theorem for the problem. The proof employs the so-called reduction technique. Then we formulate three different versions of the relaxed problem, we show that they are equivalent and finally we show that they have the same value as the original problem and their set of states captures the asymptotic behavior of the sequences of original states.

Key words

elliptic inclusion upper semicontinuous and lower semicontinuous multifunction measurable multifunction measurable selection propertyQ Young measure relaxed problem relaxability admissible relaxation weak convergence of measures 


  1. [1]
    L. Berkovitz, Optimal Control Theory. Springer-Verlag, New York, 1974.zbMATHGoogle Scholar
  2. [2]
    P. Billingsley, Convergence of Probability Measures. Wiley, New York, 1968.zbMATHGoogle Scholar
  3. [3]
    N.N. Bogoljubov, Sur quelques méthodes nouvelles en calcul des variations. Annali di Mat. Pura ed Appl.,7 (1930), 249–271.Google Scholar
  4. [4]
    G. Buttazzo, Some relaxation problems in optimal control theory. J. Math. Anal. Appl.,125 (1987), 272–287.zbMATHCrossRefMathSciNetGoogle Scholar
  5. [5]
    L. Cesari, Optimization Theory and Applications. Applications of Mathematics, Vol.17, Springer-Verlag, New York, 1983.Google Scholar
  6. [6]
    I. Ekeland, Sur le control optimal des systèmes gouvernés par des equations elliptiques. J. Punct. Anal.,9 (1972), 1–62.zbMATHCrossRefMathSciNetGoogle Scholar
  7. [7]
    A.F. Filippov, On certain questions in the theory of optimal control. SIAM J. Control,1 (1962), 76–84.zbMATHGoogle Scholar
  8. [8]
    R.V. Gamkrelidze, Principles of Optimal Control Theory. Plenum Press, New York, 1978.zbMATHGoogle Scholar
  9. [9]
    S. Hu and N.S. Papageorgiou, Handbook of Multivalued Analysis, Volume I: Theory. Kluwer, Dordrecht, The Netherlands, 1997.zbMATHGoogle Scholar
  10. [10]
    P. Lindqvist, On the equationdiv(∥Dxp−2 Dx) + λ¦x¦p−2 x = 0. Proc. AMS,109 (1990), 157–164.zbMATHCrossRefMathSciNetGoogle Scholar
  11. [11]
    J.-L. Lions, Optimal Control of Systems Governed by Partial Differential Equations. Springer-Verlag, New York, 1971.zbMATHGoogle Scholar
  12. [12]
    C. Olech, A characterization ofL 1-weak lower semicontinuity of integral functionals. Bull. Acad. Polon. Sci.,25 (1977), 135–142.zbMATHMathSciNetGoogle Scholar
  13. [13]
    N.S. Papageorgiou, Existence theory for nonlinear distributed parameter optimal control problems. Japan J. Indust. Appl. Math.,12 (1995), 457–485.zbMATHCrossRefMathSciNetGoogle Scholar
  14. [14]
    K.R. Parthasarathy, Probability Measures on Metric Spaces. Academic Press, New York, 1967.zbMATHGoogle Scholar
  15. [15]
    M.-F. Saint-Beuve, Some topological properties of vector measures with bounded variation and its applications. Annali di Math. Pura ed Appl.,116 (1978), 317–379.CrossRefGoogle Scholar
  16. [16]
    J. Warga, Optimal Control of Differential and Functional Equations. Academic Press, New York, 1972.zbMATHGoogle Scholar
  17. [17]
    L.C. Young, Generalized curves and the existence of an attained absolute minimum in the Calculus of Variations. Comptes Rendus de la Société des Sci. et des Lettres de Varsovie,30 (1937), 212–234.zbMATHGoogle Scholar
  18. [18]
    M. Struwe, Variational Methods. Springer-Verlag, Berlin, 1990.zbMATHGoogle Scholar
  19. [19]
    J.I. Diaz, Nonlinear Partial Differential Equations and Free Boundaries. Vol. 1, Elliptic Equations. Research Notes in Math., Vol.106, Pitman, London, 1985.Google Scholar
  20. [20]
    D. Zubrinic, Existence of solutions of −Δu= g(z, u, Du). Glasnik Matematički,20 (1985), 363–372.MathSciNetGoogle Scholar

Copyright information

© JJIAM Publishing Committee 2000

Authors and Affiliations

  • Nikolaos C. Kourogenis
    • 1
  • Nikolaos S. Papageorgiou
    • 1
  1. 1.Department of MathematicsNational Technical UniversityAthensGreece

Personalised recommendations