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Transportation flow problems with Radon measure variables

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Abstrakty
EN
For a multidimensional control problem $(P)_K$ involving controls $u ∈ L_∞$, we construct a dual problem $(D)_K$ in which the variables ν to be paired with u are taken from the measure space rca (Ω,𝔅) instead of $(L_∞)*$. For this purpose, we add to $(P)_K$ a Baire class restriction for the representatives of the controls u. As main results, we prove a strong duality theorem and saddle-point conditions.
Twórcy
  • Cottbus University of Technology, Institute of Mathematics, Karl-Marx-Str. 17, P.O. Box 10 13 44, D-03013 Cottbus, Germany
Bibliografia
  • [1] H.W. Alt, Lineare Funktionalanalysis, Springer, New York-Berlin 1992.
  • [2] J.-P. Aubin and H. Frankowska, Set-Valued Analysis, Birkhäuser, Boston-Basel-Berlin 1990.
  • [3] C. Carathéodory, Vorlesungen über reelle Funktionen, Chelsea, New York 1968.
  • [4] N. Dunford and J.T. Schwartz, Linear Operators. Part I: General Theory, Wiley-Interscience, New York 1988.
  • [5] R.V. Gamkrelidze, Principles of Optimal Control Theory, Plenum Press, New York-London 1978.
  • [6] F. Hüseinov, Approximation of Lipschitz functions by infinitely differentiable functions with derivatives in a convex body, Turkish J. of Math. 16 (1992), 250-256.
  • [7] R. Klötzler, On a general conception of duality in optimal control, in: Equadiff IV (Proceedings). Springer, New York-Berlin 1979. (Lecture Notes in Mathematics 703)
  • [8] R. Klötzler, Optimal transportation flows, Journal for Analysis and its Applications 14 (1995), 391-401.
  • [9] R. Klötzler, Strong duality for transportation flow problems, Journal for Analysis and its Applications 17 (1998), 225-228.
  • [10] H. Kraut, Optimale Korridore in Steuerungsproblemen, Dissertation, Karl-Marx-Universität Leipzig 1990.
  • [11] C.B. Morrey, Multiple Integrals in the Calculus of Variations, Springer, Berlin-Heidelberg-New York 1966 (Grundlehren 130).
  • [12] S. Pickenhain and M. Wagner, Critical points in relaxed deposit problems, in: A. Ioffe, S. Reich, I. Shafrir, eds., Calculus of variations and optimal control, Technion 98, Vol. II (Research Notes in Mathematics, Vol. 411), Chapman & Hall/CRC Press; Boca Raton, 1999, 217-236.
  • [13] S. Pickenhain and M. Wagner, Pontryagin's principle for state-constrained control problems governed by a first-order PDE system, BTU Cottbus, Preprint-Reihe Mathematik M-03/1999. To appear in: JOTA.
  • [14] T. Roubicek, Relaxation in Optimization Theory and Variational Calculus, De Gruyter, Berlin-New York 1997.
  • [15] M. Wagner, Erweiterungen eines Satzes von F. Hüseinov über die $C^∞$-Approximation von Lipschitzfunktionen, BTU Cottbus, Preprint-Reihe Mathematik M-11/1999.
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bwmeta1.element.bwnjournal-article-doi-10_7151_dmdico_1007
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