Differential Inclusions, Control and Optimization 20 (2000) 93-111
doi: 10.7151/dmdico.1007

[BIBTex] [PDF] [PS]

TRANSPORTATION FLOW PROBLEMS WITH RADON
MEASURE VARIABLES

Marcus Wagner

Cottbus University of Technology, Institute of Mathematics
Karl-Marx-Str. 17, P.O. Box 10 13 44, D-03013 Cottbus, Germany

e-mail: wagner@math.tu-cottbus.de

Abstract

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.

Keywords: multidimensional control problems, strong duality, saddle-point conditions, Baire classification.

1991 Mathematics Subject Classification: Primary: 49N15; Secondary: 26A21, 26E25, 28A20, 49K20, 90B06.

References

[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.

Received 19 November 1999
Revised 1 March 2000