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Pontryagin’s Maximum Principle for Multidimensional Control Problems in Image Processing

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  • M. Wagner

    (Brandenburgische Technische Universität Cottbus)

Abstract

In the present paper, we prove a substantially improved version of the Pontryagin maximum principle for convex multidimensional control problems of Dieudonné-Rashevsky type. Although the range of the operator describing the first-order PDE system involved in this problem has infinite codimension, we obtain first-order necessary conditions in a completely analogous form as in the one-dimensional case. Furthermore, the adjoint variables are subjected to a Weyl decomposition. We reformulate two basic problems of mathematical image processing (determination of optical flow and shape from shading problem) within the framework of optimal control, which gives the possibility to incorporate hard constraints in the problems. In the convex case, we state the necessary optimality conditions for these problems.

Suggested Citation

  • M. Wagner, 2009. "Pontryagin’s Maximum Principle for Multidimensional Control Problems in Image Processing," Journal of Optimization Theory and Applications, Springer, vol. 140(3), pages 543-576, March.
    • Handle: RePEc:spr:joptap:v:140:y:2009:i:3:d:10.1007_s10957-008-9460-9
    DOI: 10.1007/s10957-008-9460-9
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    References listed on IDEAS

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    1. S. Pickenhain & M. Wagner, 2000. "Pontryagin Principle for State-Constrained Control Problems Governed by a First-Order PDE System," Journal of Optimization Theory and Applications, Springer, vol. 107(2), pages 297-330, November.
    2. S. Pickenhain & M. Wagner, 2005. "Piecewise Continuous Controls in Dieudonné-Rashevsky Type Problems," Journal of Optimization Theory and Applications, Springer, vol. 127(1), pages 145-163, October.
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