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On the Projections of Multivalued Maps

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  • U. Raitums

Abstract

This paper considers analogues of the Helmholtz projections of the set Π′ of selections of a piecewise smooth multivalued map $$\prod :R^n \to 2^{R^{m \times n} } $$ , n≥2. It is shown that, for m≤n−1 (m=1), the closure of the projection of Π′ on the subspace of gradient fields (solenoidal vector fields) is a convex set. For the general case, there are given point-wise conditions on the values of the map Π which ensure that the closure of the projection of Π′ contains the zero element. Possible applications to optimal control problems are discussed.

Suggested Citation

  • U. Raitums, 1997. "On the Projections of Multivalued Maps," Journal of Optimization Theory and Applications, Springer, vol. 92(3), pages 633-660, March.
    • Handle: RePEc:spr:joptap:v:92:y:1997:i:3:d:10.1023_a:1022611608062
    DOI: 10.1023/A:1022611608062
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