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Journal of Optimization Theory and Applications

, Volume 14, Issue 5, pp 521–528 | Cite as

A lower closure theorem for abstract control problems withLp-bounded controls

  • L. D. Berkovitz
Contributed Papers

Abstract

A lower closure theorem for an abstract control problem is proved. The functional isJ(ϕ,u)=∝ G f0(t, (Mϕ)(t),u(t))dt and the state equations areNϕ(t)=f(t, (Mϕ)(t),u(t)). It is shown that, if {(ϕ k ,uk)} is a sequence of admissible controlsuk and corre-sponding trajectories ϕ k such that lim infJ k ,uk)<+∞ and such that ϕ k → ϕ weakly,Mϕ k Mϕ strongly,Nϕ k Nϕ weakly, and {uk} is bounded in someLp norm, then there is a controlu such that (ϕ,u) is admissible and lim infJ k ,uk)≧J(ϕ,u).

Key Words

Lower closure theorems generalized control theory control theory calculus of variations 

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References

  1. 1.
    Berkovitz, L. D.,Existence and Lower Closure Theorems for Abstract Control Problems, SIAM Journal on Control, Vol. 12, pp. 27–42, 1974.Google Scholar
  2. 2.
    Berkovitz, L. D.,Lower Semicontinuity of Integral Functionals, Transactions of the American Mathematical Society (to appear).Google Scholar
  3. 3.
    Krasnosel'skii, M. A., Zabreiko, P. R., Pustyl'nik, E. I., andSobolevskii, P. W.,Integral Operators in Spaces of Summable Functions (in Russian), Nauka, Moscow, USSR, 1966.Google Scholar

Copyright information

© Plenum Publishing Corporation 1974

Authors and Affiliations

  • L. D. Berkovitz
    • 1
  1. 1.Department of MathematicsPurdue UniversityLafayette

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