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Time-optimal controls of the equation of evolution in a separable and reflexive Banach space

  • Jon K. Cole
Technical Note

Abstract

Let a variable, closed, bounded, and convex subset ofX, a separable and reflexive Banach space, be denoted byG(t). Suppose thatG(t) varies upper-semicontinuously with respect to inclusion ast varies in [0,T]. We say that the strongly measurable mapu from [0,T] toX is an admissible control if, for almost everyt in [0,T],u(t) is an element ofU, a closed, bounded, and convex subset ofX, and ∥u p M1p, where p>1 andM>0.

Ifx u is the weak solution todx/dt+A(t)x=u(t), 0⩽tT, whereA(t) is as defined by Tanabe in Ref. 1, we say that the responsex u to the controlu hits the target in timeT u ifx u (0)=0 andx u (T u ) is an element ofG(T u ). If there is a control with this property, then there is a time-optimal control.

Keywords

Banach Space Weak Solution Convex Subset Admissible Control Reflexive Banach Space 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Tanabe, H.,On the Equations of Evolution in a Banach Space, Osaka Mathematical Journal, Vol. 12, No. 2, 1960.Google Scholar
  2. 2.
    Kelley, J. L.,General Topology, D. Van Nostrand Company, Princeton, New Jersey, 1955.Google Scholar
  3. 3.
    Hille, E., and Phillips, R. S.,Functional Analysis and Semi-Groups, The Waverly Press, Baltimore, Maryland, 1957.Google Scholar
  4. 4.
    Kato, T.,Integration of the Equation of Evolution in an Abstract Banach Space, Journal of the Mathematical Society of Japan, Vol. 5, No. 2, 1953.Google Scholar
  5. 5.
    McShane, E. J., andBotts, T. A.,Real Analysis, D. Van Nostrand Company, Princeton, New Jersey, 1959.Google Scholar
  6. 6.
    Filippov, A. F.,On Certain Questions in the Theory of Optimal Control, SIAM Journal on Control, Vol. 1, No. 1, 1962.Google Scholar

Copyright information

© Plenum Publishing Corporation 1968

Authors and Affiliations

  • Jon K. Cole
    • 1
  1. 1.St. John's UniversityJamaica

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