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 ⩽M 1p, where p>1 andM>0.
Ifx u is the weak solution todx/dt+A(t)x=u(t), 0⩽t⩽T, 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.
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Communicated by L. Cesari
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Cole, J.K. Time-optimal controls of the equation of evolution in a separable and reflexive Banach space. J Optim Theory Appl 2, 199–204 (1968). https://doi.org/10.1007/BF00927001
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DOI: https://doi.org/10.1007/BF00927001