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

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.