Time-optimal controls of the equation of evolution in a separable and reflexive Banach space
- 38 Downloads
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⩽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.
KeywordsBanach Space Weak Solution Convex Subset Admissible Control Reflexive Banach Space
Unable to display preview. Download preview PDF.
- 1.Tanabe, H.,On the Equations of Evolution in a Banach Space, Osaka Mathematical Journal, Vol. 12, No. 2, 1960.Google Scholar
- 2.Kelley, J. L.,General Topology, D. Van Nostrand Company, Princeton, New Jersey, 1955.Google Scholar
- 3.Hille, E., and Phillips, R. S.,Functional Analysis and Semi-Groups, The Waverly Press, Baltimore, Maryland, 1957.Google Scholar
- 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.McShane, E. J., andBotts, T. A.,Real Analysis, D. Van Nostrand Company, Princeton, New Jersey, 1959.Google Scholar
- 6.Filippov, A. F.,On Certain Questions in the Theory of Optimal Control, SIAM Journal on Control, Vol. 1, No. 1, 1962.Google Scholar