The time-optimal control problem in Banach spaces

摘要

We show here that a variant of Pontryagin's maximum principle holds for time-optimal trajectories of the systemu′ = Au + fthat begin and end at the domain ofA.HereAis the infinitesimal generator of a strongly continuous semigroupS(·) in the reflexive Banach spaceE.This partly generalizes a result of Balakrishnan (1) where no conditions are assumed on the trajectories but whereS(t)is supposed to beontofor allt≥ 0. A weakened version of the maximum principle is then proved foralltime-optimal trajectories whenAgenerates an analytic semigroup andEis a Hilbert space. Finally, the results are modified to handle two examples involving the heat equation where the spaceEis not refle