L∞-NULL CONTROLLABILITY FOR THE HEAT EQUATION AND ITS CONSEQUENCES FOR THE TIME OPTIMAL CONTROL PROBLEM

摘要

In this paper, we establish a certain L∞-null controllability for the internally con_trolled heat equation in Ω x [0, T], with the control restricted to a product set of an open nonempty subset in Ω and a subset of positive measure in the interval [0, T]. Based on this, we obtain a bang-bang principle for the time optimal control of the heat equation with controls taken from the set U_(ad) = {u(_, t) : [0, __)_úL~2(Ω) measurable; u(_, t) E U, a.e. in t}, where U is a closed and bounded subset of L~2(Ω). Namely, each optimal control u~*(_,t) of the problem satisfies necessarily the bang-bang property: u~* (_, t) E ’aU for almost all t[0, T~*], where ay denotes the boundary of the set U and T* is the optimal time. We also get the uniqueness of the optimal control when the target set S is convex and the control set U is a closed ball.