A Bang - Bang Principle of Time Optimal Internal Controls of the Heat Equation ∗

In this paper, we study a time optimal internal control problem governed by the heat equation in Ω × [0,∞). In the problem, the target set S is nonempty in L(Ω), the control set U is closed, bounded and nonempty in L(Ω) and control functions are taken from the set Uad = {u(·, t) : [0,∞) → L(Ω) measurable; u(·, t) ∈ U, a.e. in t }. We first establish a certain null controllability for the heat equation in Ω × [0, T ], with controls 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 prove that each optimal control u∗(·, t) of the problem satisfies necessarily the bang-bang property: u∗(·, t) ∈ ∂U for almost all t ∈ [0, T ∗], where ∂U denotes the boundary of the set U and T ∗ is the optimal time. We also obtain the uniqueness of the optimal control when the target set S is convex and the control set U is a closed ball.