A Bang-Bang Principle Of Time Optimal Internal Controls Of A Fractional Order Parabolic Equation

The time optimal control problems for linear system is very important in the study of control theory. In this paper, we study a time optimal internal control problem governed by a tractional order parabolic equation inΩ×[0,∞), whereΩis a bounded domain of Rⁿ with smooth boundary. The target set S is a nonempty subset in L²(Ω), the control set (?) is the closure of U which is a bounded, open and nonempty subset in L²(Ω) and control functions belong to the set U_ad={u: [0,∞)→U|u(·) is measurable}. First, we establish the null controllability result for the fractional parabolic equation inΩ×(0, T), with control restricted to a product set of an open, nonempty subset inΩand a subset of positive measure in the interval [0, T]. Then we prove that each optimal control u of the problem satisfies the bang-bang property: u(·, t)∈(?) for almost all t∈[0, T^*], where (?) denotes the boundary of the set 0 and T^* is the optimal time.