Abstract In this paper, we consider bang-bang property for a kind of timevarying. time optimal control problem of null controllable heat equation.

Transcription

1 JOTA manuscript No. (will be inserted by the editor) The Bang-Bang Property of Time-Varying Optimal Time Control for Null Controllable Heat Equation Dong-Hui Yang Bao-Zhu Guo Weihua Gui Chunhua Yang Received: date / Accepted: date Communicated by Enrique Zuazua Abstract In this paper, we consider bang-bang property for a kind of timevarying time optimal control problem of null controllable heat equation. The study is a continuation of a recent work [Systems Control Lett., 112(218), 18-23], where an approximate null controllable heat equation was considered. We Dong-Hui Yang School of Mathematics and Statistics, School of Information Science and Engineering, Central South University Changsha, China donghyang@139.com Bao-Zhu Guo, Corresponding author Department of Mathematics and Physics, North China Electric Power University, Beijing 1226, China, Key Laboratory of System and Control, Academy of Mathematics and Systems Science, Academia Sinica, Beijing, China, and School of Mathematics and Big Data, Foshan University, Foshan 528, China bzguo@iss.ac.cn Weihua Gui School of Information Science and Engineering, Central South University, Changsha, China gwh@csu.edu.cn Chunhua Yang School of Information Science and Engineering, Central South University Changsha, China ychh@csu.edu.cn

2 2 Dong-Hui Yang et al. first establish the equivalence between optimal norm control and optimal time control, and then, prove the existence of the optimal norm control and of the optimal time control. The time-varying bang-bang property for the optimal time control is finally established. Keywords Heat equation Bang-bang property Optimal time control Optimal norm control Mathematics Subject Classification (2) 35K5 49J2 1 Introduction In the past several years, the bang-bang property of optimal controls for heat equation has been attracted intensive attentions in PDE control community. Under the assumption that the control bound is essentially a constant function, the bang-bang property of an optimal time control for heat equation has been investigated in [1, 2]. Many optimal control problems including bangbang property were discussed in [3 5]. The paper [6] obtained the bang-bang property of a time optimal control for heat equation with arbitrary reachable target temperature distributions in boundary control. The maximal principle and the bang-bang property of a time optimal control for Schrödinger-type systems have been discussed in [7]. For boundary control problem, the paper [8] obtained bang-bang property for heat equation. The bang-bang property of a time-optimal control for one-dimensional heat equation with boundary control was obtained in [9]. For general controllability and optimal control problems,

3 Title Suppressed Due to Excessive Length 3 we refer to the monographes [1 13]. Very recently, the time varying bangbang property has been established in [1] for an approximately controllable heat equation, where by applying the Pontryagin maximum principle, the authors were able to relax the assumption on control bound to be a bounded function. In this paper, we shall focus on bang-bang property for a kind of time-varying time optimal control problem of null controllable heat equation. It is found that the method used in [1] is not applicable to the problem discussed in present paper. In general, for the problem discussed in this paper, it is hard to obtain the time-varying bang-bang property by using the Pontryagin maximum principle. In [14], the equivalence between time optimal control and norm optimal control for heat equation has been used to derive the bangbang property of a time optimal control problem with constant control bound, and the method has also been used in [15, 16]. In this paper, we again apply this equivalence (between optimal time control and optimal norm control) for heat equation to obtain the time-varying bang-bang property for our problem. We point out that the time-varying bang-bang property of the time optimal control problem for null controllable heat equation has not been considered in [14] and that this paper is an extension of [14]. Our problem of time-varying control bound is more extensive and realistic. For example, a blast furnace, when heated with electric heating wire, often involves a process, in which the wire needs to be heat up or cool down. For this case, it is unreasonable to limit the admissible control to be constant.

4 4 Dong-Hui Yang et al. We proceed as follows. In Section 2, the problem is formulated and the main result is stated. In Section 3, we present some auxiliary results, which are crucial to the proof of the main result. The proof of the main result is presented in Section 4, following up conclusions in Section 5. 2 Problem and Main Result Let Ω R l, < l N, be an open and bounded domain with smooth boundary, and R + :=], + [. Let M max and m be two given constants with < m < M max. Denote by T n T for T n T as n and T n T n+1 for all n N, and T n T for T n T as n and T n T n+1 for all n N. Let U := {U L (R + ; L 2 (Ω)): χ ω U(t) L 2 (Ω) M(t) for t R + a.e.}, (1) where ω Ω is a nonempty and open subset, and M( ) L (R + ) is a given function with m M(t) M max for t R + a.e., representing a time-varying bound of control in the specified domain. The system, that we consider in this paper, is described by the following heat equation t y y = χ ω U, in Ω R +, y =, on Ω R +, (2) y() = y, in Ω,

5 Title Suppressed Due to Excessive Length 5 where y L 2 (Ω) is the initial data, and U U. Throughout the paper, we assume that y, and we denote by y( ; y, U) the solution of (2). Consider the following time optimal control problem for system (2): T (M( ), y ) := inf{t R + : U U such that y(t; y, U) = }. (3) If there exists U U such that y(t (M( ), y ); y, U ) =, then, we call T (M( ), y ) the optimal time, and U an optimal time control for problem (3). In addition, when the optimal control U satisfies U (t) L 2 (Ω) = M(t) for t ], T (M( ), y )[ a.e., the optimal time control is said to satisfy the time-varying bang-bang property. When the control bound M( ) is essentially a constant function, i.e., M(t) = N (t R + a.e.) for some scalar N, the bang-bang property of an optimal time control for problem (3) has been studied in [1, 2]. Unfortunately, the method used in [1] is not applicable to problem (3). In this paper, we shall focus on time-varying bang-bang property for time optimal control problem (5), which is to be formulated later. This problem is an extension of (3) above.

6 6 Dong-Hui Yang et al. Now, let us consider the following system t y y = χ ω M( )u, in Ω R +, y =, on Ω R +, (4) y() = y, in Ω, where u L (R + ; L 2 (Ω)). We again denote the solution of system (4) by y( ; y, u). Let M( )u = U U. From (2) and (4), χ ω M(t)u(t) L 2 (Ω) M(t) for t R + a.e. Since < m M(t) M max for t R + a.e., it follows that χ ω u(t) L2 (Ω) 1 for t R + a.e. Hence, T (M( ), y ) = inf{t R + : u U 1 such that y(t; y, u) = }, where U 1 := {u L (R + ; L 2 (Ω)): χ ω u(t) L 2 (Ω) 1 for t R + a.e.}. From these discussions, we can introduce another time optimal control problem: T (N, y ) := inf{t R + : u U N such that y(t; y, u) = }, (5)

7 Title Suppressed Due to Excessive Length 7 where y( ; y, u) is the solution of system (4) and U N := {u L (R + ; L 2 (Ω)): χ ω u(t) L 2 (Ω) N for t R + a.e.}. Obviously, (3) is a special case of (5). In addition, for problem (5), if there exists u U N such that y(t (N, y ); y, u ) =, we call T (N, y ) the optimal time and u an optimal time control. When the optimal control u satisfies u (t) L 2 (Ω) = N for t ], T (N, y )[ a.e., the optimal time control is said to satisfy the time-varying bang-bang property. In order to derive the time-varying bang-bang property of problems (3) and (5), we consider a norm optimal control problem of the following: N(T, y ) := inf χ ω u L (],T [;L 2 (Ω)) : u L (], T [; L 2 (Ω)) such that y(t ; y, u) =, (6) where y( ; y, u) is the solution of the following system t y y = χ ω M( )u, in Ω ], T [, y =, on Ω ], T [, y() = y, in Ω. For problem (6), if there exists u U N(T,y) such that y(t ; y, u ) = and u L (],T [;L 2 (Ω)) = N(T, y ), N(T, y ) is called the optimal norm and u an optimal norm control. When the optimal norm control u satisfies

8 8 Dong-Hui Yang et al. u (t) L2 (Ω) = N(T, y ) for t ], T [ a.e., then, u is said to satisfy the timevarying bang-bang property. The main result of this paper is stated in Theorem 2.1. Theorem 2.1 Let y L 2 (Ω) \ {}. Then, the time optimal control problem (3) admits a solution, i.e., there exists u L (], T (M( ), y )[; L 2 (Ω)) such that y(t (M( ), y ); y, u ) =, and the optimal control u satisfies the timevarying bang-bang property: u (t) L2 (Ω) = M(t) for t ], T (M( ), y )[ a.e. 3 Auxiliary Results In this section, we give some auxiliary results that will be used later. Lemma 3.1 There exists a constant C = C(T, Ω, ω, m) > such that ϕ() L2 (Ω) C M( )χ ω ϕ L1 (],T [;L 2 (Ω)), (7) where ϕ is the solution of the following system t ϕ + ϕ =, in Ω ], T [, ϕ =, on Ω ], T [, (8) ϕ(t ) = z, in Ω. Here and in what follows, we denote by C a constant although it may have differ values in different contexts.

9 Title Suppressed Due to Excessive Length 9 Proof From (1.5) in [2] and < m M(t) for a.e. t ], T [, we obtain (7). Remark 2.1 The null controllability for system (2) (or (4)) is equivalent to the observability inequality (7) for system (8). The observability inequality can be deduced from the Carleman estimation which has been developed in many papers over the years; see, for instance, [17 2]. Now, for any T R +, set X T := {M( )χ ω ϕ( ; z): z L 2 (Ω)}, (9) where ϕ( ; z) is the solution of system (8) with the terminal value z L 2 (Ω). Obviously, X T is a linear subspace of L 1 (], T [; L 2 (Ω)). Define Y T := X T L 1 (],T [;L 2 (Ω)) (1) and denote Z T := M( )χ ω ϕ : M( )χ ω ϕ L 1 (], T [; L 2 (Ω)), for every s ], T [, there exists z s L 2 (Ω). (11) such that ϕ( ) = e (s ) z s on [, s] Theorem 3.1 For any T R +, let X T, Y T and Z T be defined as (9), (1) and (11) respectively. Then, Y T = Z T. (12)

10 1 Dong-Hui Yang et al. Proof The proof is similar to the proof of (1.1) (or Proposition 4.5) in [14] by using < m M(t) M max for a.e. t ], T [ and Lemma 3.1. Two characterizations of the optimal norm control for problem (6) are described below. Lemma 3.2 Assume y L 2 (Ω) \ {} and let T R +, and [ T J(z) := 1 2 M(t) χ ω ϕ(t; z) L 2 (Ω) dt ] 2 + y, ϕ(; z) L 2 (Ω) (13) = 1 2 M( )χ ωϕ( ; z) 2 L 1 (],T [;L 2 (Ω)) + y, ϕ(; z) L 2 (Ω), where ϕ( ; z) is the solution of (8), with the terminal data z L 2 (Ω). Denote V (T, y ) := inf J(z), (14) z L 2 (Ω) then, it holds V (T, y ) = 1 2 N(T, y ) 2, (15) where N(T, y ) is defined in (6). Proof The major steps of the proof are as follows: Step 1: Show that V (T, y ) <. Step 2: Show that inf z L 2 (Ω) J(z) exists and (15) holds by Theorem 3.1. The main idea for proof of the existence of inf z L2 (Ω) J(z) is using the minimizing sequence. Since these proofs are similar to the proof of Lemma 3.2 in [14] (or [16, p. 294]), we omit the details.

11 Title Suppressed Due to Excessive Length 11 Lemma 3.3 The optimal norm N(T, y ) of problem (6) can be characterized as N(T, y ) = sup z L 2 (Ω)\{} y, ϕ(; z) L 2 (Ω), (16) M( )χ ω ϕ( ; z) L1 (],T [;L 2 (Ω)) where ϕ( ; z) is the solution of (8) with the terminal value z L 2 (Ω). Proof The proof of this lemma is similar to the proof of Lemma 3.1 in [14] by using Lemma Proof of Main Result In this section, we give proof of our main result Theorem 2.1. Lemma 4.1 Let y L 2 (Ω) \ {}. Then, the function T N(T, y ) is continuous and strictly decreasing from R + to R +. Moreover, lim N(T, y ) = + T and lim N(T, y ) =. T Proof We carry out the proof by five steps. Step 1: Let N(T, y ) be defined in (6). Then, N(T, y ) R + for all T R +. (17)

12 12 Dong-Hui Yang et al. Suppose that (17) is not true. Then, there is t R + such that N(t, y ) =. This shows that the zero control is a control to problem (6), and hence e t y = y(t ; y, ) =. By the unique continuation of heat equation or Lemma 3.1, which implies y =. This contradicts with the assumption that y. This contradiction concludes (17). Step 2: Let < T 1 < T 2 <. Then, N(T 1, y ) > N(T 2, y ). (18) For any fixed T 1, T 2 R + with T 2 > T 1, by Lemma 3.2, problem (6) has an optimal control û 1 such that y(t 1 ; y, û 1 ) = and χ ω û 1 L (],T [;L 2 (Ω)) = N(T 1, y ). (19) By the null controllability of heat equation on the measurable set ω ], T 2 T 1 [, there exists a u 2 (T 1 + ) L (], T 2 T 1 [; L 2 (Ω)) so that the

13 Title Suppressed Due to Excessive Length 13 system t y y = χ ω M(T 1 + )u 2 (T 1 + ), in Ω ], T 2 T 1 [, y =, on Ω ], T 2 T 1 [, y() = e T1 y, in Ω, admits solution T2 T 1 e (T2 T1) (e T1 y ) + e (T2 T1 t) χ ω M(T 1 + t)u 2 (T 1 + t)dt =. (2) Since N(T 1, y ) R +, we can choose a scalar λ ], 1[ sufficiently small so that λ χ ω u 2 (T 1 + ) L (],T 2 T 1[;L 2 (Ω)) < N(T 1, y ). (21) 2 Define (1 λ)û 1 (t), t ], T 1 [, u λ (t) := λu 2 (t), t ]T 1, T 2 [. Then, y(t 2 ; y, u λ ) = e T2 y + (1 λ) T1 T2 + λ e (T2 t) χ ω M(t)u 2 (t)dt T 1 = (1 λ)e (T2 T1) (e T1 y + + λ [ e (T2 T1) (e T1 y ) + e (T2 t) χ ω M(t)û 1 (t)dt T1 T2 T 1 e (T1 t) χ ω M(t)û 1 (t)dt e (T2 T1 t) χ ω M(T 1 + t)u 2 (T 1 + t)dt ) ].

14 14 Dong-Hui Yang et al. This, together with (2) and the first equality of (19), gives y(t 2 ; y, u λ ) =, which means that u λ is a control to problem (6). By the optimality of N(T 2, y ), one has N(T 2, y ) u λ L (],T 2[;L 2 (Ω)) max { (1 λ) û 1 L (],T [;L 2 (Ω)), λ u 2 L (]T 1,T 2[;L 2 (Ω))}. From (21) and the second equality of (19), we obtain (18). Step 3: From Lemma 4.2 below, we see that N(, y ) is continuous. Step 4: By contradiction we can obtain lim T N(T, y ) = +. Step 5: Finally, it is easily verified that ˆN(y ) = lim T N(T, y ) =. This completes the proof of the lemma. Lemma 4.2 Let N(T, y ) be defined in (6). Then, N(, y ) is continuous. Proof We first show that N(, y ) is right-continuous. Let ˆT R + and T n ˆT. For every n N, there exists u n U N(Tn,y ) such that χ ω u n L (],T n[;l 2 (Ω)) = N(T n, y ) and y(t n ; y, u n ) =.

15 Title Suppressed Due to Excessive Length 15 Define u n, t ], T n [, ũ n (t) :=, t ]T n, T 1 [. Then, for every n N, χ ω ũ n L (],T 1[;L 2 (Ω)) = χ ω u n L (],T n[;l 2 (Ω)) = N(T n, y ) and y(t n ; y, ũ n ) = for all n N. (22) Since {N(T n, y )} n=1 is a bounded sequence, one can select a subsequence of {ũ n }, still denoted by itself, and ũ L (], T 1 [; L 2 (Ω)) such that χ ω ũ n χ ω ũ weakly star in L (], T 1 [; L 2 (Ω)). (23) This shows that χ ω ũ L (],T 1[;L 2 (Ω)) lim inf n N(T n, y ). (24) From (23), for every fixed n N, χ ω ũ L (]T n,t 1[;L 2 (Ω)) lim inf m χ ωũ m L (]T n,t 1[;L 2 (Ω)) =, which implies that ũ(t) = for all t ] ˆT, T 1 [. (25)

16 16 Dong-Hui Yang et al. By (23) and a standard argument, we can extract a subsequence of {y( ; y, ũ n )}, still denoted by itself, and y( ; y, ũ), such that y( ; y, ũ n ) y( ; y, ũ) strongly in C([, T 1 ]; L 2 (Ω)), (26) where y( ; y, ũ) is a solution of the following system t y y = χ ω M( )ũ, in Ω ], T 1 [, y =, on Ω ], T 1 [, y() = y, in Ω. By (22) and (26), y(t n ; y, ũ) = for all n N. Passing to the limit as n gives y( ˆT ; y, ũ) =. This, together with (24), yields N( ˆT ; y ) χ ω ũ L (], ˆT [;L 2 (Ω)) χ ω ũ L (],T 1[;L 2 (Ω)) lim inf n N(T n, y ). (27) In addition, by monotonicity (18) of N(T, y ), one has sup N(T n, y ) N( ˆT, y ). n 1

17 Title Suppressed Due to Excessive Length 17 This, together with (27), implies that N( ˆT, y ) = lim n N(T n, y ). Next, we show that N(, y ) is left-continuous. Let ˆT R + again, and {T n } n 1 ] ˆT 2, ˆT [ with T n ˆT. By (16) in Lemma 3.3, for any n N, we choose z n L 2 (Ω) \ {} such that Tn χ ω M(t)e (Tn t) z n L2 (Ω) dt = 1, (28) and N(T n, y ) 1 n y, ϕ(, z n ) L 2 (Ω) = y, e Tn z n L 2 (Ω). (29) We claim that there exists a subsequence {z kk } k 1 of {z n } and ϕ with χ ω M( )ϕ Z T such that for every δ ], T ], e (T k k (T δ)) z kk ϕ(t δ) weakly in L 2 (Ω) as k +. (3) Indeed, denote χ ω M(t)e (Tn t) z n, t ], T n ], ψ n (t) :=, t ]T n, T ]. Then, for all n N, ψ n L 1 (],T [;L 2 (Ω)) = 1. Which, together with (7) and (28), implies that for any fixed m N, there exists a constant C(T m ) such

18 18 Dong-Hui Yang et al. that e (Tn Tm) L z n 2 (Ω) = e (Tm+1 Tm) e (Tn Tm+1) L2 z n (Ω) C(T m ) χ ω M( )e (Tm+1 ) e (Tn Tm+1) L z n 1 (]T m,t m+1[;l 2 (Ω)) = C(T m ) χ ω M( )e (Tn ) L z n 1 (]T m,t m+1[;l 2 (Ω)) C(T m ) χ ω M( )e (Tn ) L1 z n (]T m,t n[;l 2 (Ω)) C(T m ) for all n m + 1. Therefore, for any fixed m N, there exist a subsequence {z mk } k 1 {z n } n m+1 and ẑ m such that e (Tm k Tm) z mk ẑ m as k. By Cantor diagonal law, there exists a subsequence {z kk } k N of {z n } such that for any m N, e (T k k T m) z kk ẑ m weakly in L 2 (Ω) as k. (31) Moreover, for fixed m 1, m 2 N, one has e (T k k T m1 ) z kk = e (Tm 1 +m 2 Tm 1 ) e (T k k T m1 +m ) 2 z kk

19 Title Suppressed Due to Excessive Length 19 for all k m 1 + m 2. This, together with (31), implies that ẑ m1 = e (Tm 1 +m 2 Tm 1 ) ẑ m1+m 2. (32) Define ϕ(t) := e (Tm t) ẑ m for t [, T m ], (33) where m N. From which, we conclude that ϕ is well defined on [, T [. (34) Now, we show that the sequence {z kk } k 1 given in (31) and ϕ defined in (33) satisfy the following properties: χ ω M( )ϕ Y T, T χ ω M( )ϕ(t) L 2 (Ω)dt 1, y, e T k k zkk L2 (Ω) y, ϕ() L2 (Ω). (35)

20 2 Dong-Hui Yang et al. From (28), (31), and Fatou s Lemma, we obtain T = lim n = lim n = lim n = lim n lim n χ ω M( )ϕ(t) L2 (Ω)dt T Tn Tn Tn Tn lim lim inf n k lim inf k = 1. χ (,Tn) χ ω M(t)ϕ(t) L 2 (Ω)dt Tkk χ ω M(t)ϕ(t) L2 (Ω)dt χ ω M(t)e (Tn t) ϕ(t n ) L 2 (Ω)dt χ ω M(t)e (Tn t) ẑ n L 2 (Ω)dt lim inf χ ωm(t)e (Tn t) e (T k k T n) z kk L2 (Ω)dt k Tn χ ω M(t)e (T k k t) z kk L 2 (Ω)dt χ ω M(t)e (T k k t) z kk L2 (Ω)dt This implies χ ω M( )ϕ Z T = Y T (by Theorem 3.1) and the second part of (35). Furthermore, by (32) and (34), we obtain (3). The third part of (35) follows from (3). Finally, by (29) and (35), we have lim sup N(T kk, y ) lim y, e T k k zkk L k + k + 2 (Ω) = y, ϕ() L 2 (Ω). (36) We claim that T lim sup N(T kk, y ) N(T, y ) M( )χ ω ϕ(t) L2 (Ω)dt N(T, y ). (37) k +

21 Title Suppressed Due to Excessive Length 21 Indeed, since χ ω M( )ϕ Z T = Y T, there exist z L 2 (Ω) and a sequence {ẑ n } n 1 L 2 (Ω) such that ϕ(t) = e ( T 2 t) z for all t [, T ] 2 (38) and χ ω M( )e (T ) ẑ n χ ω M( )ϕ( ) L1 (],T [;L 2 (Ω)) as n +. (39) On the other hand, by (7) and (38), there exists C(T ) which is independent of n such that ( ) e T 2 e T 2 ẑn z L2(Ω) C(T ) ( χ ω M( )e ( T 2 ) e T L 2 ẑn z) 1 (], T 2 [;L2 (Ω)) = C(T ) χ ω M( )e (T ) ẑ n χ ω M( )e ( T ) z 2 C(T ) χ ω M( )e (T ) ẑ n χ ω M( )ϕ( ) L1 (], T 2 [;L2 (Ω)) L1 (],T [;L 2 (Ω)), which, together with (39), implies that e T ẑ n ϕ() L 2 (Ω) as n.

22 22 Dong-Hui Yang et al. By (16), it follows that y, ϕ() L2 (Ω) = lim n y, e T ẑ n L2 (Ω) N(T, y ) lim sup χ ω M( )e (T ) ẑ n L 1 (],T [;L 2 (Ω)) n =N(T, y ) χ ω M( )ϕ L 1 (],T [;L 2 (Ω)). This, together with (36) and the second part of (35), implies that (37). From this and lim sup k N(T kk, y ) = lim sup n N(T n, y ) (i.e., the monotonicity of N(, y )), we conclude that N(, y ) = lim sup n N(T n, y ) = N(T, y ), that is, N(, y ) is left-continuous. The continuity of N(, y ) then follows from both the right-continuity and the left-continuity. Now, we present existences for time optimal control problem (5) and norm optimal control problem (6), which are straightforward. Proposition 4.1 Assume y L 2 (Ω) \ {}. Then, (i) for every N >, the time optimal control problem (5) admits a solution, i.e., there exists at least one u U N such that y(t (N, y ); y, u ) =. (ii) for any T >, the norm optimal control problem (6) admits a solution, i.e., there exists at least one u L (], T [; L 2 (Ω)) such that N(T, y ) = χ ω u L (],T [;L 2 (Ω))

23 Title Suppressed Due to Excessive Length 23 and y(t ; y, u ) =. Proof Since the proof of is standard by applying Lemmas 3.1 and 4.1, we omit the details here. Proposition 4.2 Let (N, T ) [, + [ ], + [. Then, N = N(T, y ) if and only if T = T (N, y ). Proof This will be accomplished by two steps. Step 1: If N = N(T, y ), then, T = T (N, y ). By the second part of Proposition 4.1, there exists an optimal norm control u 1 L (], T [; L 2 (Ω)) such that y(t ; y, u 1 ) = and χ ω u 1 L (],T [;L 2 (Ω)) = N(T, y ). From N = N(T, y ) and the optimality of T (N, y ), T (N, y ) T < +. (4) Since N >, by the first part of Proposition 4.1, there is a control u 2 L (], T (N, y )[; L 2 (Ω)) such that y(t (N, y ); y, u 2 ) = and χ ω u 2 L (],T (N,y )[;L 2 (Ω)) N. This shows that N(T (N, y ), y ) χ ω u 2 L (],T (N,y )[;L 2 (Ω)) N.

24 24 Dong-Hui Yang et al. From N = N(T, y ) and Lemma 4.1, it has T (N, y ) T. This, together with (4), yields T = T (N, y ). Step 2: If T = T (N, y ), then, N = N(T, y ). Since T (N, y ) = T R +, there exists u 1 L (], T [; L 2 (Ω)) such that y(t ; y, u 1 ) = and χ ω u 1 L (],T [;L 2 (Ω)) N. In addition, by Lemma 4.1 and y, N >. By Lemma 4.1, there is a ˆT R + such that N = N( ˆT, y ), From Step 1, we have ˆT = T (N, y ). This gives N = N(T (N, y ), y ). Since T = T (N, y ), we conclude that N = N(T, y ). Combining Steps 1 and 2 completes the proof of the proposition. Theorem 4.1 The optimal norm control of problem (6) satisfies the timevarying bang-bang property, i.e., the optimal norm control u L (], T [; L 2 (Ω))

25 Title Suppressed Due to Excessive Length 25 for problem (6) satisfies χ ω u (t) L 2 (Ω) = N(T, y ) for t ], T [ a.e. Furthermore, the optimal norm control is unique. Proof We prove the result by contradiction. Suppose that there were ε ], 1[ and E 1 ], T [ with E 1 > such that χ ω u (t) L 2 (Ω) < N(T, y ) ε for all t E 1. Since u L (], T [; L 2 (Ω)) is the optimal norm control of problem (6), i.e., u satisfies the following system t y y = χ ω M( )u, in Ω ], T [, y =, on Ω ], T [, y() = y, in Ω, and y(t ; y, u ) = and χ ω u L (],T [;L 2 (Ω)) = N(T, y ). (41)

26 26 Dong-Hui Yang et al. Let λ ], 1[ be a scalar which is to be determined latter. Then, the solution ψ( ; λy, λu ) of the following system t ψ ψ = χ ω M( )(λu ), in Ω ], T [, ψ =, on Ω ], T [, ψ() = λy, in Ω satisfies ψ(t ; λy, λu ) =. (42) Consider the following control problem t ψ λ ψ λ = χ ω χ E1 M( )u λ, in Ω ], T [, ψ λ =, on Ω ], T [, ψ λ () = (1 λ)y, in Ω. There exists U λ = M( )u λ L (], T [; L 2 (Ω)) such that ψ λ (T ; (1 λ)y, U λ ) =. i.e., ψ λ (T ; (1 λ)y, u λ ) =. (43)

27 Title Suppressed Due to Excessive Length 27 Moreover, there exists κ > independent of λ (Note that κ depends on m and M, as a consequence of (3.1.9) of [1]) such that χ ω u λ L (],T [;L 2 (Ω)) 1 m χ ωm( )u λ L (],T [;L 2 (Ω)) = 1 m χ ωu λ L (],T [;L 2 (Ω)) κ (1 λ)y L2 (Ω) = κ(1 λ) y L2 (Ω). Choose λ ], 1[ so that κ(1 λ) y L 2 (Ω) < ε 2, i.e., { 1 > λ > max 1 ε, 1 2κ y L 2 (Ω) 2 } and take v = λu + u λ. Then, the solution ξ( ; y, v) of the following system t ξ ξ = χ ω M( )v, in Ω ], T [, ξ =, on Ω ], T [, ξ() = y, in Ω

28 28 Dong-Hui Yang et al. satisfies ξ( ; y, v) = ψ( ; λy, λu ) + ψ λ ( ; (1 λ)y, u λ ). By (42) and (43), we obtain ξ(t ; y, v) = and χ ω v L (],T [;L 2 (Ω)) = χ ω (λu + u λ ) L (],T [;L 2 (Ω)) max{ χ ω (λu + u λ ) L (E ;L 2 (Ω)), χ ω (λu + u λ ) L (],T [ E ;L 2 (Ω))} max { λχ ω u L (E ;L 2 (Ω)) + χ ω u λ L (E ;L 2 (Ω)), } λχ ω u L (],T [ E ;L 2 (Ω)) { max λ(n(t, y ) ε ) + ε } 2, λn(t, y ) λn(t, y ) < N(T, y ). This gives N(T, y ) χ ω v L (],T [;L 2 (Ω)) < N(T, y ), which is a contradiction. Finally, by the time-varying bang-bang property, it is clear that the optimal norm control is unique. This completes the proof of the theorem. Theorem 4.2 The time optimal control problem (5) satisfies the time-varying bang-bang property. i.e., the optimal time control u U N for problem (5) satisfies χ ω u (t) L 2 (Ω) = N for a.e. t ], T (N, y )[. (44)

29 Title Suppressed Due to Excessive Length 29 Proof Let u L (], T (N, y )[; L 2 (Ω)) be an optimal time control for problem (5). Then, the solution y := y( ; y, u ) of the following system t y y = χ ω M( )u, in Ω ], T (N, y )[, y =, on Ω ], T (N, y )[, y () = y, in Ω satisfies y (T (N, y ); y, u ) = and χ ω u L (],T (N,y )[;L 2 (Ω)) N. If Ñ := χ ω u L (],T (N,y )[;L 2 (Ω)) < N, let T := T (N, y ), then, N(T, y ) Ñ < N, which, by virtue of Proposition 4.2, implies that T = T (N(T, y ), y ) > T (N, y ) = T. This is a contradiction. It therefore must have χ ω u L (],T (N,y )[;L 2 (Ω)) = N. Thus, it suffices to show χ ω u (t) L 2 (Ω) = N for t ], T (N, y )[ a.e.

30 3 Dong-Hui Yang et al. Indeed, let T (N, y ) and u U N be the optimal time and the optimal time control respectively for problem (5). Then, N = N(T (N, y ), y ) is the optimal norm for problem (6) with T = T (N, y ) by Theorem 4.1. Hence, u is also an optimal norm control for problem (6) with T = T (N, y ). By the uniqueness of optimal norm control, u U N is the unique optimal time control for problem (5). Furthermore, by Theorem 4.1, u satisfies the timevarying bang-bang property, i.e., (44) holds. Proof of Theorem 2.1 Since (3) is a special case of (5), Theorem 2.1 is deduced from Theorem 4.2 directly. 5 Conclusions This paper presents the bang-bang property of time-varying optimal time control for a heat equation, without requiring the admissible control with essentially constant upper bound. Instead, we only assume that the admissible control has different given bound at different time moment and has a low bound greater than zero. We note that, by the present technique, the lower bound cannot be relaxed to be zero, a case which deserves future studies. The time-varying bang-bang property for heat equation with potential may also be considered in the future work. More general framework about time-varying bang-bang property should also be considered. It is also intriguing to see what kind of time-varying control operators enjoys the bang-bang property. Acknowledgements The authors would like to thank Dr. Yubiao Zhang of Tianjin University for his helpful discussion during preparation of this work. The authors are grateful

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