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The existence of time optimal control of semilinear parabolic equations
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Systems & Control Letters 53 (2004) 171 – 175
www.elsevier.com/locate/sysconle
The existence of time optimal control of semilinear parabolic equations Gengsheng Wang∗ Department of Mathematics, Wuhan University, The Center for Optimal Control and Discrete Mathematics, Huazhong Normal University, Wuhan 430072, PR China Received 27 April 2003; received in revised form 31 March 2004; accepted 2 April 2004
Abstract This work is concerned with the existence of the time optimal control for some semilinear parabolic di0erential equations with control distributed in a subdomain. c 2004 Elsevier B.V. All rights reserved. Keywords: Time optimal control; Semilinear parabolic equation; Controllability; Stabilization
1. Introduction Let ⊂ Rn ; n ∈ N , be a bounded domain with the boundary @ su6ciently smooth. Consider the controlled semilinear parabolic equations yt (x; t) − 8y(x; t) + f(y(x; t)) =m(x)u(x; t) y(x; t) = 0
in × (0; ∞);
on @ × (0; ∞);
y(x; 0) = y0 (x)
in ;
(1.1)
∞
where y0 ∈ L () is a given function, m is the characteristic function of an open subset ! ∈ ; f : R → R
is monotonically nondecreasing and f ∈ C 1 (R), and u is a control taken from a given set U = { u ∈ L∞ ( × (0; ∞)); |u(x; t)| 6 a:e in × (0; ∞)}; where ¿ 0 is an arbitrary but Fxed positive constant. As we know that for each u ∈ U , Eq. (1.1) has a unique solution y ∈ C([0; T ]; L2 ()) ∩ W 1; 1 ((0; T ]; L2 ()) for each T ¿ 0. Let ye ∈ L∞ () ∩ H01 () be a steady-state solution to (1.1), i.e. −8ye (x) + f(ye (x)) = 0 y(x) = 0
in ;
on @:
(1.2)
In this paper, we shall study the following time optimal control problem: This work was supported by The New Century Excellent Teacher Plan of The Ministry of National Education of China and by the Grant of Key Laboratory—Optimal Control and Discrete Mathematics of Hubei Province. ∗ Tel.: +86-276-786-8075; fax: +86-276-786-8067. E-mail address: [email protected] (G. Wang).
(P)
Min{ T ; y(T ) = ye ; u ∈ U and y is the solution to (1:1) corresponding to u}:
u ∈ U is called admissible if the corresponding solution y to (1.1) satisfying y(T ) = ye for some
c 2004 Elsevier B.V. All rights reserved. 0167-6911/$ - see front matter doi:10.1016/j.sysconle.2004.04.002
172
G. Wang / Systems & Control Letters 53 (2004) 171 – 175
T ¿ 0; T ∗ ≡ Min{T ; y(T ) = ye ; u ∈ U } is called the minimal time for (P) and a control u∗ ∈ U such that the corresponding solution y∗ to (1.1) satisfying y∗ (T ∗ ) = ye is called a time optimal control for problem (P). The time optimal control problem of di0erential equations was studied Frst for Fnite-dimensional case, i.e. the controlled equations are ordinary di0erential equations (cf. [7]). The main purpose of the problem is to study the control in a given control set which makes the corresponding solution to the controlled equation reaching a given target in the shortest time. There are two main subjects on the time optimal control problem. One is the necessary conditions for a time optimal control while another is the existence of a time optimal control which is also relatively straightforward to the necessary conditions. Then the problem was developed to inFnite-dimensional case., i.e. the controlled equations are partial di0erential equations (cf. [2,5,8]). For both Fnite- and inFnite-dimensional time optimal control problems, the key to get the existence of a time optimal control is to show the existence of an admissible control which is related to a type of controllability of the equations with some kind of control constraint (for instance, u ∈ U in this paper). This is the really di6cult part, even for the Fnite-dimensional time optimal control problems (cf. [7]). So most works dealing with existence of time optimal control of differential equations start by assuming the existence of an admissible control (cf. [5,7,8]). In [2], the existence of the time optimal control for some controlled parabolic variational inequalities, where the control is distributed in the whole domain , was obtained without assumption of the existence of an admissible control. The method used these to get the existence of an admissible control is that one constructs a closed-loop equation from the original controlled equation with control in some feedback form, and then obtains the result by proving the existence of the solution to the closed-loop equation. However, this method is suitable only for the case where the control is distributed in the whole domain . In [9], without the assumption of the existence of an admissible control, the existence of a time optimal control for phase–Feld systems with control distributed in a subdomain ! ⊂ was obtained via a modiFed Carleman inequality based on the form established in [6]. But in [9], the condition that initial data y0 is close to the target ye is required.
This means that the problem discussed in [9] is a kind of local time optimal control problem. In problem (P) considered here, the control is distributed in an arbitrary but Fxed subdomain ! ⊂ and the initial data y0 can be any function in L∞ () which is independent of the target ye . The methods used in [2,9] are not suitable for problem (P). In this paper, we develop an approach, based on the local null controllability and some special type of feedback stabilization of equation yt − 8y + f(y + ye ) − f(ye ) =mu y=0
in × (0; ∞); on @ × (0; ∞);
y(x; 0) = y0 ≡ y0 − ye
in ;
(1.3)
to get the existence of an admissible control. (Where and throughout the paper, we shall omit all x; t in the functions of x and t if there is no any ambiguity.) More precisely, we take a control in the feedback form of u = − sgn y, where 1 if y ¿ 0; if y ¡ 0; sgny = −1 (1.4) [ − 1; 1] if y = 0; to make Eq. (1.3), where u = − sgn y, is exponentially stable, i.e. there exist constants ¿ 0 and C0 ¿ 0 such that |y(t)|L1 () 6 C0 e−t |y0 |L1 ()
∀t ¿ 0:
(1.5)
This result was obtained in [1] for the same equations with Neumann boundary condition. Next we use the local null controllability for Eq. (1.3) obtained in [3] to get the existence of such a u ∈ U that the corresponding solution y to Eq. (1.3) satisFes y(T ) = 0 for any T ¿ 0. Then we combine these with the smooth e0ect of the solution to Eq. (1.3) to obtain the existence of an admissible control. We believe that this method is suitable for large class of problems of the existence for time optimal controls. 2. Main results In this section we shall present the main results and their proofs of the paper. We Frst point out that problem (P) is equivalent to the following
G. Wang / Systems & Control Letters 53 (2004) 171 – 175
173
problem:
satis;es
(P)
y(t)L1 () 6 C1 e−t y0 (x)L1 ()
Min{ T ; u ∈ U ; y(T ) = 0; y is the solution to (1:3) corresponding to u}:
0
(2.1)
then it is clear that (F1 ) (F2 ) (F3 ) (F4 )
F : × R → R is continuous; r → F(x; r) is monotonically nondecreasing; F(x; 0) = 0 ∀ x ∈ R; F(x; r) = G(x; r)r, where G(x; r) satisFes |G(x; r)| 6 CM for all x ∈ and r ∈ R with |r| 6 M , here CM is a positive constant dependent on M .
Theorem 1. There exists at least one admissible control for problem (P). Proof. By the previous discussion, we are lead to prove the existence of an admissible control for prob which will be divided into several steps. lem (P), Step 1: Feedback stabilization of Eq. (1.3). For feedback stabilization of Eq. (1.3), we have the following result. Lemma 1. Let y0 ∈ L∞ (). Then the feedback control u(x; t) = − sgn y(x; t);
+ m sgn y(x; #) 0 y=0
in × (t − 1; t);
on @ × (t − 1; t);
y(x; t − 1) = yN 0 (x)
in ;
(2.4)
where yN 0 (x) ∈ L1 (). By the Proposition 3.3 in chapter 4 of [2] (where (F2 ) was needed), the unique solution yN to (2.4) satisFes y(x; N t)L∞ () 6 C2 yN 0 L1 () ; where C2 ¿ 0 is independent of t, which together with Lemma 1 implies the following estimate: y(t)L∞ () 6 C1 C2 e−(t−1) y0 L1 ()
∀t ¿ 1; (2.5)
if y ¡ 0;
where y is the solution to (1.3) with u = − sgn y. Step 3: Local null controllability for Eq. (1.3). Let t ¿ 0 be an arbitrary but Fxed number. Consider the equation
if y = 0
y# (x; #) − 8y(x; #) + F(x; y(x; #)) = m(x)u(x; #);
if y ¿ 0;
yt − 8y + F(x; y) + m sgn y 0
in × (0; ∞);
on @ × (0; ∞);
y(x; 0) = y0 (x)
y# (x; #) − 8y(x; #) + F(x; y(x; #))
(x; t) ∈ × R ;
exponentially stabilizes Eq. (1.3), i.e., the solution y to the closed-loop equation
y=0
The same result on the feedback stabilization for Eq. (1.3) with Neumann boundary condition @y=@"=0 on @ × (0; ∞) was obtained in [1] via comparison principle for parabolic variational inequality and the Carleman inequality in L1 () established [4] under conditions (F1 )–(F3 ). We may use exactly the same methods as those used in [1] to prove Lemma 1. We omit the proof here. Step 2: Smooth e
+
where
1 sgn y = −1 [ − 1; 1]
(2.3)
for some constants C1 , ¿ 0 independent of t.
Note that if we deFne F : × R → R by F(x; r) = f(r + ye (x)) − f(ye (x)) 1 = f (ye (x) + r) dr ≡ G(x; r)r;
∀t ¿ 0
in
(2.2)
in Qt ≡ × (t; t + 1); y(x; #) = 0;
on @ × (t; t + 1);
0 (x) y(x; t) = y
in :
(2.6)
We have the following local null controllability result for Eq. (2.6).
174
G. Wang / Systems & Control Letters 53 (2004) 171 – 175
Lemma 2. There exists a constant 0 ¿ 0 such that 0 ∈ L∞ (); y0 L∞ () 6 0 , there are funcfor any y ∞ tions u ∈ L (Qt ) with |u(x; #)| 6 a:e: in Qt , and y ∈ C([t; t +1]; L2 ())∩L2 (t; t +1; H01 ())∩L∞ (Qt ) satisfying Eq. (2.6) and y(x; t + 1) = 0. Under more general conditions on F which may be implied by (F1 ), (F2 ) and (F4 ), Lemma 2 was proved in [3] via the Carleman inequality (cf. [5]) and Kakutani Fxed point theorem (cf. [2]). Step 4: Existence of an admissible control. By (2.5) we imply that for any y0 ∈ L∞ () Fxed, there exists t0 ¿ 0 such that the solution y to (1:3), where u=− sgn y, satisFes |y(t0 )|L∞ () 6 0 ; where 0 was given by Lemma 2. Then it follows from Lemma 2 that there exists a control u ∈ L∞ (×(t0 ; t0 + 1)) with | u(x; t)| 6 ; a:e: in × (t0 ; t0 + 1) such to equation that the solution y t − 8 ) = m y y + F(x; y u =0 y
in × (t0 ; t0 + 1);
on @ × (t0 ; t0 + 1);
(x; t0 ) = y(x; t0 ) y
n be the solution to (1.1), where u = un , then and y n (Tn ) = ye . Now for any T such that T ∗ ¡ T ¡ ∞, y we have that Tn ¡ T for all n large enough. Without loss of generality we may assume that un → u∗
weakly star in L2 (QT );
(2.7)
∗
which implies | u (x; t)| 6 ; a:e: (x; t) ∈ QT . By monotonicity of f, we may obtain that on a subsequence of { yn }, still denoted in the same way, n → y ∗ weakly in W 1; 2 ([0; T ]; L2 ()) y ∩L2 0; T ; H 2 () ∩ H01 () ; strongly in C([0; T ]; L2 ()):
(2.8)
It follows from (2.7) and (2.8) that ∗ − 8 ∗ ) = m y y∗ + f(x; y u∗ in QT ; t ∗ = 0 y on @ × (0; T ); ∗ (x; 0) = y0 (x) y in and n (Tn )L2 () y∗ (T ∗ ) − y
in
n (T ∗ )L2 () 6 y∗ (T ∗ ) − y
(x; t0 + 1) = 0. Hence that if we let satisFes y sgn y(x; t); 0 6 t 6 t0 ; x ∈ ; u(x; t); t0 ¡ t 6 t0 + 1; x ∈ ; u∗ (x; t) = 0; t0 + 1 ¡ t ¡ ∞; x ∈ ; where y is the solution to Eq. (2.2) on × (0; t0 ), and y∗ be the solution to Eq. (1.1) corresponding to u∗ , then u∗ ∈ U and y∗ (x; t0 + 1) = 0 for almost all x ∈ . This completes the proof. Theorem 2. There exists at least one time optimal control for problem (P).
n (T ∗ )L2 () → 0; + yn (Tn ) − y as n → ∞;
(2.9) ∗
∗
which implies y (T ) = ye . Finally, we let ∗ u (x; t); 0 6 t 6 T ∗ ; x ∈ ; ∗ u (x; t) = 0; t ¿ T ∗ ; x ∈ ; and y∗ be the solution to (1.1), where u=u∗ , it follows from (2.7) and (2.9) that u∗ ∈ U and y∗ (T ∗ ) = ye . Hence, u∗ is a time optimal control for problem (P). This completes the proof. Acknowledgements
∗
Proof. The argument is standard. Let T = Min(P). By Theorem 1, 0 6 T ∗ ¡ ∞ and there exist sequences {Tn } with Tn → T ∗ ; Tn ¿ T ∗ and {un } ⊂ U such that the solutions yn to (1:1), where u = un , satisFes yn (Tn ) = ye for all n. We let un (x; t); 0 6 t 6 Tn ; x ∈ ; un (x; t) = 0; t ¿ Tn ; x ∈
The author would like to thank Dr. L. Cartalin for his good suggestion on this paper. References [1] Gh. Aniculaesei, S. Anita, Stabilization of the heat equation via internal feedback control, Nonlinear Functional Anal. Appl. 6 (2001) 12–24.
G. Wang / Systems & Control Letters 53 (2004) 171 – 175 [2] V. Barbu, Analysis and Control of Nonlinear InFnite Dimensional Systems, Academic Press, Boston, 1993. [3] V. Barbu, Controllability of parabolic and Navier–Stokes equations, Sci. Math. Japonica 6 (2002) 143–211. [4] V. Barbu, The Carleman inequality for linear parabolic equation in Lq norm, Di0erential Integral Equations 15 (5) (2002) 513–525. [5] H.O. Fattorini, InFnite-dimensional Optimization and Control Theory, Encyclopaedia of Mathematics and its Applications, Vol. 62, Cambridge University Press, Cambridge, UK, 1999.
175
[6] A.V. Fursifov, O.Yu. Imanuvilov, Controllability of Evolution Equations, Lecture Notes Series, Vol. 34, RIM, Seoul University, Korea, 1996. [7] J.P. LaSalle, The time optimal control problem, Contributions to the Theory of Nonlinear Oscillations 5, Princeton University Press, Princeton, 1960, pp. 1–24. [8] X. Li, J. Yong, Optimal Control Theory for InFnite Dimensional Systems, Birkhauser, Boston, 1995. [9] L. Wang, G. Wang, The time optimal control of phase-Feld system, SIAM J. Control Optim. 42 (4) (2003) 1483–1508.
Systems & Control Letters 53 (2004) 171 – 175
www.elsevier.com/locate/sysconle
The existence of time optimal control of semilinear parabolic equations Gengsheng Wang∗ Department of Mathematics, Wuhan University, The Center for Optimal Control and Discrete Mathematics, Huazhong Normal University, Wuhan 430072, PR China Received 27 April 2003; received in revised form 31 March 2004; accepted 2 April 2004
Abstract This work is concerned with the existence of the time optimal control for some semilinear parabolic di0erential equations with control distributed in a subdomain. c 2004 Elsevier B.V. All rights reserved. Keywords: Time optimal control; Semilinear parabolic equation; Controllability; Stabilization
1. Introduction Let ⊂ Rn ; n ∈ N , be a bounded domain with the boundary @ su6ciently smooth. Consider the controlled semilinear parabolic equations yt (x; t) − 8y(x; t) + f(y(x; t)) =m(x)u(x; t) y(x; t) = 0
in × (0; ∞);
on @ × (0; ∞);
y(x; 0) = y0 (x)
in ;
(1.1)
∞
where y0 ∈ L () is a given function, m is the characteristic function of an open subset ! ∈ ; f : R → R
is monotonically nondecreasing and f ∈ C 1 (R), and u is a control taken from a given set U = { u ∈ L∞ ( × (0; ∞)); |u(x; t)| 6 a:e in × (0; ∞)}; where ¿ 0 is an arbitrary but Fxed positive constant. As we know that for each u ∈ U , Eq. (1.1) has a unique solution y ∈ C([0; T ]; L2 ()) ∩ W 1; 1 ((0; T ]; L2 ()) for each T ¿ 0. Let ye ∈ L∞ () ∩ H01 () be a steady-state solution to (1.1), i.e. −8ye (x) + f(ye (x)) = 0 y(x) = 0
in ;
on @:
(1.2)
In this paper, we shall study the following time optimal control problem: This work was supported by The New Century Excellent Teacher Plan of The Ministry of National Education of China and by the Grant of Key Laboratory—Optimal Control and Discrete Mathematics of Hubei Province. ∗ Tel.: +86-276-786-8075; fax: +86-276-786-8067. E-mail address: [email protected] (G. Wang).
(P)
Min{ T ; y(T ) = ye ; u ∈ U and y is the solution to (1:1) corresponding to u}:
u ∈ U is called admissible if the corresponding solution y to (1.1) satisfying y(T ) = ye for some
c 2004 Elsevier B.V. All rights reserved. 0167-6911/$ - see front matter doi:10.1016/j.sysconle.2004.04.002
172
G. Wang / Systems & Control Letters 53 (2004) 171 – 175
T ¿ 0; T ∗ ≡ Min{T ; y(T ) = ye ; u ∈ U } is called the minimal time for (P) and a control u∗ ∈ U such that the corresponding solution y∗ to (1.1) satisfying y∗ (T ∗ ) = ye is called a time optimal control for problem (P). The time optimal control problem of di0erential equations was studied Frst for Fnite-dimensional case, i.e. the controlled equations are ordinary di0erential equations (cf. [7]). The main purpose of the problem is to study the control in a given control set which makes the corresponding solution to the controlled equation reaching a given target in the shortest time. There are two main subjects on the time optimal control problem. One is the necessary conditions for a time optimal control while another is the existence of a time optimal control which is also relatively straightforward to the necessary conditions. Then the problem was developed to inFnite-dimensional case., i.e. the controlled equations are partial di0erential equations (cf. [2,5,8]). For both Fnite- and inFnite-dimensional time optimal control problems, the key to get the existence of a time optimal control is to show the existence of an admissible control which is related to a type of controllability of the equations with some kind of control constraint (for instance, u ∈ U in this paper). This is the really di6cult part, even for the Fnite-dimensional time optimal control problems (cf. [7]). So most works dealing with existence of time optimal control of differential equations start by assuming the existence of an admissible control (cf. [5,7,8]). In [2], the existence of the time optimal control for some controlled parabolic variational inequalities, where the control is distributed in the whole domain , was obtained without assumption of the existence of an admissible control. The method used these to get the existence of an admissible control is that one constructs a closed-loop equation from the original controlled equation with control in some feedback form, and then obtains the result by proving the existence of the solution to the closed-loop equation. However, this method is suitable only for the case where the control is distributed in the whole domain . In [9], without the assumption of the existence of an admissible control, the existence of a time optimal control for phase–Feld systems with control distributed in a subdomain ! ⊂ was obtained via a modiFed Carleman inequality based on the form established in [6]. But in [9], the condition that initial data y0 is close to the target ye is required.
This means that the problem discussed in [9] is a kind of local time optimal control problem. In problem (P) considered here, the control is distributed in an arbitrary but Fxed subdomain ! ⊂ and the initial data y0 can be any function in L∞ () which is independent of the target ye . The methods used in [2,9] are not suitable for problem (P). In this paper, we develop an approach, based on the local null controllability and some special type of feedback stabilization of equation yt − 8y + f(y + ye ) − f(ye ) =mu y=0
in × (0; ∞); on @ × (0; ∞);
y(x; 0) = y0 ≡ y0 − ye
in ;
(1.3)
to get the existence of an admissible control. (Where and throughout the paper, we shall omit all x; t in the functions of x and t if there is no any ambiguity.) More precisely, we take a control in the feedback form of u = − sgn y, where 1 if y ¿ 0; if y ¡ 0; sgny = −1 (1.4) [ − 1; 1] if y = 0; to make Eq. (1.3), where u = − sgn y, is exponentially stable, i.e. there exist constants ¿ 0 and C0 ¿ 0 such that |y(t)|L1 () 6 C0 e−t |y0 |L1 ()
∀t ¿ 0:
(1.5)
This result was obtained in [1] for the same equations with Neumann boundary condition. Next we use the local null controllability for Eq. (1.3) obtained in [3] to get the existence of such a u ∈ U that the corresponding solution y to Eq. (1.3) satisFes y(T ) = 0 for any T ¿ 0. Then we combine these with the smooth e0ect of the solution to Eq. (1.3) to obtain the existence of an admissible control. We believe that this method is suitable for large class of problems of the existence for time optimal controls. 2. Main results In this section we shall present the main results and their proofs of the paper. We Frst point out that problem (P) is equivalent to the following
G. Wang / Systems & Control Letters 53 (2004) 171 – 175
173
problem:
satis;es
(P)
y(t)L1 () 6 C1 e−t y0 (x)L1 ()
Min{ T ; u ∈ U ; y(T ) = 0; y is the solution to (1:3) corresponding to u}:
0
(2.1)
then it is clear that (F1 ) (F2 ) (F3 ) (F4 )
F : × R → R is continuous; r → F(x; r) is monotonically nondecreasing; F(x; 0) = 0 ∀ x ∈ R; F(x; r) = G(x; r)r, where G(x; r) satisFes |G(x; r)| 6 CM for all x ∈ and r ∈ R with |r| 6 M , here CM is a positive constant dependent on M .
Theorem 1. There exists at least one admissible control for problem (P). Proof. By the previous discussion, we are lead to prove the existence of an admissible control for prob which will be divided into several steps. lem (P), Step 1: Feedback stabilization of Eq. (1.3). For feedback stabilization of Eq. (1.3), we have the following result. Lemma 1. Let y0 ∈ L∞ (). Then the feedback control u(x; t) = − sgn y(x; t);
+ m sgn y(x; #) 0 y=0
in × (t − 1; t);
on @ × (t − 1; t);
y(x; t − 1) = yN 0 (x)
in ;
(2.4)
where yN 0 (x) ∈ L1 (). By the Proposition 3.3 in chapter 4 of [2] (where (F2 ) was needed), the unique solution yN to (2.4) satisFes y(x; N t)L∞ () 6 C2 yN 0 L1 () ; where C2 ¿ 0 is independent of t, which together with Lemma 1 implies the following estimate: y(t)L∞ () 6 C1 C2 e−(t−1) y0 L1 ()
∀t ¿ 1; (2.5)
if y ¡ 0;
where y is the solution to (1.3) with u = − sgn y. Step 3: Local null controllability for Eq. (1.3). Let t ¿ 0 be an arbitrary but Fxed number. Consider the equation
if y = 0
y# (x; #) − 8y(x; #) + F(x; y(x; #)) = m(x)u(x; #);
if y ¿ 0;
yt − 8y + F(x; y) + m sgn y 0
in × (0; ∞);
on @ × (0; ∞);
y(x; 0) = y0 (x)
y# (x; #) − 8y(x; #) + F(x; y(x; #))
(x; t) ∈ × R ;
exponentially stabilizes Eq. (1.3), i.e., the solution y to the closed-loop equation
y=0
The same result on the feedback stabilization for Eq. (1.3) with Neumann boundary condition @y=@"=0 on @ × (0; ∞) was obtained in [1] via comparison principle for parabolic variational inequality and the Carleman inequality in L1 () established [4] under conditions (F1 )–(F3 ). We may use exactly the same methods as those used in [1] to prove Lemma 1. We omit the proof here. Step 2: Smooth e
+
where
1 sgn y = −1 [ − 1; 1]
(2.3)
for some constants C1 , ¿ 0 independent of t.
Note that if we deFne F : × R → R by F(x; r) = f(r + ye (x)) − f(ye (x)) 1 = f (ye (x) + r) dr ≡ G(x; r)r;
∀t ¿ 0
in
(2.2)
in Qt ≡ × (t; t + 1); y(x; #) = 0;
on @ × (t; t + 1);
0 (x) y(x; t) = y
in :
(2.6)
We have the following local null controllability result for Eq. (2.6).
174
G. Wang / Systems & Control Letters 53 (2004) 171 – 175
Lemma 2. There exists a constant 0 ¿ 0 such that 0 ∈ L∞ (); y0 L∞ () 6 0 , there are funcfor any y ∞ tions u ∈ L (Qt ) with |u(x; #)| 6 a:e: in Qt , and y ∈ C([t; t +1]; L2 ())∩L2 (t; t +1; H01 ())∩L∞ (Qt ) satisfying Eq. (2.6) and y(x; t + 1) = 0. Under more general conditions on F which may be implied by (F1 ), (F2 ) and (F4 ), Lemma 2 was proved in [3] via the Carleman inequality (cf. [5]) and Kakutani Fxed point theorem (cf. [2]). Step 4: Existence of an admissible control. By (2.5) we imply that for any y0 ∈ L∞ () Fxed, there exists t0 ¿ 0 such that the solution y to (1:3), where u=− sgn y, satisFes |y(t0 )|L∞ () 6 0 ; where 0 was given by Lemma 2. Then it follows from Lemma 2 that there exists a control u ∈ L∞ (×(t0 ; t0 + 1)) with | u(x; t)| 6 ; a:e: in × (t0 ; t0 + 1) such to equation that the solution y t − 8 ) = m y y + F(x; y u =0 y
in × (t0 ; t0 + 1);
on @ × (t0 ; t0 + 1);
(x; t0 ) = y(x; t0 ) y
n be the solution to (1.1), where u = un , then and y n (Tn ) = ye . Now for any T such that T ∗ ¡ T ¡ ∞, y we have that Tn ¡ T for all n large enough. Without loss of generality we may assume that un → u∗
weakly star in L2 (QT );
(2.7)
∗
which implies | u (x; t)| 6 ; a:e: (x; t) ∈ QT . By monotonicity of f, we may obtain that on a subsequence of { yn }, still denoted in the same way, n → y ∗ weakly in W 1; 2 ([0; T ]; L2 ()) y ∩L2 0; T ; H 2 () ∩ H01 () ; strongly in C([0; T ]; L2 ()):
(2.8)
It follows from (2.7) and (2.8) that ∗ − 8 ∗ ) = m y y∗ + f(x; y u∗ in QT ; t ∗ = 0 y on @ × (0; T ); ∗ (x; 0) = y0 (x) y in and n (Tn )L2 () y∗ (T ∗ ) − y
in
n (T ∗ )L2 () 6 y∗ (T ∗ ) − y
(x; t0 + 1) = 0. Hence that if we let satisFes y sgn y(x; t); 0 6 t 6 t0 ; x ∈ ; u(x; t); t0 ¡ t 6 t0 + 1; x ∈ ; u∗ (x; t) = 0; t0 + 1 ¡ t ¡ ∞; x ∈ ; where y is the solution to Eq. (2.2) on × (0; t0 ), and y∗ be the solution to Eq. (1.1) corresponding to u∗ , then u∗ ∈ U and y∗ (x; t0 + 1) = 0 for almost all x ∈ . This completes the proof. Theorem 2. There exists at least one time optimal control for problem (P).
n (T ∗ )L2 () → 0; + yn (Tn ) − y as n → ∞;
(2.9) ∗
∗
which implies y (T ) = ye . Finally, we let ∗ u (x; t); 0 6 t 6 T ∗ ; x ∈ ; ∗ u (x; t) = 0; t ¿ T ∗ ; x ∈ ; and y∗ be the solution to (1.1), where u=u∗ , it follows from (2.7) and (2.9) that u∗ ∈ U and y∗ (T ∗ ) = ye . Hence, u∗ is a time optimal control for problem (P). This completes the proof. Acknowledgements
∗
Proof. The argument is standard. Let T = Min(P). By Theorem 1, 0 6 T ∗ ¡ ∞ and there exist sequences {Tn } with Tn → T ∗ ; Tn ¿ T ∗ and {un } ⊂ U such that the solutions yn to (1:1), where u = un , satisFes yn (Tn ) = ye for all n. We let un (x; t); 0 6 t 6 Tn ; x ∈ ; un (x; t) = 0; t ¿ Tn ; x ∈
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