Null controllability of the semilinear heat equation
Abstract
This paper is concerned with the null controllability of systems governed by semilinear parabolic equations. The control is exerted either on a small subdomain or on a portion of the boundary. We prove that the system is null controllable when the nonlinear term f(s) grows slower than s . log|s| as |s| → ∞.
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ESAIM: Control, Optimisation and Calculus of Variations URL: http://www.emath.fr/cocv/ April 1997, Vol. 2, pp. 87{103 NULL CONTROLLABILITY OF THE SEMILINEAR HEAT EQUATION E. FERN ANDEZ-CARA Abstract . This pap er is concerned with the null controllabil ity of systems governed by semilinear parab olic equations. The control is exerted either on a small sub domain or on a p ortion of the b oundary.Weprove that the system is null controllabl e when the nonlinear term f ( s ) grows slower than s log j s j as j s j! + 1 . 1. Introduction Assume a b ounded and connected op en set R N with regular b oundary @ , a nonempty op en subset O and a p ositivenumber T are given. We will use the following notation: Q = (0 T ), = @ (0 T ), H = L 2 () j : j and ( : : ) will stand for the usual norm and scalar pro duct in H , resp ectively C will denote a generic constant. In this pap er, we consider systems of the form 8 < : @ t y ; y + f ( y )= 1 O v in Q y =0 on y (0) = y 0 (1.1) where y 0 2 H and v 2 L 2 (0 T H ) are given. Here, 1 O is the characteristic function of the set O and it is assumed that f : R 7! R is lo cally Lipschitzcontinuous and satises f (0) = 0. It will b e said that (1.1) is null controllable if there exists a control v 2 L 2 (0 T H ) such that the corresp onding initial-b ound ary problem p ossesses a solution y 2 C 0 (0 T H ) with y ( T ) = 0. In other w ords, if the set R ( y 0 T ) formed by all reachable states at time T satises R ( y 0 T ) 3 0. Of course, the existence of a solution to (1.1) dened in the whole interval 0 T ] is not assured for arbitrary y 0 and v , unless something is imp osed to f .However, for the functions f considered b elow, at least lo cal existence holds. During the last years, the null controllability of systems governed byparab olic PDE's has b een intensively studied. Nowadays, the situation seems to b e the following: 1. In the linear case ( f ( s )= as for some a ), (1.1) is null controllable with no restriction on y 0 , T or O .For the pro of of this result there exist, at least, two approaches. The rst one is due to G. Leb eau Department of Dierential Equations and Numerical Analysis, University of Sevilla, Tara s/n, E-41012 Sevilla, Spain. E-mail: [email protected] . Received by the journal January 8, 1997. Accepted for publicati on January 31, 1997. This work has b een partially supp orted by D.G.I.C.Y.T. (Spain), Proyecto PB95{1242. c So ciete de Mathematiques Appliquees et Industrielles. Typ eset byL A T E X.
88 E. FERN ANDEZ-CARA and L. Robbiano. It connects null controllability to the observability through O (0 T ) of the solutions to the adjointsystem 8 < : ; @ t ' ; ' + a' =0 in Q ' =0 on ' ( T )= ' 0 (1.2) i.e. to the fact that, for some C only dep ending on , O and T ,one has j ' (0) j 2 C ZZ O (0 T ) j ' j 2 8 ' 0 2 H: (1.3) In the particular case a = 0, a pro of of (1.3) is given in 8] (see also 9]). A second approach,due toA.V. Fursikov and O. Yu. Imanuvilov, is based on some global Carleman's inequalities that can b e deduced for the solutions to (1.2) (and lead in particular to (1.3)). This metho d can b e applied to quite general linear parab olic problems and provides a control v such that y ( T ) = 0 and y is of minimal norm (see 5], 3]). It will b e revisited b elow, in section 5. 2. When f is sublinear at innity,i.e. j f ( s ) j C j s j + C (1.4) system (1.1) is again null controllable with no restriction on y 0 , T or O . This is due to O. Yu. Imanuvilov (see 6]). The argument uses a xed-p oint reformulation of the problem and the null controllability prop erties of linear problems. The assumption (1.4) leads to suitable uniform estimates and allows to apply Schauder's theorem. 3. In general, when f is sup erlinear at innity, (1.1) is not null controllable. Indeed, recall that when f ( s )= ; s (log j s j ) r ( r> 1) blow-up phenomena may o ccur. Even when f has \the go o d sign", null controllabilitymay fail. More precisely, when C j s j r +1 ; C f ( s ) s C j s j r +1 + C ( r> 1) and y 0 6 = 0, there exists T 0 > 0 such that, if T<T 0 , (1.1) is not null controllable (however, under this assumption on f , there exists T 1 such that, for all y 0 2 H , (1.1) is null controllable whenever T>T 1 see 5] and 6]). Hence, the natural question is: What happ ens if j f ( s ) j grows faster than j s j but slower than all j s j r ( r> 1) as j s j! 1 ? We are also interested in questions of this kind for systems controlled on a p ortion of the b oundary. More precisely, let b e a nonempty op en subset of @ and let us put ;= @ n : For each h 2 L 2 (), we consider the parab olic problem 8 < : @ t y ; y + f ( y )= 0 in Q y =1 h on y (0) = y 0 (1.5) Esaim: Cocv, April 1997, Vol. 2, pp. 87{103
NULL CONTROLLABILITY OF THE SEMILINEAR HEAT EQUATION 89 where f is as b efore and 1 is the characteristic function of .We will say that (1.5) is null controllable if there exists h such that the corresp onding initial-b oundary problem p ossesses a solution y 2 C 0 (0 T H ) satisfying y ( T )= 0. As for system (1.1), the existence of a global solution to (1.5) is not always assured. Moreover, the sense in which a function y can solve (1.5) has to b e sp ecied for general h 2 L 2 (). Nevertheless, we will work with appropriate functions f and we will nd control functions h which are regular enough to guarantee that (1.5) can b e solved (at least) in the usual weak sense. Again, under condition (1.4), system (1.5) is null controllable (cf. 6] see also 1] for the case of Neumann b oundary controllability). It is also true that, for sup erlinear nonlinearities, (1.5) is not null controllable in general. Consequently, it is also natural to study the case in which j f ( s ) j grows sup erlinearly but slower than all j s j r . 2. The main results and some remarks In this pap er, w e prove that systems (1.1) and (1.5) are null controllable when y 0 2 C 0 () and the b ehavior of f ( s )as j s j! 1 is, at most, almost critical. More precisely, one has: Theorem 2.1. Assume f : R 7! R is local ly Lipschitz-continuous, f (0) = 0 and lim j s j!1 f ( s ) s log j s j =0 : (2.1) Also, assume that y 0 2 C 0 () . Then system (1.1) is nul l control lable. Theorem 2.2. Under the assumptions of theorem 2.1, system (1.5) is nul l control lable. Moreprecisely, there exists h ,with 1 h 2 L 2 (0 T H 3 = 2 ( @ )) \ L 1 () such that the corresponding system (1.5) possesses at least one solution y 2 L 2 (0 T H 1 ()) \ C 0 (0 T H ) \ L 1 ( Q ) satisfying y ( T )= 0 . Remarks. 1. Assumption (2.1) can b e slightly weakened. Indeed, from the pro ofs in sections 3 and 4, we see there exists a p ositive constant " = " ( O T )such that theorems 2.1 and 2.2 still hold when j f ( s ) j " j s j log j s j for large j s j . On the other hand, the requirement y 0 2 C 0 () is unnecessary in practice. It is sucient to assume the following: The initial data y 0 and the function f are such that, when v = 0, system (1.1) p ossesses a solution y 2 C 0 (0 H ) which satises y ( ) 2 C 0 (). This happ ens for all y 0 2 H and for all functions f as in theorems 2.1 and 2.2, as a consequence of the regularizing e ect. Esaim: Cocv, April 1997, Vol. 2, pp. 87{103
90 E. FERN ANDEZ-CARA 2. We are dealing with quasi-optimal assumptions: recall again that, for f ( s )= j s j r ; 1 s ( r> 1), systems (1.1) and (1.5) are not null controllable in general. Also, recall that blow-up may o ccur for f ( s )= ; s (log j s j ) r . 3. Theorems 2.1 and 2.2 hold for more general problems. Thus, one can replace ; by a general linear elliptic op erator A ,with Ay = ; @ i ( a ij ( x ) @ j y )+ @ i ( b i ( x ) y )+ c ( x ) y: In this case, the co ecients have to b e regular enough and the a ij must satisfy the usual ellipticity condition. The nonlinear term can also b e of the form f ( x t y ), with f : Q R 7! R b eing a Carath!eo dory function satisfying (for instance): f ( x t 0) 0 s 7! f ( x t s ) is lo cally Lipschitz and lim j s j!1 f ( x t s ) s log j s j =0 uniformly in ( x t ) 2 Q . Furthermore, a nonvanishing additional term F = F ( x t ) can b e put in the left in the equations in (1.1) and (1.5). The ab ove results will still b e true provided F is suciently small near t = T .For instance, it will suce to have ZZ Q F ( x t ) 2 e 1 ( T ; t ) 2 < + 1 : Let us also mention that, for semilinear parab olic equations of higher order, similar results can b e obtained. 4. By adapting the arguments in sections 3 and 4, one is led to a new pro of of known lo cal results. These must hold for arbitrary lo cally Lipschitz functions f satisfying f (0) = 0. More precisely,if f is given, there exists = ( O Tf ) > 0such that, for every y 0 2 C 0 () with k y 0 k C 0 ,a control v 2 L 2 (0 T H ) can b e found such that the corresp onding problem (1.1) p ossesses a solution y 2 C 0 (0 T H ) which satises y ( T ) = 0 (compare with the lo cal results in 1], 4] and 6]). 5. System (1.1) is approximately controllable (in H ) if the set R ( y 0 T ) formed by all reachable states at time T is dense in H . It has b een proved in 2] that, under condition (1.4), system (1.1) is approximately controllable with no restriction on y 0 , T or O .However, to our knowledge, the approximate controllability of (1.1) under assumption (2.1) is an op en question. A similar remark concerning the approximate controllability of system (1.5) can b e made. 3. The proof of the boundary controllability result In this section, weprove theorem 2.2. We will rst consider the case in which, for instance, f is C 1 in the op en interval ( ; 1 1) and y 0 2 C 0 () for some 2 (0 1). We will use the standard norm in C 0 (), given as follows: k z k 1 + z ] := k z k 1 +sup j z ( x ) ; z ( x 0 ) j j x ; x 0 j Under these assumptions, we are going to rewrite (1.5) together with the equality y ( T ) = 0 as a xed-p oint equation in the Banachspace Y = L 1 ( Q ). Esaim: Cocv, April 1997, Vol. 2, pp. 87{103
NULL CONTROLLABILITY OF THE SEMILINEAR HEAT EQUATION 91 Then, using (2.1) and some sp ecic prop erties of linear parab olic equations, we will see that Schauder's theorem can b e applied. Arguments of this kind have already b een used in this and other similar contexts (see 2], 6], 10], ::: ). Here, since f is not necessarily globally Lipschitz, one sees that the range of the mapping arising in the xed-p oint formulation is not b ounded (this is also the case for the questions considered in 11]). Thus, the metho ds in 1]and6] cannot b e directly applied and some extra work has to b e made. Let us put g ( s )= 8 < : f ( s ) s for s 6 =0 f 0 (0) at s =0 : (3.1) Then, in view of assumption (2.1), one has: 8 > 0 9 C such that j g ( s ) j C + j log j s jj 8 s 2 R : (3.2) For each y 2 Y ,we consider the following null controllability problem 8 < : @ t u ; u + g ( y ) u =0 in Q u =1 h on u (0) = y 0 u ( T )= 0 (3.3) where the state equation is linear. A solution u ="( y ) to (3.3) can b e obtained by adapting the metho d of A.V. Fursiko v and O.Yu. Imanuvilov (cf. 5], 3]). More precisely, for the construction of "( y ), we do the following: a) Weintro duce z = z ( y ), with 8 < : @ t z ; z + g ( y ) z =0 in Q z =0 on z (0) = y 0 : (3.4) b) We x a function 2 C 1 (0 T )such that ( t ) = 1 near t =0, ( t )=0 near t = T and 0 1. Then, weset"( y )= z ( y )+ w ( y ), with w ( y ) b eing (together with h ) the solution furnished by lemma 3.2 to the following problem (see b elow): 8 < : @ t w ; w + g ( y ) w = ; 0 ( t ) z ( x t y ) in Q w =1 h on w (0) = 0 w ( T )= 0 : Our aim is to nd a xed-p ointin Y of the mapping y 7! "( y ). Obviously, if we are able to provethat such a xed-p oint exists, theorem 2.2 will have b een demonstrated (at least when f is C 1 in ( ; 1 1) and y 0 2 C 0 ()). First, we will show that the mapping y 7! "( y )is well dened and, also, that it maps a ball of Y into itself. We need the following elementary result: Lemma 3.1. Let a 2 Y and z 0 2 L 1 () be given. Then the linear problem 8 < : @ t z ; z + a ( x t ) z =0 in Q z =0 on z (0) = z 0 (3.5) Esaim: Cocv, April 1997, Vol. 2, pp. 87{103
92 E. FERN ANDEZ-CARA possesses exactly one solution z 2 L 2 (0 T H 1 0 ()) \ Y ,with k z ( t ) k 1 e t k a k Y k z 0 k 1 8 t 2 0 T ] : (3.6) We will also use the following result, whose pro of is given in section 5: Lemma 3.2. Toeach couple f a k g ,with a k 2 Y such that k =0 for 0 <t<b and T ; b< t < T one can assign a solution f w h g to the nul l linear control lability problem 8 < : @ t w ; w + a ( x t ) w = k ( x t ) in Q w =1 h on w (0) = 0 w ( T )= 0 (3.7) in such a way that w 2 L 2 (0 T H 2 ()) \ Y @ t w 2 L 2 ( Q ) (3.8) 1 h 2 L 2 (0 T H 3 = 2 ( @ )) \ L 1 () (3.9) and k w k L 2 (0 T H 2 ()) + k @ t w k L 2 ( Q ) + k w k Y e C (1+ k a k Y ) k k k Y : (3.10) Here, the constant C can depend on , , T and b , but not on k . Remark. Lemma 3.2 will also b e used in section 4, for the pro of of theorem 2.1. It remains true under more general conditions on k . From lemmas 3.1 and 3.2, it is clear that y 7! "( y )isawell dened mapping from Y into itself. Moreover, one has k "( y ) k Y e C (1+ k g ( y ) k Y ) k y 0 k 1 8 y 2 Y (3.11) where C can only dep end on , and T . Hence, in view of (3.2), one sees that " maps a ball B R of Y into itself. It is clear that y 7! "( y )is acontinuous mapping. Let us now prove that, for some 2 (0 1), it maps b ounded sets in Y into b ounded sets in C 0 = 2 ( Q ). This will suce to our purp ose, since this space is compactly emb edded in Y . Recall that C 0 = 2 ( Q ) is the Banach space formed byall functions u 2 C 0 ( Q ) such that u ] = 2 := sup Q j u ( x t ) ; u ( x 0 t ) j j x ; x 0 j + sup Q j u ( x t ) ; u ( x t 0 ) j j t ; t 0 j = 2 < + 1 : The (natural) norm in C 0 = 2 ( Q ) is as follows: k u k = 2 := k u k Y + u ] = 2 We will use the following lemma, which is proved in 7] (see theorems 7.1 and 10.1, Ch. I I I): Esaim: Cocv, April 1997, Vol. 2, pp. 87{103
NULL CONTROLLABILITY OF THE SEMILINEAR HEAT EQUATION 93 Lemma 3.3. Assume a k 2 Y , e h 2 L 2 (0 T H 1 = 2 ( @ )) \ L 1 () and z 0 2 C 0 () ,where 2 (0 1) . Then the linear problem 8 > < > : @ t u ; u + a ( x t ) u = k ( x t ) in Q u = e h on u (0) = z 0 possesses exactly one solution u 2 L 2 (0 T H 1 ()) \ Y .Furthermore, there exists 2 (0 ) (only depending on , T and ) such that u 2 C 0 = 2 ( Q ) and there exists a function M 1 (increasing with respect to its last argument) such that u ] = 2 M 1 ( T k z 0 k 1 + z 0 ] k u k Y ) : (3.12) We deduce that, for some , all functions "( y ) b elong to C 0 = 2 ( Q ). Combining (3.12) with (3.11), we also deduce that, for some M 2 (again increasing with resp ect to its last argument), one has: k "( y ) k = 2 M 2 ( T k y 0 k 1 + y 0 ] k g ( y ) k Y ) : This shows that " maps b ounded sets of Y into b ounded sets of C 0 = 2 ( Q ). Wehavethus seen that Schauder's theorem can b e applied to the xedpoint equation y ="( y ), y 2 Y . This proves theorem 2.2 when f is C 1 in ( ; 1 1) and y 0 2 C 0 (). In the general case, one can always put f = lim n !1 f n uniformly in R , for some lo cally Lipschitz functions f n which are C 1 in ( ; 1 1) and satisfy f n (0) = 0, and f n = f outside ( ; 2 2). One can also put y 0 = lim n !1 y n 0 uniformly in , for some functions y n 0 2 C 0 (). For each n 1, let f y n h n g b e a couple satisfying 8 < : @ t y n ; y n + f n ( y n )=0 in Q y n =1 h n on y n (0) = y n 0 y n ( T )= 0 (3.13) constructed as ab ove. For g n ( s )= f n ( s ) s for s 6 =0 f 0 n (0) at s =0 it is not dicult to see using (2.1) that 8 > 0 9 C such that j g n ( s ) j C + j log j s jj 8 s 2 R 8 n 1 : Consequently, one has k y n k Y Const. (use the corresp onding estimates (3.11)). Notice that y n can b e written in the form y n = z n + w n , where z n is uniformly b ounded in L 2 (0 T H 1 0 ()) \ Y and @ t z n is uniformly b ounded in L 2 (0 T H ; 1 ()), see (3.5) and (3.6). w n can b e estimated as in (3.10). Esaim: Cocv, April 1997, Vol. 2, pp. 87{103
94 E. FERN ANDEZ-CARA Hence, it can b e assumed that y n converges strongly in L 2 ( Q ) and a.e. and 1 h n converges (at least) weakly in L 2 (0 T H 3 = 2 ( @ )). We can thus take limits in (3.13) as n !1 and obtain (1.5) together with the equality y ( T )= 0. This proves theorem 2.2. 4. The proof of the internal controllability result In this section, we will prove theorem 2.1. As in the previous section, we will rst assume that f is C 1 intheopeninterval ( ; 1 1) and y 0 2 C 0 () for some 2 (0 1). Again, we will use the function g given by(3.1). We will intro duce a mapping y 7! $( y ) such that, for each y 2 Y , u = $( y ) is, together with some v 2 L 2 (0 T H ), a solution to the following problem: 8 < : @ t u ; u + g ( y ) u =1 O v in Q u =0 on u (0) = y 0 u ( T )= 0 : (4.1) This is made as follows: a) First, we x a nonempty op en, simply connected and regular set O 0 O ,we put = @ O 0 ;= @ and we denote by D the op en set b ounded by and ;, i.e. D = n O 0 . b) Using lemma 3.2 in D (0 T ), we assign to y a couple f ^ w ( y ) ^ h ( y ) g satisfying 8 > < > : @ t ^ w ; ^ w + g ( y )^ w = ; 0 ( t ) z ( x t y ) in D (0 T ) ^ w =1 ^ h on @D (0 T ) ^ w (0) = 0 and ^ w ( T )= 0 in D . (4.2) Here, z = z ( y ) is the (unique) solution to (3.4). One has ^ w 2 L 2 (0 T H 2 ( D )) \ L 1 ( D (0 T )) @ t ^ w 2 L 2 ( D (0 T )) (4.3) 1 ^ h 2 L 2 (0 T H 3 = 2 ( @D )) \ L 1 ( @D (0 T )) (4.4) and k ^ w k L 1 ( D (0 T )) e C (1+ k g ( y ) k Y ) k y 0 k 1 (4.5) where C can dep end only on , O 0 and T .We will denote by& w the extensionby-zero of ^ w to the whole set Q . c) Let us set w = & w and v = ; 2 r r & w ; ( )& w +(1 ; ) 0 z ( y ). Here, = ( x )is a C 1 cut-o function satisfying: =1 in a neighb orho o d of nO , =0 in a neighb orho o d of O 0 and 0 1. Then, we put $( y )= z ( y )+ w ( y ), with being as in section 3, i.e. a function satisfying 2 C 1 (0 T ), ( t )=1near t =0, ( t )=0 near t = T and 0 1. Esaim: Cocv, April 1997, Vol. 2, pp. 87{103
NULL CONTROLLABILITY OF THE SEMILINEAR HEAT EQUATION 95 It is not dicult to see that u =$( y ) is (together with v ) a solution to (4.1). From (4.3){(4.5) and (3.6), one has u 2 L 2 (0 T H 1 0 ()) \ Y v 2 L 2 (0 T H ) and k $( y ) k Y e C (1+ k g ( y ) k Y ) k y 0 k 1 Consequently, $ maps a ball B R of Y into itself. It is also clear that y 7! $( y ) is a continuous mapping. Furthermore, we can apply lemma 3.3 to (4.2). This leads easily to the facts that $( y ) 2 C 0 = 2 ( Q )and k $( y ) k = 2 M 3 ( T k y 0 k 1 + y 0 ] k g ( y ) k Y ) for all y 2 Y and for some 2 (0 ) (once more, M 3 is increasing with resp ect to its last argument). Hence, $ maps b ounded sets of Y into b ounded sets of C 0 = 2 ( Q ) and we can apply Schauder's theorem to the xed-p oint equation y =$( y ), y 2 Y . This proves theorem 2.1 when f is C 1 in ( ; 1 1) and y 0 2 C 0 (). In order to prove the same result in the general case, wecan argueasin section 3. Thus, intro ducing the functions f n and y n 0 ,we nd for each n 1 a couple f y n v n g satisfying 8 < : @ t y n ; y n + f n ( y n )= 1 O v n in Q y n =0 on y n (0) = y n 0 y n ( T )= 0 (4.6) and k y n k Y Const. k v n k L 2 (0 T H ) Const. Hence, it can b e assumed that y n converges strongly in L 2 ( Q ) and a.e. and 1 v n converges (at least) weakly in L 2 ( Q ). Taking limits in (4.6), we obtain (1.1) and y ( T ) = 0. This proves theorem 2.1. 5. The proof of lemma 3.2 We will rst analyze a null controllability problem, similar to (3.7), in a suitable larger domain G (0 T ). We will present an argument, taken from 5] and 3], which leads to a function e w that solves this problem in a generalized sense. Then, we will assign to e w a couple f w h g satisfying (3.7) and we will check that (3.8), (3.9) and (3.10) are satised. Accordingly, the pro of will b e divided in several steps. Step 1: The analysis of an auxiliaryproblem { Let and @ b e as b efore and assume that a k 2 Y , with k =0 for 0 <t<b and also for T ; b<t<T .Let G R N b e a b ounded, regular and op en set with G and @ \ @G =;= @ n .Let S 0 @G b e an op en neighborhood of;( S 0 6 = @G ) and let us put S 1 = @G n S 0 e = @G (0 T ) and e i = S i (0 T ) for i =1 2. We will denote by e a and e k the extensions-by-zero to the whole set G (0 T ) of the functions a and k , resp ectively. We will need the following Esaim: Cocv, April 1997, Vol. 2, pp. 87{103
102 E. FERN ANDEZ-CARA Also, 1 s ZZ jr ' j ; 1 j j 2 + s 3 ZZ jr ' j 3 j j 2 As 1 = 2 ZZ jr ' j jr j 2 (6.7) for all s s 6 for some s 6 s 5 . This can b e seen as follows: s 1 = 2 ZZ jr ' jjr j 2 = s 1 = 2 ZZ jr ' j ( ; ) ; 1 2 s 1 = 2 ZZ rjr ' jr ( 2 ) 1 2 s ZZ jr ' j ; 1 j j 2 + 1 2 s 3 ZZ jr ' j 3 j j 2 + 1 2 s 1 = 2 ZZ ( jr ' j ) j j 2 : From (6.5), (6.6) and (6.7), we see that k M 1 k 2 + 1 s ZZ jr ' j ; 1 j j 2 + s 1 = 2 ZZ jr ' jjr j 2 + s 3 ZZ jr ' j 3 j j 2 C k w s k 2 + Cs ZZ ( @ i @ j ' ) @ i @ j for all s s 6 + s 4 k a k 2 = 3 Y and all 1 .We can thus cho ose (and x) suciently large (dep ending on G , S 1 and T )tocontrol the last term in the right with the fourth term in the left. After this, we can write k M 1 k 2 + 1 s ZZ jr ' j ; 1 j j 2 + s ZZ jr ' jjr j 2 + s 3 ZZ jr ' j 3 j j 2 C k w s k 2 : It is also true that, for some s 7 s 6 and for all s s 7 , k M 1 k 2 + s ZZ jr ' jjr j 2 A s ZZ jr ' j ; 1 j @ t j 2 : Indeed, when s is suciently large, one has: 1 s ZZ jr ' j ; 1 j @ t j 2 = 1 s ZZ jr ' j ; 1 ( M 1 ; 2 s r ' r ) 2 k M 1 k 2 + Cs ZZ jr ' j 3 j j 2 : Consequently, 1 s ZZ jr ' j ; 1 ; j @ t j 2 + j j 2 + s ZZ jr ' jjr j 2 + s 3 ZZ jr ' j 3 j j 2 C k w s k 2 (6.8) Esaim: Cocv, April 1997, Vol. 2, pp. 87{103
NULL CONTROLLABILITY OF THE SEMILINEAR HEAT EQUATION 103 for all s s 7 + s 4 k a k 2 = 3 Y . Replacing by ; s q in (6.8), it is not dicult to deduce (5.1) for s s 8 + s 4 k a k 2 = 3 Y . Hence, lemma 5.2 is proved. The author is indebted to J.-P. Puel and E. Zuazua for several fruitful discussions. References 1] D. Chae, O.Yu. Imanuvilov and S.M. Kim: Exact controllabil ity for semilinear parab olic equations with Neumann b oundary conditions, to app ear. 2] C. Fabre, J.P. Puel and E. Zuazua: Approximate controllabili ty of the semilinear heat equation, Proc. Royal Soc. Edinbourgh , 125 A , 1995, 31{61. 3] A.V. Fursikov and O.Yu. Imanuvilov: On controllabil ity of certain systems simulating auid ow, in Flow Control, IMA Vol. Math. Appl. , 68 , Ed. by M.D. Gunzburger, Springer-Verlag, New York, 1995. 4] A.V. Fursikov and O.Yu. Imanuvilov: Lo cal exact controllabi li ty for 2-D Navier- Stokes equations, to app ear. 5] O.Yu. Imanuvilov: Thesis, Moscow 1991 (in russian) see also: Exact b oundary controllabili ty of the parab olic equation, Russian Math. Surveys , 48 , No. 3, 1993, 211{ 212. 6] O.Yu. Imanuvilov: Boundary controllabi li ty of parab olic equations, Russian Acad. Sci. Sb. Math. , 186 , No. 6, 1995, 109{132. 7] O.A. Ladyzhenska ja, V.A. Solonnikov and N.N. Ural'ceva: Linear and quasilinear equations of parab olic typ e, Trans. Math. Monographs ,Vol. 23, AMS, Providence, 1968. 8] G. Leb eau and L. Robbiano: Contr^ole exact de l'equation de la chaleur, Comm. P.D.E. , 20 , 1995, 335{356. 9] G. Leb eau and E. Zuazua: Null controllabili ty of a system of linear thermo elasticity, Arch. Rat. Mech. Anal. , to app ear. 10] E. Zuazua: Exact b oundary controllabil ity for the semilinear wave equation, in Nonlinear dierential equations and their applications , H. Brezis and J.L. Lions eds, Research Notes in Math. 10, Pitman, London, 1991, 357{391. 11] E. Zuazua: Exact controllabil ity for semilinear wave equations in one space dimension, Ann. Inst. Henri Poincar e ,Vol. 10, No. 1, 1993, 109{129. Esaim: Cocv, April 1997, Vol. 2, pp. 87{103