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. FERNANDEZ-CARA
Abstract. This paper is concerned with the null controllability of sys- tems 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 termf(s) grows slower thans logjsjasjsj!+1.
1. Introduction
Assume a bounded and connected open set RN with regular bound- ary @, a nonempty open subsetO and a positive numberT are given.
We will use the following notation: Q = (0T), = @(0T),
H =L2()j:jand (::) will stand for the usual norm and scalar product in
H, respectivelyC will denote a generic constant.
In this paper, we consider systems of the form
8
<
:
@
t
y;y+f(y) = 1Ov in Q
y= 0 on
y(0) =y0 (1.1)
where y0 2H and v2L2(0TH) are given. Here, 1O is the characteristic function of the set O and it is assumed thatf :R7!Ris locally Lipschitz- continuous and satises f(0) = 0.
It will be said that (1.1) is null controllable if there exists a control v 2
L
2(0TH) such that the corresponding initial-boundary problem possesses a solutiony2C0(0TH) withy(T) = 0. In other words, if the setR(y0T) formed by all reachable states at time T satises R(y0T)30.
Of course, the existence of a solution to (1.1) dened in the whole interval 0T] is not assured for arbitrary y0 and v, unless something is imposed to
f. However, for the functions f considered below, at least local existence holds.
During the last years, the null controllability of systems governed by par- abolic PDE's has been intensively studied. Nowadays, the situation seems to be the following:
1. In the linear case (f(s) = as for some a), (1.1) is null controllable with no restriction on y0, T or O. For the proof of this result there exist, at least, two approaches. The rst one is due to G. Lebeau
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 publication January 31, 1997.
This work has been partially supported by D.G.I.C.Y.T. (Spain), Proyecto PB95{1242.
c
Societe de Mathematiques Appliquees et Industrielles. Typeset by LATEX.
88 E. FERNANDEZ-CARA
and L. Robbiano. It connects null controllability to the observability throughO(0T) of the solutions to the adjoint system
8
<
:
;@
t
';'+a'= 0 in Q
'= 0 on
'(T) ='0 (1.2)
i.e. to the fact that, for some C only depending on , O and T, one has
j'(0)j2C
ZZ
O (0T) j'j
2
8' 0
2H : (1.3)
In the particular case a = 0, a proof of (1.3) is given in 8] (see also 9]).A second approach, due to A. V. Fursikov and O. Yu. Imanuvilov, is based on some global Carleman's inequalities that can be deduced for the solutions to (1.2) (and lead in particular to (1.3)). This method can be applied to quite general linear parabolic problems and provides a controlv such that y(T) = 0 andy is of minimal norm (see 5], 3]).
It will be revisited below, in section 5.
2. When f is sublinear at innity, i.e.
jf(s)jCjsj+C (1.4)
system (1.1) is again null controllable with no restriction on y0, T or
O. This is due to O. Yu. Imanuvilov (see 6]). The argument uses a xed-point reformulation of the problem and the null controllability properties 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 superlinear at innity, (1.1) is not null control- lable. Indeed, recall that when f(s) = ;s(logjsj)r (r >1) blow-up phenomena may occur. Even when f has \the good sign", null con- trollability may fail. More precisely, when
Cjsj r +1
;C f(s)sCjsjr +1+C (r>1)
and y0 6= 0, there exists T0 >0 such that, ifT <T0, (1.1) is not null controllable (however, under this assumption onf, there existsT1such that, for ally0 2H, (1.1) is null controllable whenever T >T1 see 5]
and 6]).
Hence, the natural question is: What happens if jf(s)jgrows faster than
jsj but slower than all jsjr (r>1) asjsj!1 ?
We are also interested in questions of this kind for systems controlled on a portion of the boundary. More precisely, let be a nonempty open subset of @ and let us put
; =@n:
For each h2L2(), we consider the parabolic problem
8
<
:
@
t
y;y+f(y) = 0 in Q
y= 1h on
y(0) =y0 (1.5)
Esaim: Cocv, April 1997, Vol. 2, pp. 87{103
where f is as before and 1 is the characteristic function of . We will say that (1.5) is null controllable if there exists h such that the corresponding initial-boundary problem possesses a solution y 2 C0(0TH) 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 be specied for generalh2L2(). Nevertheless, we will work with appropriate functionsf and we will nd control functionshwhich are regular enough to guarantee that (1.5) can be 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 boundary controllability). It is also true that, for superlinear nonlinearities, (1.5) is not null controllable in general.
Consequently, it is also natural to study the case in which jf(s)j grows superlinearly but slower than all jsjr.
2. The main results and some remarks
In this paper, we prove that systems (1.1) and (1.5) are null controllable when y0 2C0() and the behavior off(s) as jsj! 1 is, at most, almost critical. More precisely, one has:
Theorem 2.1. Assume f :R7!Ris locally Lipschitz-continuous, f(0) = 0 and
lim
jsj!1
f(s)
slogjsj = 0: (2.1)
Also, assume that y0 2C0(). Then system (1.1) is null controllable.
Theorem 2.2. Under the assumptions of theorem 2.1, system (1.5) is null controllable. More precisely, there exists h, with
1h2L2(0TH3=2(@))\L1()
such that the corresponding system (1.5) possesses at least one solution
y2L
2(0TH1())\C0(0TH)\L1(Q) satisfying y(T) = 0.
Remarks.
1. Assumption (2.1) can be slightly weakened. Indeed, from the proofs in sections 3 and 4, we see there exists a positive constant"="(O T) such that theorems 2.1 and 2.2 still hold when
jf(s)j"jsjlogjsj for largejsj.
On the other hand, the requirement y0 2C0() is unnecessary in practice.
It is sucient to assume the following: The initial datay0and the functionf are such that, when v= 0, system (1.1) possesses a solution y2C0(0 H) which satises y( ) 2 C0(). This happens for all y0 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. FERNANDEZ-CARA
2. We are dealing with quasi-optimal assumptions: recall again that, for
f(s) =jsjr ;1s(r>1), systems (1.1) and (1.5) are not null controllable in general. Also, recall that blow-up may occur for f(s) =;s(logjsj)r.
3. Theorems 2.1 and 2.2 hold for more general problems. Thus, one can replace ; by a general linear elliptic operatorA, with
Ay =;@i(aij(x)@jy) +@i(bi(x)y) +c(x)y:
In this case, the coecients have to be regular enough and the aij must satisfy the usual ellipticity condition. The nonlinear term can also be of the formf(xty), withf :QR7!Rbeing a Carath!eodory function satisfying (for instance):
f(xt0)0 s7!f(xts) is locally Lipschitz
and lim
jsj!1
f(xts)
slogjsj = 0 uniformly in (xt)2Q.
Furthermore, a nonvanishing additional termF =F(xt) can be put in the left in the equations in (1.1) and (1.5). The above results will still be true provided F is suciently small near t =T. For instance, it will suce to
have ZZ
Q
F(xt)2e(T;t)1 2 <+1:
Let us also mention that, for semilinear parabolic equations of higher order, similar results can be obtained.
4. By adapting the arguments in sections 3 and 4, one is led to a new proof of known local results. These must hold for arbitrary locally Lipschitz functions f satisfying f(0) = 0. More precisely, if f is given, there exists
= (O Tf) > 0 such that, for every y0 2 C0() with ky0kC0 , a control v 2L2(0TH) can be found such that the corresponding problem (1.1) possesses a solution y2C0(0TH) which satisesy(T) = 0 (compare with the local results in 1], 4] and 6]).
5. System (1.1) is approximately controllable (in H) if the set R(y0T) formed by all reachable states at time T is dense in H. It has been proved in 2] that, under condition (1.4), system (1.1) is approximately controllable with no restriction on y0,T orO. However, to our knowledge, the approxi- mate controllability of (1.1) under assumption (2.1) is an open question. A similar remark concerning the approximate controllability of system (1.5) can be made.
3. The proof of the boundary controllability result In this section, we prove theorem 2.2. We will rst consider the case in which, for instance, f is C1 in the open interval (;11) and y0 2 C0() for some 2 (01). We will use the standard norm in C0(), given as follows:
kzk
1+ z] :=kzk1+ sup
jz(x);z(x0)j
jx;x 0
j
Under these assumptions, we are going to rewrite (1.5) together with the equalityy(T) = 0 as a xed-point equation in the Banach spaceY =L1(Q).
Esaim: Cocv, April 1997, Vol. 2, pp. 87{103
Then, using (2.1) and some specic properties of linear parabolic equations, we will see that Schauder's theorem can be applied.
Arguments of this kind have already been 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-point formulation is not bounded (this is also the case for the questions considered in 11]). Thus, the methods in 1] and 6] cannot be directly applied and some extra work has to be made.
Let us put
g(s) =
8
<
: f(s)
s
fors6= 0
f
0(0) ats= 0: (3.1)
Then, in view of assumption (2.1), one has:
8 >0 9C such that jg(s)jC+ jlogjsjj 8s2R: (3.2) For each y2Y, we consider the following null controllability problem
8
<
:
@
t
u;u+g(y)u= 0 in Q
u= 1h on
u(0) =y0 u(T) = 0 (3.3)
where the state equation is linear. A solution u = "(y) to (3.3) can be obtained by adapting the method of A.V. Fursikov and O.Yu. Imanuvilov (cf. 5], 3]). More precisely, for the construction of "(y), we do the following:
a) We introduce z=z(y), with
8
<
:
@
t
z;z+g(y)z= 0 in Q
z= 0 on
z(0) =y0: (3.4)
b) We x a function 2C1(0T) such that(t) = 1 neart= 0,(t) = 0 near t = T and 0 1. Then, we set "(y) = z(y) +w(y), with
w(y) being (together with h) the solution furnished by lemma 3.2 to the following problem (see below):
8
<
:
@
t
w;w+g(y)w=;0(t)z(xty) in Q
w= 1h on
w(0) = 0 w(T) = 0:
Our aim is to nd a xed-point inY of the mappingy7!"(y). Obviously, if we are able to prove that such a xed-point exists, theorem 2.2 will have been demonstrated (at least when f isC1 in (;11) and y02C0()).
First, we will show that the mapping y7!"(y) is well dened and, also, that it maps a ball of Y into itself. We need the following elementary result:
Lemma 3.1. Leta2Y and z0 2L1() be given. Then the linear problem
8
<
:
@
t
z;z+a(xt)z= 0 in Q
z= 0 on
z(0) =z0 (3.5)
Esaim: Cocv, April 1997, Vol. 2, pp. 87{103
92 E. FERNANDEZ-CARA
possesses exactly one solution z2L2(0TH01())\Y, with
kz(t)k1etkakYkz0k1 8t20T]: (3.6) We will also use the following result, whose proof is given in section 5:
Lemma 3.2. To each couple fak g, with ak2Y such that
k= 0 for 0<t<b and T;b<t<T
one can assign a solution fw hgto the null linear controllability problem
8
<
:
@
t
w;w+a(xt)w=k(xt) in Q
w= 1h on
w(0) = 0 w(T) = 0 (3.7)
in such a way that
w2L
2(0TH2())\Y @tw2L2(Q) (3.8) 1h2L2(0TH3=2(@))\L1() (3.9) and
kw k
L 2
(0TH 2
())+k@tw kL2(Q)+kw kY eC(1+kakY)kk kY: (3.10) Here, the constant C can depend on ,,T and b, but not on k.
Remark. Lemma 3.2 will also be used in section 4, for the proof 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) is a well dened mapping fromY into itself. Moreover, one has
k"(y)kY eC(1+kg (y )k Y
)
ky
0 k
1
8y 2Y (3.11)
where C can only depend on , and T. Hence, in view of (3.2), one sees that " maps a ball BR of Y into itself.
It is clear that y 7! "(y) is a continuous mapping. Let us now prove that, for some 2(01), it maps bounded sets in Y into bounded sets in
C
0=2(Q). This will suce to our purpose, since this space is compactly embedded in Y. Recall that C0=2(Q) is the Banach space formed by all functions u2C0(Q) such that
u]=2 := sup
Q
ju(xt);u(x0t)j
jx;x 0
j
+ sup
Q
ju(xt);u(xt0)j
jt;t 0
j
=2
<+1:
The (natural) norm in C0=2(Q) is as follows:
kuk
=2 :=kukY + u]=2
We will use the following lemma, which is proved in 7] (see theorems 7.1 and 10.1, Ch. III):
Esaim: Cocv, April 1997, Vol. 2, pp. 87{103
Lemma 3.3. Assume ak 2 Y, eh 2 L2(0TH1=2(@))\L1() and z0 2
C
0(), where 2(01). Then the linear problem
8
>
<
>
:
@
t
u;u+a(xt)u=k(xt) in Q
u=eh on
u(0) =z0
possesses exactly one solution u2L2(0TH1())\Y. Furthermore, there exists 2(0) (only depending on , T and ) such that u2C0=2(Q) and there exists a function M1 (increasing with respect to its last argument) such that
u]=2M1(Tkz0k1+ z0]kukY): (3.12) We deduce that, for some , all functions "(y) belong to C0=2(Q).
Combining (3.12) with (3.11), we also deduce that, for some M2 (again increasing with respect to its last argument), one has:
k"(y)k=2M2(Tky0k1+ y0]kg(y)kY):
This shows that " maps bounded sets ofY into bounded sets ofC0=2(Q).
We have thus seen that Schauder's theorem can be applied to the xed- point equation y = "(y),y 2Y. This proves theorem 2.2 when f is C1 in (;11) and y02C0().
In the general case, one can always put
f = lim
n!1 f
n uniformly in R,
for some locally Lipschitz functions fn which are C1 in (;11) and satisfy
f
n(0) = 0, and fn =f outside (;22). One can also put
y
0 = lim
n!1 y
n
0 uniformly in ,
for some functions y0n2 C0(). For each n 1, let fynhng be a couple satisfying
8
<
:
@
t y
n
;yn+fn(yn) = 0 in Q
y
n= 1hn on
y
n(0) =yn0 yn(T) = 0 (3.13) constructed as above. For
g
n(s) =
f
n(s)
s
fors6= 0
f 0
n(0) ats= 0 it is not dicult to see using (2.1) that
8 >0 9C such that jgn(s)jC+ jlogjsjj 8s2R8n1: Consequently, one has kynkY Const. (use the corresponding estimates (3.11)). Notice that yn can be written in the form yn= zn+wn, where
z
n is uniformly bounded in L2(0TH01())\Y and@tzn is uniformly bounded in L2(0TH;1()), see (3.5) and (3.6).
w
ncan be estimated as in (3.10).
Esaim: Cocv, April 1997, Vol. 2, pp. 87{103
94 E. FERNANDEZ-CARA
Hence, it can be assumed thatyn converges strongly in L2(Q) and a.e. and 1hn converges (at least) weakly in L2(0TH3=2(@)). We can thus take limits in (3.13) asn!1and obtain (1.5) together with the equalityy(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 thatf isC1 in the open interval (;11) andy0 2C0() for some 2(01). Again, we will use the function ggiven by (3.1).
We will introduce a mapping y 7! $(y) such that, for each y 2 Y, u =
$(y) is, together with some v 2 L2(0TH), a solution to the following problem:
8
<
:
@
t
u;u+g(y)u= 1Ov in Q
u= 0 on
u(0) =y0 u(T) = 0: (4.1)
This is made as follows:
a) First, we x a nonempty open, simply connected and regular setO0
O, we put
=@O0 ; =@
and we denote by Dthe open set bounded by and ;, i.e.D= nO0. b) Using lemma 3.2 in D(0T), we assign toy a couplefw^(y)^h(y)g satisfying
8
>
<
>
:
@
tw^; ^w+g(y)^w=;0(t)z(xty) in D(0T)
^
w= 1^h on @D(0T)
^
w(0) = 0 and ^w(T) = 0 in D. (4.2)
Here, z=z(y) is the (unique) solution to (3.4). One has
^
w2L
2(0TH2(D))\L1(D(0T)) @tw^2L2(D(0T)) (4.3) 1^h2L2(0TH3=2(@D))\L1(@D(0T)) (4.4) and
kw k^ L1(D (0T))eC(1+kg (y )k Y
)
ky
0 k
1 (4.5)
whereCcan depend only on ,O0andT. We will denote by &wthe extension- by-zero of ^w to the whole set Q.
c) Let us setw=w& andv=;2rrw;& ()&w+(1;)0z(y). Here,
=(x) is aC1 cut-o function satisfying:
= 1 in a neighborhood of nO,
= 0 in a neighborhood ofO0
and 0 1. Then, we put $(y) = z(y) +w(y), with being as in section 3, i.e. a function satisfying 2 C1(0T), (t) = 1 near t = 0,
(t) = 0 neart=T and 01.
Esaim: Cocv, April 1997, Vol. 2, pp. 87{103
It is not dicult to see that u = $(y) is (together with v) a solution to (4.1). From (4.3){(4.5) and (3.6), one has
u2L
2(0TH01())\Y v2L2(0TH) and
k$(y)kY eC(1+kg (y )k Y
)
ky
0 k
1
Consequently, $ maps a ball BR 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 $(y)2C0=2(Q) and
k$(y)k=2M3(Tky0k1+ y0]kg(y)kY)
for all y2Y and for some 2(0) (once more,M3 is increasing with re- spect to its last argument). Hence, $ maps bounded sets ofY into bounded sets of C0=2(Q) and we can apply Schauder's theorem to the xed-point equation y= $(y),y2Y. This proves theorem 2.1 whenf isC1 in (;11) and y0 2C0().
In order to prove the same result in the general case, we can argue as in section 3. Thus, introducing the functions fnandy0n, we nd for eachn1 a couple fynvng satisfying
8
<
:
@
t y
n
;yn+fn(yn) = 1Ovn in Q
y
n = 0 on
y
n(0) =y0n yn(T) = 0 (4.6) and
ky
n k
Y
Const. kvnkL2(0TH)Const.
Hence, it can be assumed thatyn converges strongly in L2(Q) and a.e. and 1vnconverges (at least) weakly inL2(Q). Taking limits in (4.6), we obtain (1.1) andy(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(0T). We will present an argument, taken from 5] and 3], which leads to a function we that solves this problem in a generalized sense. Then, we will assign towea couple fw hgsatisfying (3.7) and we will check that (3.8), (3.9) and (3.10) are satised. Accordingly, the proof will be divided in several steps.
Step 1: The analysis of an auxiliary problem { Let and @ be as before and assume that ak 2 Y, with k= 0 for 0 <t <b and also for T ;b< t< T. Let G RN be a bounded, regular and open set with
G and @\@G= ; =@n. LetS0 @Gbe an open neighborhood of ; (S0 6=@G) and let us put
S
1 =@GnS0 =e @G(0T) and ei=Si(0T)
for i= 12. We will denote by eaand ek the extensions-by-zero to the whole set G(0T) of the functions a and k, respectively. We will need the following
Esaim: Cocv, April 1997, Vol. 2, pp. 87{103
96 E. FERNANDEZ-CARA
Lemma 5.1. There exists a positive function0 2C2(G) such thatr06= 0 in Gand @n00 on S0.
Here (and also in the sequel),@nstands for the outward normal derivative.
Lemma 5.1 is proved in 6] (see 1] for another similar but di erent result).
Let us introduce the space
Z
0 =fq2C2(G0T]) q= 0 on ,e @nq= 0 on e1g:
We will need a second lemma:
Lemma 5.2. There exists a positive function (which is of class C2 in the set G(0T) and depends only on G, S1 and T) and there exist constants
s
(GS1TkakY) and C(GS1T) such that the following inequalities hold for all ss and for all q2Z0:
8
>
>
>
>
>
>
<
>
>
>
>
>
>
: 1
s ZZ
;2s
t(T ;t);j@tqj2+jqj2 +s
ZZ
;2s
t
;1(T;t);1jrqj2+s3
ZZ
;2s
t
;3(T;t);3jqj2
C
ZZ
;2s
j@
t
q+ q;eaqj2:
(5.1)
Here, the integrals are extended to G(0T). Furthermore,s can be chosen of the form 1+2kak2=3Y , where 1 and2 only depend onG, S1 and T.
The inequalities (5.1) are global Carleman's estimates associated to the adjoint system (1.2). The main ideas needed for the proof of lemma 5.2 are presented in 5] and 3]. Nevertheless, in order to clarify the situation as much as possible, we will give a complete proof of this lemma in the following section. This will serve to show that we can chooses as indicated above. Also, it will be seen that an admissible choice for is
(xt) = exp
1(x)
t(T ;t)
where 1 = exp(&);exp(0), and & are large enough and 0 is the function furnished by lemma 5.1.
Let us come back to the proof of lemma 3.2. Let us x sandsuch that (5.1) holds. Then
pq] =
ZZ
G(0T)
;2s(@tp+ p;eap)(@tq+ q;eaq)
is a scalar product in Z0. LetZ be the completion of Z0 for ]. Ifq 2Z, the functions ;2st(T;t)j@tqj2, ;2st(T ;t)jqj2, etc. are integrable and (5.1) is satised. Let us put
hl qi=;
ZZ
G(0T) e
kq 8q2Z :
Then, l is a bounded linear form on Z. Indeed, using H(older's inequality and (5.1), one has:
ZZ
G(0T) e
kq E(GS1Tk s)qq]1=2 8q2Z (5.2)
Esaim: Cocv, April 1997, Vol. 2, pp. 87{103
