Boundary stabilization of parabolic nonlinear equations

0.22 MB·English

by  Munteanu Ionut

STABILIZATION OF PARABOLIC SEMILINEAR EQUATIONS
6
IONUT¸ MUNTEANU
1
0
2

r
p
Abstract. Wedesignhereafinite-dimensionalfeedbackstabilizingDirichlet
A
boundarycontrollerfortheequilibriumsolutionstoparabolicequations. These
9 results extend that ones in[1], whichprovide afeedback controller expressed
1 in terms of the eigenfunctions φj corresponding to the unstable eigenvalues
{λj}Nj=1 of the operator corresponding to the linearized equation. In [1], the
] stabilizability result is conditioned by the require of linear independence of
C N
O n∂∂νφjoj=1,onthepartoftheboundarywherecontrolacts. Inthiswork,we
designasimilarcontrol asin[1],andshowthat itassuresthestabilityof the
. system. This time,wedropthe requireof linearindependence andany other
h
additional hypothesis. Some examples are provided in order to illustrate the
t
a acquired results. More exactly, boundary stabilization of the heat equation
m andtheFitzhugh-Nagumoequation.
[ Keywords: semilinear parabolic equations, feedback controller, eigenvalue.

2000 MSC: 35K91; 93D15; 80A20.
2
v
7 1. introduction
3
7 Weconsiderthefollowingparabolicboundarystabilizationprobleminabounded
5 domainΩ⊂Rd, with a smooth boundary∂Ω, split in two,i.e., ∂Ω=Γ ∪Γ , such
1 2
0
that the Lebesgue measure σ(Γ )6=0:
. 1
1
0 pt(t,x)−∆p(t,x)+f(x,p)=0, t>0, x∈Ω,
5 p(t,x)=u(t,x), t>0, x∈Γ ,
 1 (1.1)
1 ∂ p(t,x)=0, t>0, x∈Γ ,
v:  p∂(ν0,x)=po(x), x∈Ω. 2
i
X Here,ν standsfortheunitoutwardnormalontheboundary∂Ω;po ∈L2(Ω)isgiven
r (theinitialdata); f isanonlinearfunctionforwhich,dependingonthecontext,we
a
choose from the following two hypotheses
(f ) f,f ∈C(Ω×R);
1 p
(f ) f,f ,f ∈ C(Ω×R), and there exist C > 0, q ∈ N, α > 0, i = 1,...,q,
2 p pp 1 i
when d=1,2, and 0<α ≤1, i=1,...,q, when d=3, such that
i
q
|f (x,p)|≤C |p|αi +1 , ∀p∈R.
pp 1
!
i=1
X
Here,f ,f denotethepartialderivativesoff withrespecttoitssecondargument.
p pp
On the part of the boundary Γ it is applied the control u, while Γ is insulated.
1 2
Let p ∈C1(Ω) be an equilibrium solution to (1.1), that is, p satisfies
e e
∂
−∆p +f(x,p )=0 in Ω; p =0 on Γ , p =0 on Γ .
e e e 1 ∂ν e 2
1
2 IONUT¸ MUNTEANU
Then,defining the fluctuationvariablew :=p−p ,equation(1.1)canbe rewritten
e
as
w (t,x)−∆w(t,x)+f(x,w+p )−f(x,p )=0, t>0, x∈Ω,
t e e
w(t,x)=v:=u(t,x)−p (x), t>0, x∈Γ ,
 e 1 (1.2)
∂ w(t,x)=0, t>0, x∈Γ ,
 w∂ν(0,x)=w (x):=p (x)−p2 (x), x∈Ω.
o o e

Our main concern here is the design of a boundary feedback controller u that
stabilizes exponentially the equilibrium state p in (1.1), or, equivalently, the zero
e
solution in (1.2).
The problem of boundary feedback stabilization of this kind of parabolic-like
equations was first solved in the pioneering work [14]. Then, several others meth-
ods were proposed for deriving new types of controls, like backstepping approach,
see [4, 5, 8, 15], or the proportionaltype controllersdesignedin [1, 2, 13]. The last
ones provide a simple form feedback stabilizer expressed in terms of the unstable
eigenfunctions of the operator given by the linearized equation aroundthe equilib-
rium p (that is the operator A below), which is easily implementable in practice.
e
However, the stabilizing procedure is conditioned by the require that the system
∂ φ N is linearly independent in L2(Γ ), where, φ , i = 1,...,N, are the cor-
∂ν i i=1 1 i
responding first N eigenfunctions. In the present work, we extend these results in
(cid:8) (cid:9)
thesensethatwedropthe hypothesisoflinearindependence (which,infact,seems
tobe quiterestrictive,asshownin[1]). Moreprecisely,wedesignasimilartype of
controlas that one in [1], and show that it assures the stability of the steady-state
p , without needing the linear independence assumption. Roughly saying,the idea
e
is as follows: in the case when the system ∂ φ N is not linearly independent,
∂ν i i=1
the corresponding Gram matrix is singular, and, consequently, the approach in [1]
(cid:8) (cid:9)
fails because the controller is not well-defined anymore. However, we overcome
this in the next manner: for the beginning, in order to clarify the procedure, we
work under the assumption that the first N ∈ N unstable eigenvalues are simple.
We involve N matrices obtained from the Gramian, multiplied by some diagonal
matrices, and show that their sum is invertible, even if each one is not. Therefore,
we may well-define u as the sum of N different controllers obtained from those
matrices, and arrive to similar stabilization results as in [1]. The case of general
eigenvalues involves a similar boundary feedback, as the one described above. The
idea is to artificially make the eigenvalues simple again by slightly changing the
linear operator. Finally, via a fixed point argument, locally stabilization results
may be deduced for the fully nonlinear system (1.1).
On this subject, it should be mentioned also the work [6], where controllers, of
the abovetype, aredesigned. Insteadofthe eigenfunctions system, they showthat
there exists a (possible different) family of functions for which, the corresponding
proportional controller assures the stability. That result is based on some unique
continuationpropertyfortheinvolvedoperators. Also,thework[11],ontheNavier-
Stokes equations, provide similar proportional feedbacks, but with the coefficients
givenbyRiccatialgebraicequations. However,incomparisontothoseresults,here,
the form of the feedback is given explicitly, and no additional assumption to the
classical context is needed.
STABILIZATION OF PARABOLIC SEMILINEAR EQUATIONS 3
2. The design of the feedback law and the main results
Inthissection,wedetailthedesignofthenewfeedback,showhowitdiffersfrom
that one in [1], and state the main stabilization results. To this end, let us define
the operator A:D(A)→L2(Ω) as
∂
Ay :=−∆y+a(x)y, ∀y ∈D(A)= y ∈H2(Ω):y| =0, y| =0 ,
Γ1 ∂ν Γ2
(cid:26) (cid:27)
where a(x) := f (x,y (x)), x ∈ Ω. Notice that A is self-adjoint, besides this, it
y e
can be shown that A has compact resolvent, therefore, it has a countable set of
∞
eigenvalues, denoted by {λ } (repeated accordingly to their multiplicity) such
j j=1
that, given ρ > 0, there exists only a finite number of eigenvalues {λ }N with
i i=1
λ < ρ, i = 1,...,N (we call them the unstable eigenvalues). So, λ ≥ ρ, for all
i i
i≥N+1(for more details,see [1]). As we see below,the largerρ is, the faster the
exponential decay of the solution is. (But, on the other hand, the larger ρ is, the
∞
largerdimension,N,ofthefeedbackis.) Letusdenoteby{φ } thecorresponding
i i=1
eigenfunctions, that can be chosen in such a way to form an orthonormal basis of
the space D(A).
2.1. Thecaseofsimpleeigenvalues. Forthebeginning,inordertomakeclearer
our approach, we add to the above context the following hypothesis:
(H) The eigenvalues λ , i=1,...,N, are mutually distinct.
i
In other words, we assume that the first N eigenvalues are simple. So, we can
(eventually) rearrange them such that
λ <λ <...<λ .
1 2 N
Then,inthesubsectionbelow,weshowhowwemaydropthisassumption,inorder
to obtain stabilization results in the framework of general eigenvalues.
Oneofthetricksusedhereisto”lift”theboundaryconditionsintotheequations.
For this, we introduce the so-called Dirichlet operator as: given α ∈ L2(Γ ) and
1
γ >0, we denote by D α:=y, the solution to the equation
γ
N
−∆y(x)+a(x)y(x)−2 λ hy,φ iφ (x)+γy(x)=0, for x∈Ω,
k k k
 k=1 (2.1)
 y =α, on Γ1, X
∂ y =0, on Γ .
∂ν 2
Forγ > 0 large enough equation (2.1) has a solution, defining so the map Dγ ∈
L(L2(Γ1),H21(Ω)). Besides this, we have
C
kD αk ≤ kαk , ∀α∈L2(Γ ).
γ H12 γ L2(Γ1) 1
(see, e.g., [10, p.6, line 16]) We choose ρ < γ < γ < ... < γ , N constants such
1 2 N
that equation (2.1) is well-posedfor eachof them, and denote by D , i=1,...,N,
γi
the corresponding Dirichlet operator.
Next,letusdenotebyBtheGrammatrixofthesystem ∂ φ N intheHilbert
∂ν i i=1
space L2(Γ ), with the standard scalar product hg,hi := f(x)g(x)dσ (σ being
1 0 (cid:8)Γ1 (cid:9)
R
4 IONUT¸ MUNTEANU
the corresponding Lebesgue measure on the boundary Γ ). That is
1
∂ φ , ∂ φ ∂ φ , ∂ φ ... ∂ φ , ∂ φ
∂ν 1 ∂ν 1 0 ∂ν 1 ∂ν 2 0 ∂ν 1 ∂ν N 0
∂ φ , ∂ φ ∂ φ , ∂ φ ... ∂ φ , ∂ φ
B:= (cid:10)∂ν 2 ∂ν 1(cid:11)0 (cid:10)∂ν 2 ∂ν 2(cid:11)0 (cid:10)∂ν 2 ∂ν N(cid:11)0 . (2.2)
... ... ... ...
(cid:10) (cid:11) (cid:10) (cid:11) (cid:10) (cid:11)
 ∂ φ , ∂ φ ∂ φ , ∂ φ ... ∂ φ , ∂ φ 
 ∂ν N ∂ν 1 0 ∂ν N ∂ν 2 0 ∂ν N ∂ν N 0 
 
Further, we in(cid:10)troduce the m(cid:11)atr(cid:10)ices (cid:11) (cid:10) (cid:11)
1 0 ... 0
γk−λ1
0 1 ... 0
Λγk := ... γk.−..λ2 ... ... , k =1,...,N, (2.3)
 0 0 ... 1 
 γk−λN 
 
1 ∂ φ (x) 1 ∂ φ (x) ... 1 ∂ φ (x)
γ1−λ1∂ν 1 γ1−λ2∂ν 2 γ1−λN ∂ν N
1 ∂ φ (x) 1 ∂ φ (x) ... 1 ∂ φ (x)
T := γ2−λ1∂ν 1 γ2−λ2∂ν 2 γ2−λN ∂ν N , (2.4)
... ... ... ...
 1 ∂ φ (x) 1 ∂ φ (x) ... 1 ∂ φ (x) 
 γN−λ1∂ν 1 γN−λ2∂ν 2 γN−λN ∂ν N 
 
and
A=(B +B +...+B )−1, (2.5)
1 2 N
where
B :=Λ BΛ , k =1,...,N. (2.6)
k γk γk
Atthisstage,weareabletogivetheformofthestabilizingfeedback,but,before
this let us take a pause and make some comments on the invertibility of the sum
B +...+B ,whichis,infact,the keyresultofthispaper. Moreexactly,sincethe
1 N
system ∂ φ N isnotnecessarilylinearlyindependentinL2(Γ ),the gramianB
∂ν i i=1 1
may be singular, and, therefore, each B ,k = 1,...,N, may be singular. However,
k
(cid:8) (cid:9)
by Lemma 5.2 in the Appendix, the sum B +...+B is an invertible matrix,
1 N
provided that hypothesis (H) holds true. Therefore, the matrix A is well-defined.
Now, let us introduce the feedbacks
hp(t)−pe,φ1i γk−1λ1∂∂νφ1(x)
hp(t)−p ,φ i 1 ∂ φ (x)
uk(t,x)= A ... e 2 , γk−λ2.∂.ν. 2  +pe, t≥0, x∈Γ1,
* +
 hp(t)−pe,φNi   γk−1λN ∂∂νφN(x)  N
    (2.7)
for k = 1,2,...,N. Here h·,·i denotes the standard scalar product in L2(Ω); while
h·,·i denotes the standard scalar product in RN.
N
In the case where the Gram matrix B is nonsingular, one may consider only
one feedback from the above list, for example u (since, in this case, B−1 exists),
1 1
take u=u and show that it assures the stability stated in the main results of the
1
paper, Theorems 2.1, 2.2 below. In this case, in fact, we stumble on the feedback
designed in [1].
When B is not invertible anymore, we take u to be the sum
u=u +u +...+u ,
1 2 N
STABILIZATION OF PARABOLIC SEMILINEAR EQUATIONS 5
which, in a condensed form, can be written as
hp(t)−p ,φ i 1
e 1
hp(t)−p ,φ i 1
u= T A ... e 2 , ...  +pe. (2.8)
* +
 hp(t)−pe,φNi   1  N
   
We claim that the above feedback assures the stability of the steady-state p in
e
(1.1). In what follows,we focus to provethis is indeed so. The main ingredientto-
wardtheproposedgoalisthestabilizationofthelinearizedequationcorresponding
to (1.2), given by,
y (t,x)−∆y(t,x)+a(x)y(t,x)=0, t>0, x∈Ω,
t
y(t,x)=v(t,x), t>0, x∈Γ ,
 1 (2.9)
∂ y(t,x)=0, t>0, x∈Γ ,
 y∂(ν0,x)=y , x∈Ω. 2
o
The followingresults amounts to saying that the feedback u achieves global expo-
nential stability in the linear system (2.9), and local exponential stability in (1.1).
More precisely
Theorem2.1. Assumethathypothesis(H)holdsandf obeys(f ). Thefeedbacku,
1
given by (2.8), exponentially stabilizes the linearized equation (2.9). More exactly,
the solution y to the equation
y (t,x)−∆y(t,x)+f (x,y (x))y(t,x)=0, t>0, x∈Ω,
t y e
hy(t),φ i 1
1

hy(t),φ i 1
 y(t,x)=*T A hy(t)..,.φN2i , .1.. +N, t>0, x∈Γ1, (2.10)
∂ y(t,x)=0, t>0, x∈Γ ,   
∂ν 2
satisfiesyth(0e,exx)p=onyeon,tiaxl∈deΩca,y
ky(t)k2 ≤Ce−µtky k2, t≥0, (2.11)
o
for a prescribed µ > 0, and a constant C > 0. Here T is introduced in relation
(2.4), while A is introduced in relation (2.5). k·k denotes the norm in L2(Ω).
Theorem 2.2. Assume that hypotheses (H),(f ) hold true. Then, the solution to
2
the closed-loop nonlinear equation
p (t,x)−∆p(t,x)+f(x,p)=0, t>0, x∈Ω,
t
hp(t)−p ,φ i 1
e 1

hp(t)−p ,φ i 1
 p(t,x)=*T A hp(t)−..p.ee,φN2i , .1.. +N +pe(x), t>0, x∈Γ1,
∂ p(t,x)=0, t>0, x∈Γ ,   
∂ν 2
satisfipe(s0,txhe)=exppoon,exnt∈iaΩl ,decay (2.12)
kp(t)−p k2 ≤Ce−µtkp −p k2, t≥0,
e o e
6 IONUT¸ MUNTEANU
for a prescribed µ > 0, and a constant C > 0, provided that kp −p k is small
o e
enough. Here T is introduced in relation (2.4), while A is introduced in relation
(2.5).
The proofs of these two theorems will be given latter, in Section 4.
2.2. Thecaseofgeneraleigenvalues. Asannouncedabove,weseethat,slightly
perturbing the linear operator A, we are able to show that the present approach
works for the case of general unstable eigenvalues (i.e., not necessarily simple).
Indeed, let us assume, for example, that the first unstable eigenvalue has its mul-
tiplicity equal to 2, i.e., λ = λ , and the others unstable eigenvalues are simple
1 2
(other cases can be treated in a similar manner as below).
This time, the Dirichlet operator is introduced as follows: for any α ∈ L2(Γ ),
1
we define D α:=y, where y is solution to
γ
N
−∆y(x)+a(x)y(x)−2 λ hy,φ iφ (x)−δhy,φ iφ +γy(x)=0, x∈Ω,
k k k 1 1

k=1
 y =α on Γ1, X
∂y =0 on Γ ,
∂ν 2
for some δ >0. We choose ρ<γ <γ :=γ + 1 <γ :=γ + 1 ...<γ(2.1:3=)
1 2 1 N−1 3 1 N−2 N
γ +1,with γ largeenough,suchthat, forδ = 1 equation(2.13)is well-posedfor
1 1 γ4
1
each γ , i=1,...,N, and such that λ +δ,λ ,...,λ are mutually distinct.
i 1 2 N
Now, our feedback has the form
hp(t)−p ,φ i 1
e 1
hp(t)−p ,φ i 1
u= T A ... e 2 , ...  +pe. (2.14)
* +
 hp(t)−pe,φNi   1  N
T is defined as    
1 ∂ φ (x) 1 ∂ φ (x) ... 1 ∂ φ (x)
γ1−δ−λ1∂ν 1 γ1−λ2∂ν 2 γ1−λN ∂ν N
1 ∂ φ (x) 1 ∂ φ (x) ... 1 ∂ φ (x)
T := γ2−−δ−λ1∂ν 1 γ2−λ2∂ν 2 γ2−λN ∂ν N  (2.15)
... ... ... ...
 1 ∂ φ (x) 1 ∂ φ (x) ... 1 ∂ φ (x) 
 γN−δ−λ1∂ν 1 γN−λ2∂ν 2 γN−λN ∂ν N 
 
and
A=(B +B +...+B )−1, (2.16)
1 2 N
where
B :=Λ BΛ , k =1,...,N, (2.17)
k γk γk
for
∂ φ , ∂ φ ∂ φ , ∂ φ ... ∂ φ , ∂ φ
∂ν 1 ∂ν 1 0 ∂ν 1 ∂ν 2 0 ∂ν 1 ∂ν N 0
∂ φ , ∂ φ ∂ φ , ∂ φ ... ∂ φ , ∂ φ
B:= (cid:10)∂ν 2 ∂ν 1(cid:11)0 (cid:10)∂ν 2 ∂ν 2(cid:11)0 (cid:10)∂ν 2 ∂ν N(cid:11)0  (2.18)
... ... ... ...
(cid:10) (cid:11) (cid:10) (cid:11) (cid:10) (cid:11)
 ∂ φ , ∂ φ ∂ φ , ∂ φ ... ∂ φ , ∂ φ 
 ∂ν N ∂ν 0 ∂ν N ∂ν 3 0 ∂ν N ∂ν N 0 
 
and (cid:10) (cid:11) (cid:10) (cid:11) (cid:10) (cid:11)
1 0 ... 0
γk−δ−λ1
0 1 ... 0
Λγk := ... γk.−..λ2 ... ... , k =1,...,N. (2.19)
 0 0 ... 1 
 γk−λN 
 
STABILIZATION OF PARABOLIC SEMILINEAR EQUATIONS 7
By the above definitions, taking into account that λ +δ,λ ,...,λ are mutually
1 2 N
distinctandγ ,γ ,...,γ arealsomutually distinct, onemay showthatthe matrix
1 2 N
A is well-defined, that is, the sum B +B +...+B is indeed invertible (this
1 2 N
follows similarly as in Lemma 5.1 and Lemma 5.2 in the Appendix).
The main results, in the context of general eigenvalues, are stated below.
Theorem 2.3. Assumethat f satisfies (f ). Then, thefeedback u, given by(2.14),
1
exponentially stabilizes the linearized system (2.9). More precisely, the solution y
to the system
y (t,x)−∆y(t,x)+f (x,y (x))y(t,x)=0, t>0, x∈Ω,
t y e
hy(t),φ i 1
1

hy(t),φ i 1
 y(t,x)=*T A hy(t)..,.φN2i , .1.. +N, t>0, x∈Γ1, (2.20)
∂ y(t,x)=0, t>0, x∈Γ ,   
∂ν 2
satisfiesyth(0e,exx)p=onyeon,tiaxl∈deΩca,y
ky(t)k2 ≤Ce−µtky k2, t≥0, (2.21)
o
for a prescribed µ > 0, and a constant C > 0. Here T is introduced in relation
(2.15), while A is introduced in relation (2.16). k·k denotes the norm in L2(Ω).
And, concerning the nonlinear equation.
Theorem 2.4. Let 1≤d≤3 and f obeying assumption (f ). The solution to the
2
closed-loop nonlinear system
p (t,x)−∆p(t,x)+f(x,p)=0, t>0, x∈Ω,
t
hp(t)−p ,φ i 1
e 1

hp(t)−p ,φ i 1
 p(t,x)=*T A hp(t)−..p.ee,φN2i , .1.. +N +pe(x), t>0, x∈Γ1,
∂ p(t,x)=0, t>0, x∈Γ ,   
∂ν 2
satisfipe(s0,txhe)=exppoon,exnt∈iaΩl ,decay (2.22)
kp(t)−p k2 ≤Ce−µtkp −p k2, t≥0,
e o e
for a prescribed µ > 0, and a constant C > 0, provided that kp −p k is small
o e
enough. Here T is introduced in relation (2.15), while A is introduced in relation
(2.16).
The proofs will be given latter, in Section 4.
3. Examples
In order to illustrate the acquired results, let us consider some examples arising
fromthermodynamics and biology. These examples are treatedbriefly, since we do
not want to oversize the present work. However, more details and also numerical
simulations can be found in the work [12] on the stabilization of the Fischer’s
equation by similar proportional feedbacks as above.
8 IONUT¸ MUNTEANU
Example 1. We consider first a problem discussed in [4], and also in [1]. More
precisely, the stabilization of the heat equation on the rod (0,1)
y −y −λy, x∈(0,1), t>0,
t xx (3.1)
y (t,0)=0, y(t,1)=u(t), t>0,
x
(cid:26)
with the Dirichlet actuation u in x=1 and with λ a positive constant parameter.
First of all, we notice that the stabilizing results using the backsteppting method
in [4] work for arbitrarly level of instability, while in [1] they are applicable un-
der the condition that the parameter λ do not exceed the bound 3π 2. In the
2
present paper, Theorem 2.1 above holds true without any a priori condition on
λ. This is indeed so, because, in our case, the operator Ay = −y(cid:0)′′ −(cid:1)λy, ∀y ∈
D(A) = y ∈H2(0,1): y′(0)=y(1)=0 has the eigenvalues λ = (2j−1)2π2 −λ
j 4
with the corresponding eigenfunctions φ = cos(2j−1)πx, j = 1,2,.... It is clear
(cid:8) j(cid:9) 2
that hypothesis (H) holds true for any N ∈ N. Let us pick some N such that
λ >ρ>0, ∀j =N +1,N +2,.... Since system (3.1) is linear and Theorem 2.1 is
j
applicable we get that the feedback
1y(x)cosπxdx 1
0 2
1y(x)cos3πxdx 1
u= TA R0 2 ,  ,
... ...
* +
 01Ry(x)cos(2N−21)πxdx   1  N
   
where R
− 2π 6π ... (−1)N 2(2N−1)π
l1−π2 l1−(3π)2 l1−[(2N−1)π]2
− 2π 6π ... (−1)N 2(2N−1)π
T = l2−π2 l2−(3π)2 l2−[(2N−1)π]2 ,
... ... ... ...
 − 2π 6π ... (−1)N 2(2N−1)π 
 lN−π2 lN−(3π)2 lN−[(2N−1)π]2 
 
and
A=
1 −3 ... (−1)N+1(2N−1) −1
(lk−π2)2 [lk−π2][lk−(3π)2] [lk−π2][lk−(2N−1)2π2]
 N  −3 9 ... (−1)N3(2N−1) 
(h=ereXkw=e14hπa2vede[lnko−[(ltπk−e−2d1]π)[lNb2k]+−y[.l1.(k.(l2−2kNN(3−:−=π1)1)2)24]π(2γ]k +−λ[l)k,−k((3−π[=1l)k)2N−]1[+(l.,k3.1..π−3..)((,222N]NN2−−),11))a2sπs2u]res......the[lgkl−o([b3lkaπ−l)2((e]22[xNNlk.p−.−−.o1(1-2))2N2π−2]12)2π2]  ,
nential stability of the null solution of (3.1). Here, ρ<γ <γ <...<γ are any
1 2 N
positive constants.
Example 2. We consider now the classical Fitzhugh-Nagumo equation that de-
scribes the dynamics of electrical potential across cell membrane (for details, see
[9])
y −y +y(y−1)(y−a)=0, 0<x<l, t>0,
t xx (3.2)
y (t,l)=0, y(t,0)=u(t), t>0,
x
(cid:26)
where 0 < a < 1. It is known that the equilibrium y = a is unstable. The
2 e
linearized operator A is given by
Ay =−y′′−a(1−a)y, ∀y ∈D(A)= y ∈H2(0,l): y(0)=y′(l)=0 .
(cid:8) (cid:9)
STABILIZATION OF PARABOLIC SEMILINEAR EQUATIONS 9
It has the simple eigenvalues λ = (2j−1)2π2 − a(1 − a), j = 1,2,..., with the
j 4l2
corresponding eigenfunctions φ = sin(2j−1)πx, j = 1,2,.... Since hypothesis (H)
j 2l
isfullfiledforanyN ∈N, wechoseonesuchthatλ >ρ>0, j =N+1,N+2,....
j
It is easy to see that f(y)=y(y−1)(y−a) obeys the hypothesis of Theorem 2.2,
and so we get that the feedback
l(y(x)−a)sin πxdx 1
0 2l
u= TA R0l(y(x)−a)sin32πlxdx , 1  +a,
... ...
* +
 l(Ry(x)−a)sin(2N−1)πxdx   1 
 0 2l    N
   
where R
2lπ 6lπ ... 2(2N−1)l
l1−π2 l1−(3π)2 l1−(2N−1)2π2]
2lπ 6lπ ... 2(2N−1)l
T = l2−π2 l2−(3π)2 l2−(2N−1)2π2] ,
... ... ... ...
 2lπ 6lπ ... 2(2N−1)l 
 lN−π2 lN−(3π)2 lN−(2N−1)2π2] 
with l :=4l2(γ +a(1−a)), k =1,2,...,N; and 
k k
2 −1
1 3 ... 2N−1
lk−π2 [lk−π2][lk−(3π)2] [lk−π2][lk−(2N−1)2π2]
A=(2l)2π2kXN=1 [lk−(cid:16)π22]N[.l3.k−.−1(cid:17)(3π)2] (cid:16)lk3(−2(N3.3..π−)12)(cid:17)2 ......... [lk−(3π)23]2([N2lkN.−−..−1(21N) −1)22π2]  ,
locally stabilizes the s[olkl−uπti2o][nlk−ye(2=N−a1)i2nπ2(]3.2)[l.k−(3π)2][lk−(2N−1)2π2] (cid:16)lk−(2N−1)2π2(cid:17) 
Example 3. Finally, we consider the periodic heat equation in (0,π)2
y −∆y−µy =0, x∈(0,π)2, t>0,
t
y(t,x ,0)=u(t), y (t,x ,π)=0, x ∈(0,π), (3.3)
 1 x2 1 1
y (t,0,x )=y (t,π,x )=0, x ∈(0,π).
 x1 2 x1 2 2
In this case the operator

Ay =−∆y−µy,
for all
y ∈D(A)= y ∈H2((0,π)2):y(x ,0)=y (x ,π)=0, y (0,x )=y (π,x )=0 ,
1 x2 1 x1 2 x1 2
has the eigen(cid:8)values (cid:9)
2k +1 2
λ =k2+ 2 −µ, ∀k =(k ,k )∈N2,
k 1 2 1 2
(cid:18) (cid:19)
with the corresponding eigenfunctions
2k +1
φ =cosk x sin 2 x , ∀k ∈N2.
k 1 1 2
2
Ordering the eigenvalues set as an increasing sequence and redefine them, we have
λ = 1.25− µ, φ = cosx sin1x ; λ = 3.25−µ, φ = cosx sin3x ; λ =
1 1 1 2 2 2 2 1 2 2 3
4.25−µ, φ = cos2x sin1x ; λ = 6.25−µ, φ = cos2x sin3x ; λ = 7.25−
3 1 2 2 4 4 1 2 2 5
µ,φ = cosx sin5x ; λ = 9.25−µ,φ = cos3x sin1x ; λ = 10.25−µ,φ =
5 1 2 2 6 6 1 2 2 7 7
cos2x sin5x ; λ =11.25−µ,φ =cos3x sin3x ; λ =13.25−µ,φ =cosx sin7x ;
1 2 2 8 8 1 2 2 9 9 1 2 2
λ =15.25−µ,φ =cos3x sin5x ; λ =16.25−µ,φ =cos4x sin1x ; λ =
10 10 1 2 2 11 11 1 2 2 12
10 IONUT¸ MUNTEANU
16.25−µ,φ =cos2x sin7x .Itisclearthatsinceλ =λ ,hypothesis(H)fails
12 1 2 2 11 12
tohold. So,weapplythis time the resultinTheorem2.3. Thus,thecorresponding
control u of the form (2.14) assures the stability of the null solution in the system
(3.3). We will not write it down here explicitly since it involves three 11th order
square matrices.
4. Proofs of the results
Proof of Theorem 2.1. The proof is based on similar ideas with those ones
intheproofof[1,Theorem4.1]. WeshallshowonlythefactthatthefirstN modes
of the solution y are exponentially decaying. The rest will be omitted because it
can be deduced by almost identical arguments as in the proof of [1, Theorem 4.1].
The main difference between the present proof and that one in [1] is that while in
[1], the stability of the unstable modes is showed for each one independently, here
we consider the system formed by them, and show that it is stable. That is why,
the present feedback involves matrices, rather than vectors as it does in [1].
Firstly, recall the Dirichlet operators D , i = 1,...,N, defined after relation
γi
(2.1). In the following, we need to compute the scalar product D α,φ , for all
γj i
i,j ∈{1,...,N}. To this end, we scalarlymultiply equation (2.1), corresponding to
(cid:10) (cid:11)
γ , by φ , to get, via Green’s formula, that
j i
0=− ∆y(x)φ (x)dx+ a(x)y(x)φ (x)dx+(γ −2λ ) y(x)φ (x)dx
i i j i i
ZΩ ZΩ ZΩ
∂
= α(x) φ (x)dσ+ y(x)(−∆φ (x)+a(x)φ (x))dx+(γ −2λ ) y(x)φ (x)dx.
i i i j i i
∂ν
ZΓ1 ZΩ ZΩ
(4.1)
It yields that
1 ∂
D α,φ =− α, φ , i,j =1,...,N. (4.2)
γj i γ −λ ∂ν i
j i (cid:28) (cid:29)0
(cid:10) (cid:11)
Next, we introduce the feedbacks
hy(t),φ1i γk−1λ1∂∂νφ1(x)
hy(t),φ i 1 ∂ φ (x)
vk(t,x)= A ... 2 , γk−λ2.∂.ν. 2  , t≥0, x∈Γ1, (4.3)
* +
 hy(t),φNi   γk−1λN ∂∂νφN(x)  N
   
for k =1,2,...,N; and v=v +v +...+v . It is easy to see that we have
1 2 N
hy(t),φ i 1
1
hy(t),φ i 1
v= T A 2 ,  ,
... ...
* +
 hy(t),φNi   1  N
   
thatisexactlytheboundaryfeedbackpluggedin(2.10). Here,T isdefinedin(2.4),
while A is defined in (2.5).
For latter purpose, we show that
hD v ,φ i hy(t),φ i
γk k 1 1
hD v ,φ i hy(t),φ i
 γk..k. 2 =−BkA ... 2 , (4.4)
 hDγkvk,φNi   hy(t),φNi 
   