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Stabilization of Second Order Evolution Equations by Unbounded Nonlinear Feedback
Copn-i).(hl © IFA C COlllrol o r Di Sl rihlll e d Para me te r Syste ms. Per pi g- na ll . Fra ll ct'. 1~ I H9
ST ABILIZA TION OF SECOND ORDER EVOLUTION EQUATIONS BY UNBOUNDED NONLINEAR FEEDBACK F. Conrad*,
J.
Leblond** and
J.
P. Marmorat**
"' [ 'lIh 'IT , il" ril' .\'11111\' 2 1'1 CA, CXHS 1511 , JU', 5-1511iJ \ '11 Ilriol'lIl ,rt' /1'.1 ,\'(1111'\', F mllo'
2JCJ,
ril' ,\II1I/II;IIWliqlll'., AIIII/iqll"" .,, F ,\ 'S.\IP' S/JIJ/II II -A Il1iIJ/J !i." ()iJ5iJ5 \'ri/h/J II/I/', F ml/(I'
""" (,'1' 11 11."
Abstract. We give decay estimates for the solutions of abstract evolution equations of the form y + Ay + C* g(C y) = 0, where 9 is typically sublinear, The basic assumption concerns C-observability of the uncontrolled system. In some cases, this hypothesis is also proved to be necessary in order to get exponential stability. These results apply in particular to the control of flexible structures by nonlinear feedback located on the boundary. Keywords. Partial differential equations; nonlinear control systems; boundary feedback; stabilization; decay rates.
(HI) The uncon trolled system (1) is C-observable: :.ITa > 0 , M > 0, such
INTRODUCTION
that
Let A be an unbounded, self-adjoint, coerCIve, linear operator on a Hilbert space H, V D(A 1 / 2 ), with inner product 1 a(u,v) = (A / 2 u, A 1 / 2 v). We denote also by A E £(V, V') the unique operator such that a(u,v) = (Au,v) . Let C E £(v,Rm), and 9 : R m _ R m , be continuous, monotone as a graph, and such that g(O) = o.
1~(0)11- + 111>(0) 11~ for any solution
:S M
laToIC~(tW dt
1> of (1).
(H2) The open-loop controlled system (2) is
We consider the second order evolution equations:
well-posed: for any u E L 2(0,T;R m ),(za,zl) E V X H, (2) admits a unique solution (z, z) E L2(O, T; V X H) such that C z E L2(0, T; Rm) and the mapping (U,Za,Zl) - (z,z,C z) is continuous .
(1)
(H3) The closed-loop system (3) defines a
0
(2)
nonlinear semi-group of contractions on V X H associated with a maximal monotone operator .4 on V X H with dense domain D(.4) .
!i+Ay+C*g(Cy)= 0
(3)
1>+A1>= 0 (uncontrolled system)
z + Az + C* u = (open-loop controlled system)
(H4)(a) (g(x), x) ~ a Ixlr + f3lxl 2 (b) (g(x),x) + ilxls ~ 8Ig(x)12
(feedback controlled system)
with positive constants a, f3, i, 8, and o < s :S 2 (r - 1) :S r :S 2.
with the following assumptions:
51
52
F. Conrad .
.I.
Le blond a nd
Remark 1. Assume that A has a Hilbert basis of eigenfunctions 4>p, with associated eigenvalues Then (HI) is satisfied iff
w;.
~ A/
/,
2 2
2
)'" < ~ wp (a p +bp pEN
l~ w,(b,oo,w,t - a"inw,t)c~,r dt.
If (i) IC4>pl 2: ao > 0 and (ii) liminfp---+oo(wp+l - w p ) 2: f30 > 0, then (HI) is satisfied (for any To > 2'Tf-jf3o) . Note that (i) and (ii) are part of a sufficient condition for (H2) to be true (Leblond and Marmorat, 1987).
Remark 2. The functions 9 satisfying (H4) are typically of the form
g(x) '" Ixl<-l x, as Ixl- 0, g(x) '" x, as Ixl- 00 with 0 < I: :S 1, s :S 21: :S 1: + 1 :S r . Note also that (H4) implies (i) "'I11xl :S Ig(x)1 :S "'I21xl at infinity, and that s = riff s = r = 2, in which case (i) holds for any x.
THE MAIN RESULT Let E(t) denote the energy associated with any solution y of (3) at time t : 1
E(t) = 2Iy(t)l~
1
+ 21Iy(t)l l ~
J.
P. I\larllloral
Remark 4. An abstract result concerning exponential stabilization has been obtained by Haraux (1988) in the case of a feedback from H to H . Our proof is based on the ideas contained in his paper. For nonlinear feedback from V to V', assumed to be H-coerciv€' , 'l,uazua (1988a), using suitable Liapunov functionals, gives estimates for the decay rate similar to (i) and (ii) which apply in a superlinear framework. Moreover, he obtains decay rates in the case of nonlinear boundary feedback for the wave equation (Zuazua, 1988b). We are currently attempting to apply the methods of (Zuazua, 1988a, 1988b) t o the sublinear case . Remark 5 . Nakao (1987) gives estimates of the energy decay rate for the wave equation, with a distributed sublinear feedback . With assumptions similar to (H4) , he obtains the same estimate as in Theorem 1.
APPLICATION Our result applies for instance to an EulerBernoulli beam clamped at one end and controlled at the other end by a force which depends on the transversal velocity. This system is modelled by the following equation
a2 y
at 2 (x , t)
a4 y
+ ax4 (x , t) = 0
ay y(O,t) = ax(O,t) = 0 Under hypotheses (H1) J (H2) J (llS) J and (HO J the foLLowing stabilization result holds for the feedback controlled system (3): (i) if s = r : ::lA/o > O,Jl > 0 such that E(t) :S A/o e- lJt E(O) where A/o and Jl are constants (uniform exponential stabilization). (ii) if s < r : ::lA/1 > 0, T1 > 0 such that
Theorem 1.
E(t) :S A/1 .~. E(O), where A/1 and T1 depend only on E(O).
C+1TJ
Remark 3 . In case (ii), an equivalent estimate is given by E(t) :S A/2
(1+ ;1) .~. E(O), 1
where A/2 is constant (see Remark 6).
a3y ay ax3(I ,t) =g(a-t(1,t)) X Ej O,I [, t E [O,T j where 9 : R -
R is continuous , monotone,
g(O) = 0 , g(x) '" Ixl<-l X ,0 < { 9 (x) '" x,
I:
:S 1 for x - 0, for x - 00
For this problem, assumptions (HI) and (H2) have been proved by Leblond and Marmorat (1987) . (H3) has been proved by Conrad and Pierre (1989) in the general case of a
53
Stabilizatioll of Second Orde r E,"olutioll Equatiolls
maximal monotone graph in R2. Then Theorem 1 gives the following estimate of the energy:
Next we proceed as in (Haraux, 1988) . Let = kTo, kEN . By Lemma 2, there exists pE {D,-. -, k - 1} such that:
T
!,
(P+1)TO
E(D) (g(C y(t)), C iI(t)) dt :S -k-
(4)
l,To
Let v be the solution of the uncontrolled problem (1) which satisfies This generalizes a result of Chen and coworkers (1987), where 9 was linear. As also noticed in (Zuazua, 1988a, 1988b), the estimation of the decay rate depends only on the behaviour of 9 at the origin, though some assumptions on the global behaviour of 9 is necessary to set up our proof.
It is enough to establish the result when (Yo, yt) E D(A) with constants depending continuously on E(D). Then, by density of D(A) and the contraction property, the result holds for (Yo, yt) E VxH. Since (Yo, yt) E D(A), by standard theory, (H3) implies that (3) admits a unique solution (y(t), iI(t)) E D(A), (y, iI) E W1,00(0, 00; V x H) and
where, by definition of D(A), Ay+C' g(CiI) is an element of H such that V E V
(Ay
w+Aw+C'g(CiI)=O { w(pTo) = w(pTo) = 0
+ (g(C iI),C
well defined in
ICw(tW dt
l pTo
!,
:S Co
(P+1)TO
Ig(C iJ(t)) 12 dt
pTo
Since C
v = C iJ - C w, we get
!,
(P+l)TO
Ig(C y(t)W dt
(5)
pTo
By hypothesis (H4)(a) and (4) (P+l)To
+ C' g(C iJ),
= (Ay,
w is
(P+l)To
+ 2 Co
= 0 a.e . t
y(pTo) iI(pTo)
and set w = y - v; w is solution of
By hypothesis (H2) , C £2(0, T; Rm) and
PROOF OF THEOREM 1
~iI+ Ay + C' g(C iI)
v(pTo) = { v(pTo) =
l pTo
ICy(tW dt
m
1 l(P+ l)To
:S -{3
Taking = iI E £00 (0, Tj V) leads to
(g(C y(t)) , C y(t)) dt
pTo
(:t iI, iJ) H
+ a(y, iI) + (g(C iI), C iJ)R m
a .e. t; thus by the Lebesgue theorem which is justified here by the regularity of (y, iI) :
E(T) - E(O) = -
-
l
foT (g(C iI(t)),C iI(t)) dt:S E(D)
(P+l)To
Ig(C iJ(t)W dt
pTo
1 l(P+1)TO
:S c
(g(C iJ(t)),C y(t)) dt
pTo
U
Lemma 2 _ VT 2: 0, V(YO,Yl) E D(A),
{3 k
and by (H4)(b)
foT (g(C iJ(t)) , C iI(t)) dt
In particular, the energy is non increasing and
(6)
< E(D)
= 0
+~
l
(P+l)To
pTo
E(O) :S +.1 8k
8
1
IC iJ(tW dt
(p+1)To
pTo
ICiJ(tW dt
(7)
F. (:onrad.
54
J.
Le blond and
l
2
IC iJ (t ) I
pTo
+2
T /
(3 Co + 15 E(O) d t ::; 2 '----'::---::{315 k
j(P+1)TO
C
pTo
IC iJ(tW dt
(8)
We then apply Holder's inequality to the last term of inequality (8): (p+1)To
1
pTo
(1
<
(p+1)To
,
IC iJ(tw dt
ICiJ(t)lrdt
)
~
P. i\larmorat
The rate of decay JL is proportional to l/To, and the "best" rate of decay is
Injecting (6) and (7) into (5), (P+l)To
J.
Let us suppose now s < r. First, we look for a constant C such that M
E(O) (E(O))f] < C (E(O))f -k[C l -k- + C 2 -k-
for all kEN. We take
C = M
r-.
To-;-
[Cl
(E(O))l-f
p1 0
::;
1 l(P+1)To
-
( a
pTo
(g(C y(t)), C iJ(t)) dt
)
~
= M
[Cl
(E(O))l-f
+ C2]
+ c3 (TO)l-f]
(10)
r-.
To-;-
with constants Cl and C3 depending on a,{3,/,15 (C3 = 2/co/15a f).
by (H4)(a) ,
< (E(O))~ ~r;. -
ak
We consider the following inequality implied by (9) and (10):
0
E(k To) ::; C (EiO)
by Lemma 2, so that (8) becomes
l
(P+1)TO
pTo
IC iJ(tW dt r-.
(3 Co + 15 E(O) ::; 2 (3 15 -k-
+2
/
Co
To-;- (Ek(O)) ~
15 a ~
We then apply hypothesis (HI) to the particular solution v of the autonomous system
(1) :
r
(11)
with p = ~ < 1. Let qn = sup{q E N/E(qTo) ~ Jl- n E(O)} where we have fixed Jl > 1. Obviously, (i) qo = 0 if E(O)
i=
0
(ii) qn is finite if E(O)
(qn ::;
i= 0
C; /~ E(O)l-;)
(iii) qn increases with nand qn /
00
E(pTo) Then, applying (11),
and, since E(t) is nonincreasing,
E(O) E(kTo) ::; M [Cl -k-
+ C2
(E(O)) -k- ;] (9)
where
= c,
Cn- l = C(E(qn-lTO)) ::; C(E(O)) Vk E N, where
so that
_ 2 {3 Co + 15 (315
Cl -
If s = r = 2, uniform exponential stability holds by a usual straightforward argument.
1
::; CP
Jl-(n-2)
E(O) 1
[Jl- n
E(O)]p
1
::;
1
!!.i!.=.cl
1
CP E(O) -p
Jl2 Jl
P
Stahilizatiol1 of Second Orde r E\olutioll EqllatiollS
by (iv) and definition of qn. By summation
qn ~
n
C; E(O)l-; 1-'2 :L
= I-' l:::£
Note that T1, M1
qn
~
1
1
1
Cp E(O) -p I-'
.!±e w n p
[1 +T~-P E(OY-l] l~P
--+ 00
if E(O)
--+
o.
Remark 6 • M1 and T1 depend on the choice of r-11., but the exponent _3_ P 1 does not. r-3 = --P Note also that the constant M2 of Remark 3 is just 1-'2.
> 1; then
p
I-'G....I!.(w-I)l-p
wk
k=l
with w
= (Toc4)~
3-p
1
--;
w-l
thus
A PARTIAL CONVERSE TO THEOREM 1
which gives
1 I-' -n l:::£ p < ___--,_.,...,-_ 1+ q,,(w-1)
(12)
.l.!±.e Gp E(O)l- P I-' p .1
Let us now take tn
= qn To.
By (iv)
and using (12) ....I!.-
~ I-'E(O) [ 1+.!±eI~(W-l)
jl_P I
To I-' p (G E(O)p-1 JP
For any time t, there exists n E N such that tn-l < t ~ t n , and E(t) ~ E(tn-d ~ I-'-n 1-'2 E(O) by (iv) . Moreover (12) implies
I-' -n l:::£ p
~
1 I(w-l)
_ _ _---,-,_:-;--_ _
1+
1
=
s
2.
Hypothesis
(H4) (a) (g(x), x) ~ alxl2 (b) (g(x), x) + i lxl2 ~ 8Ig(x)12
E(t n ) ~ 1-'-(n-1) E(O) = I-'-n I-' E(O)
E(t n )
Suppose now r (H4) becomes
l.!±.e
To Gp E(O)l-p I-' p
We recall that, in that case, illxl ~ Ig(x)1 :S for any x . As in (Haraux, 1988), we claim that (HI) is also a necessary condition for uniform exponential stability. Let E(t) denote the energy of a solution of Equation
i21xl,
(3) . Theorem 3 . Assume (HE) I (H3) I (H4) I and E(t) ~ f(t) E(O) where f does not depend on E(O) and liml-->oo f(t) = o. Then (H1) is satisfied (and thus uniform exponential decay holds) .
hence,
~
Jl2 E(O)
r1+
I~W-l)
lli
J.
Proof of Theorem 3 Again we take regular initial conditions. Let
1~
Tol-' p (GE(O)p-l)p
(ii - ~) + A (y -
so that, using the expression (10) of C E(t) ~
2
I-' E(O)
l
1+
1
Hi
I(w-l) J.
Toc41-' p (1+T~-P E(O)p-l)p 1
l
~
!±.e
To C4 ~ w-l
and by (H2) , (H4)(b) and (H4)(a) respectively,
faT
with C4 = [M max( Cl, C3) 1p . This is result (ii) of Theorem 1 with p = ~ < I, and 1
[1 + T~-P E(Oy-l] P
= 0
IC (y-~)(tW dt
:S Co faT Ig(C y(t)W dt :S
c; [loT (g(Cy(t)), Ci/(t))dt + i
faT ICi/(tWdt]
:S Co (1+.2) rT(g(Cy(t)),Ci/(t))dt 8 a lo so that by Lemma 2
(13)
F. Conrad .
56
with Ob
.J .
Leblo nd an d
= 1 + Iln.
.I .
P. Mannorat
REFERENCES
We substract Equation (1) written as
~+A4>+C·g(C~) =C·g(C~)
from Equation (3) and choose y(t) - ~(t) as test-function :
liI(T) - ~(T)lk +
+ Ily(T) -
4>(T)II~
loT (g(C y(t))-g(C ~(t)), C i;(t)-C ~(t)) dt
using hypothesis (H4) and (13). Then 2 E(O) =
liI(O)lk + Ily(O)II~
= 1~(O)lk + 1 1 4>(O)II~ = 1~(T)lk + 114>(T)II~ ~ 21i1(T) - ~(T)lk
+ 21Iy(T) -
+21i1(T)lk
+ 21Iy(T)II~
4>(T)II~
Now we choose T such that E(T) ~ E(0)/8, and we take). ;::: 2 b Co. Finally
Remark 7 . Hypothesis (H4) implies
alxl 2 blxl,
(H4') (aJ (g(x), x) ;:::
(b)
Ig(x)1
~
with 8 b = 1 + I I a, which is sufficient for the proof of Theorem 3.
Acknowledgments. We would like to thank A. Haraux and E. Zuazua for valuable references and comments.
Chen, G ., M. Delfour, A.M. Krall and G. Payre (1987). Modeling, stabilization and control of serially connected beams. SIAM .r J. Contr. Opt., 25(3), pp. 526-546. Conrad, F . and M. Pierre (1989). Stabilisation d'une poutre par feedback frontiere non lineaire (in preparation) . Haraux, A. (1988). Une remarque sur la stabilisation de certains systemes du deuxieme ordre en temps (to appear) . Ingham, A.E. (1936). Some trigonometrical inequalities with applications to the theory of series, Mathematische Zeitschrift, 41, pp . 367-379. Leblond, J . and J.P. Marmorat (1987) . Stabilization of a vibrating beam: a regularity result. Proceedings of the Workshop on Stabilization of flexible structures, COMCON, Optimization Software Inc., pp. 162183. Nakao, M . (1987). On solutions of the wave equation with a sublinear dissipative term. J. Differential Equations, 69, pp . 204215 . Pazy, A. (1982) . Initial value problems for nonlinear differential equations in Banach spaces. Proc . Sem. Coll. France, 84(5). Zuazua, E. (1988a). Stability and decay for a class of non linear hyperbolic problems . Asymptotic Analysis, 1, pp . 161-185. Zuazua, E. (1988b). Uniform stabilization of the wave equation by nonlinear boundary feedback (submitted for publication to SIAM J . Contr. Opt .).
ST ABILIZA TION OF SECOND ORDER EVOLUTION EQUATIONS BY UNBOUNDED NONLINEAR FEEDBACK F. Conrad*,
J.
Leblond** and
J.
P. Marmorat**
"' [ 'lIh 'IT , il" ril' .\'11111\' 2 1'1 CA, CXHS 1511 , JU', 5-1511iJ \ '11 Ilriol'lIl ,rt' /1'.1 ,\'(1111'\', F mllo'
2JCJ,
ril' ,\II1I/II;IIWliqlll'., AIIII/iqll"" .,, F ,\ 'S.\IP' S/JIJ/II II -A Il1iIJ/J !i." ()iJ5iJ5 \'ri/h/J II/I/', F ml/(I'
""" (,'1' 11 11."
Abstract. We give decay estimates for the solutions of abstract evolution equations of the form y + Ay + C* g(C y) = 0, where 9 is typically sublinear, The basic assumption concerns C-observability of the uncontrolled system. In some cases, this hypothesis is also proved to be necessary in order to get exponential stability. These results apply in particular to the control of flexible structures by nonlinear feedback located on the boundary. Keywords. Partial differential equations; nonlinear control systems; boundary feedback; stabilization; decay rates.
(HI) The uncon trolled system (1) is C-observable: :.ITa > 0 , M > 0, such
INTRODUCTION
that
Let A be an unbounded, self-adjoint, coerCIve, linear operator on a Hilbert space H, V D(A 1 / 2 ), with inner product 1 a(u,v) = (A / 2 u, A 1 / 2 v). We denote also by A E £(V, V') the unique operator such that a(u,v) = (Au,v) . Let C E £(v,Rm), and 9 : R m _ R m , be continuous, monotone as a graph, and such that g(O) = o.
1~(0)11- + 111>(0) 11~ for any solution
:S M
laToIC~(tW dt
1> of (1).
(H2) The open-loop controlled system (2) is
We consider the second order evolution equations:
well-posed: for any u E L 2(0,T;R m ),(za,zl) E V X H, (2) admits a unique solution (z, z) E L2(O, T; V X H) such that C z E L2(0, T; Rm) and the mapping (U,Za,Zl) - (z,z,C z) is continuous .
(1)
(H3) The closed-loop system (3) defines a
0
(2)
nonlinear semi-group of contractions on V X H associated with a maximal monotone operator .4 on V X H with dense domain D(.4) .
!i+Ay+C*g(Cy)= 0
(3)
1>+A1>= 0 (uncontrolled system)
z + Az + C* u = (open-loop controlled system)
(H4)(a) (g(x), x) ~ a Ixlr + f3lxl 2 (b) (g(x),x) + ilxls ~ 8Ig(x)12
(feedback controlled system)
with positive constants a, f3, i, 8, and o < s :S 2 (r - 1) :S r :S 2.
with the following assumptions:
51
52
F. Conrad .
.I.
Le blond a nd
Remark 1. Assume that A has a Hilbert basis of eigenfunctions 4>p, with associated eigenvalues Then (HI) is satisfied iff
w;.
~ A/
/,
2 2
2
)'" < ~ wp (a p +bp pEN
l~ w,(b,oo,w,t - a"inw,t)c~,r dt.
If (i) IC4>pl 2: ao > 0 and (ii) liminfp---+oo(wp+l - w p ) 2: f30 > 0, then (HI) is satisfied (for any To > 2'Tf-jf3o) . Note that (i) and (ii) are part of a sufficient condition for (H2) to be true (Leblond and Marmorat, 1987).
Remark 2. The functions 9 satisfying (H4) are typically of the form
g(x) '" Ixl<-l x, as Ixl- 0, g(x) '" x, as Ixl- 00 with 0 < I: :S 1, s :S 21: :S 1: + 1 :S r . Note also that (H4) implies (i) "'I11xl :S Ig(x)1 :S "'I21xl at infinity, and that s = riff s = r = 2, in which case (i) holds for any x.
THE MAIN RESULT Let E(t) denote the energy associated with any solution y of (3) at time t : 1
E(t) = 2Iy(t)l~
1
+ 21Iy(t)l l ~
J.
P. I\larllloral
Remark 4. An abstract result concerning exponential stabilization has been obtained by Haraux (1988) in the case of a feedback from H to H . Our proof is based on the ideas contained in his paper. For nonlinear feedback from V to V', assumed to be H-coerciv€' , 'l,uazua (1988a), using suitable Liapunov functionals, gives estimates for the decay rate similar to (i) and (ii) which apply in a superlinear framework. Moreover, he obtains decay rates in the case of nonlinear boundary feedback for the wave equation (Zuazua, 1988b). We are currently attempting to apply the methods of (Zuazua, 1988a, 1988b) t o the sublinear case . Remark 5 . Nakao (1987) gives estimates of the energy decay rate for the wave equation, with a distributed sublinear feedback . With assumptions similar to (H4) , he obtains the same estimate as in Theorem 1.
APPLICATION Our result applies for instance to an EulerBernoulli beam clamped at one end and controlled at the other end by a force which depends on the transversal velocity. This system is modelled by the following equation
a2 y
at 2 (x , t)
a4 y
+ ax4 (x , t) = 0
ay y(O,t) = ax(O,t) = 0 Under hypotheses (H1) J (H2) J (llS) J and (HO J the foLLowing stabilization result holds for the feedback controlled system (3): (i) if s = r : ::lA/o > O,Jl > 0 such that E(t) :S A/o e- lJt E(O) where A/o and Jl are constants (uniform exponential stabilization). (ii) if s < r : ::lA/1 > 0, T1 > 0 such that
Theorem 1.
E(t) :S A/1 .~. E(O), where A/1 and T1 depend only on E(O).
C+1TJ
Remark 3 . In case (ii), an equivalent estimate is given by E(t) :S A/2
(1+ ;1) .~. E(O), 1
where A/2 is constant (see Remark 6).
a3y ay ax3(I ,t) =g(a-t(1,t)) X Ej O,I [, t E [O,T j where 9 : R -
R is continuous , monotone,
g(O) = 0 , g(x) '" Ixl<-l X ,0 < { 9 (x) '" x,
I:
:S 1 for x - 0, for x - 00
For this problem, assumptions (HI) and (H2) have been proved by Leblond and Marmorat (1987) . (H3) has been proved by Conrad and Pierre (1989) in the general case of a
53
Stabilizatioll of Second Orde r E,"olutioll Equatiolls
maximal monotone graph in R2. Then Theorem 1 gives the following estimate of the energy:
Next we proceed as in (Haraux, 1988) . Let = kTo, kEN . By Lemma 2, there exists pE {D,-. -, k - 1} such that:
T
!,
(P+1)TO
E(D) (g(C y(t)), C iI(t)) dt :S -k-
(4)
l,To
Let v be the solution of the uncontrolled problem (1) which satisfies This generalizes a result of Chen and coworkers (1987), where 9 was linear. As also noticed in (Zuazua, 1988a, 1988b), the estimation of the decay rate depends only on the behaviour of 9 at the origin, though some assumptions on the global behaviour of 9 is necessary to set up our proof.
It is enough to establish the result when (Yo, yt) E D(A) with constants depending continuously on E(D). Then, by density of D(A) and the contraction property, the result holds for (Yo, yt) E VxH. Since (Yo, yt) E D(A), by standard theory, (H3) implies that (3) admits a unique solution (y(t), iI(t)) E D(A), (y, iI) E W1,00(0, 00; V x H) and
where, by definition of D(A), Ay+C' g(CiI) is an element of H such that V E V
(Ay
w+Aw+C'g(CiI)=O { w(pTo) = w(pTo) = 0
+ (g(C iI),C
well defined in
ICw(tW dt
l pTo
!,
:S Co
(P+1)TO
Ig(C iJ(t)) 12 dt
pTo
Since C
v = C iJ - C w, we get
!,
(P+l)TO
Ig(C y(t)W dt
(5)
pTo
By hypothesis (H4)(a) and (4) (P+l)To
+ C' g(C iJ),
= (Ay,
w is
(P+l)To
+ 2 Co
= 0 a.e . t
y(pTo) iI(pTo)
and set w = y - v; w is solution of
By hypothesis (H2) , C £2(0, T; Rm) and
PROOF OF THEOREM 1
~iI+ Ay + C' g(C iI)
v(pTo) = { v(pTo) =
l pTo
ICy(tW dt
m
1 l(P+ l)To
:S -{3
Taking = iI E £00 (0, Tj V) leads to
(g(C y(t)) , C y(t)) dt
pTo
(:t iI, iJ) H
+ a(y, iI) + (g(C iI), C iJ)R m
a .e. t; thus by the Lebesgue theorem which is justified here by the regularity of (y, iI) :
E(T) - E(O) = -
-
l
foT (g(C iI(t)),C iI(t)) dt:S E(D)
(P+l)To
Ig(C iJ(t)W dt
pTo
1 l(P+1)TO
:S c
(g(C iJ(t)),C y(t)) dt
pTo
U
Lemma 2 _ VT 2: 0, V(YO,Yl) E D(A),
{3 k
and by (H4)(b)
foT (g(C iJ(t)) , C iI(t)) dt
In particular, the energy is non increasing and
(6)
< E(D)
= 0
+~
l
(P+l)To
pTo
E(O) :S +.1 8k
8
1
IC iJ(tW dt
(p+1)To
pTo
ICiJ(tW dt
(7)
F. (:onrad.
54
J.
Le blond and
l
2
IC iJ (t ) I
pTo
+2
T /
(3 Co + 15 E(O) d t ::; 2 '----'::---::{315 k
j(P+1)TO
C
pTo
IC iJ(tW dt
(8)
We then apply Holder's inequality to the last term of inequality (8): (p+1)To
1
pTo
(1
<
(p+1)To
,
IC iJ(tw dt
ICiJ(t)lrdt
)
~
P. i\larmorat
The rate of decay JL is proportional to l/To, and the "best" rate of decay is
Injecting (6) and (7) into (5), (P+l)To
J.
Let us suppose now s < r. First, we look for a constant C such that M
E(O) (E(O))f] < C (E(O))f -k[C l -k- + C 2 -k-
for all kEN. We take
C = M
r-.
To-;-
[Cl
(E(O))l-f
p1 0
::;
1 l(P+1)To
-
( a
pTo
(g(C y(t)), C iJ(t)) dt
)
~
= M
[Cl
(E(O))l-f
+ C2]
+ c3 (TO)l-f]
(10)
r-.
To-;-
with constants Cl and C3 depending on a,{3,/,15 (C3 = 2/co/15a f).
by (H4)(a) ,
< (E(O))~ ~r;. -
ak
We consider the following inequality implied by (9) and (10):
0
E(k To) ::; C (EiO)
by Lemma 2, so that (8) becomes
l
(P+1)TO
pTo
IC iJ(tW dt r-.
(3 Co + 15 E(O) ::; 2 (3 15 -k-
+2
/
Co
To-;- (Ek(O)) ~
15 a ~
We then apply hypothesis (HI) to the particular solution v of the autonomous system
(1) :
r
(11)
with p = ~ < 1. Let qn = sup{q E N/E(qTo) ~ Jl- n E(O)} where we have fixed Jl > 1. Obviously, (i) qo = 0 if E(O)
i=
0
(ii) qn is finite if E(O)
(qn ::;
i= 0
C; /~ E(O)l-;)
(iii) qn increases with nand qn /
00
E(pTo) Then, applying (11),
and, since E(t) is nonincreasing,
E(O) E(kTo) ::; M [Cl -k-
+ C2
(E(O)) -k- ;] (9)
where
= c,
Cn- l = C(E(qn-lTO)) ::; C(E(O)) Vk E N, where
so that
_ 2 {3 Co + 15 (315
Cl -
If s = r = 2, uniform exponential stability holds by a usual straightforward argument.
1
::; CP
Jl-(n-2)
E(O) 1
[Jl- n
E(O)]p
1
::;
1
!!.i!.=.cl
1
CP E(O) -p
Jl2 Jl
P
Stahilizatiol1 of Second Orde r E\olutioll EqllatiollS
by (iv) and definition of qn. By summation
qn ~
n
C; E(O)l-; 1-'2 :L
= I-' l:::£
Note that T1, M1
qn
~
1
1
1
Cp E(O) -p I-'
.!±e w n p
[1 +T~-P E(OY-l] l~P
--+ 00
if E(O)
--+
o.
Remark 6 • M1 and T1 depend on the choice of r-11., but the exponent _3_ P 1 does not. r-3 = --P Note also that the constant M2 of Remark 3 is just 1-'2.
> 1; then
p
I-'G....I!.(w-I)l-p
wk
k=l
with w
= (Toc4)~
3-p
1
--;
w-l
thus
A PARTIAL CONVERSE TO THEOREM 1
which gives
1 I-' -n l:::£ p < ___--,_.,...,-_ 1+ q,,(w-1)
(12)
.l.!±.e Gp E(O)l- P I-' p .1
Let us now take tn
= qn To.
By (iv)
and using (12) ....I!.-
~ I-'E(O) [ 1+.!±eI~(W-l)
jl_P I
To I-' p (G E(O)p-1 JP
For any time t, there exists n E N such that tn-l < t ~ t n , and E(t) ~ E(tn-d ~ I-'-n 1-'2 E(O) by (iv) . Moreover (12) implies
I-' -n l:::£ p
~
1 I(w-l)
_ _ _---,-,_:-;--_ _
1+
1
=
s
2.
Hypothesis
(H4) (a) (g(x), x) ~ alxl2 (b) (g(x), x) + i lxl2 ~ 8Ig(x)12
E(t n ) ~ 1-'-(n-1) E(O) = I-'-n I-' E(O)
E(t n )
Suppose now r (H4) becomes
l.!±.e
To Gp E(O)l-p I-' p
We recall that, in that case, illxl ~ Ig(x)1 :S for any x . As in (Haraux, 1988), we claim that (HI) is also a necessary condition for uniform exponential stability. Let E(t) denote the energy of a solution of Equation
i21xl,
(3) . Theorem 3 . Assume (HE) I (H3) I (H4) I and E(t) ~ f(t) E(O) where f does not depend on E(O) and liml-->oo f(t) = o. Then (H1) is satisfied (and thus uniform exponential decay holds) .
hence,
~
Jl2 E(O)
r1+
I~W-l)
lli
J.
Proof of Theorem 3 Again we take regular initial conditions. Let
1~
Tol-' p (GE(O)p-l)p
(ii - ~) + A (y -
so that, using the expression (10) of C E(t) ~
2
I-' E(O)
l
1+
1
Hi
I(w-l) J.
Toc41-' p (1+T~-P E(O)p-l)p 1
l
~
!±.e
To C4 ~ w-l
and by (H2) , (H4)(b) and (H4)(a) respectively,
faT
with C4 = [M max( Cl, C3) 1p . This is result (ii) of Theorem 1 with p = ~ < I, and 1
[1 + T~-P E(Oy-l] P
= 0
IC (y-~)(tW dt
:S Co faT Ig(C y(t)W dt :S
c; [loT (g(Cy(t)), Ci/(t))dt + i
faT ICi/(tWdt]
:S Co (1+.2) rT(g(Cy(t)),Ci/(t))dt 8 a lo so that by Lemma 2
(13)
F. Conrad .
56
with Ob
.J .
Leblo nd an d
= 1 + Iln.
.I .
P. Mannorat
REFERENCES
We substract Equation (1) written as
~+A4>+C·g(C~) =C·g(C~)
from Equation (3) and choose y(t) - ~(t) as test-function :
liI(T) - ~(T)lk +
+ Ily(T) -
4>(T)II~
loT (g(C y(t))-g(C ~(t)), C i;(t)-C ~(t)) dt
using hypothesis (H4) and (13). Then 2 E(O) =
liI(O)lk + Ily(O)II~
= 1~(O)lk + 1 1 4>(O)II~ = 1~(T)lk + 114>(T)II~ ~ 21i1(T) - ~(T)lk
+ 21Iy(T) -
+21i1(T)lk
+ 21Iy(T)II~
4>(T)II~
Now we choose T such that E(T) ~ E(0)/8, and we take). ;::: 2 b Co. Finally
Remark 7 . Hypothesis (H4) implies
alxl 2 blxl,
(H4') (aJ (g(x), x) ;:::
(b)
Ig(x)1
~
with 8 b = 1 + I I a, which is sufficient for the proof of Theorem 3.
Acknowledgments. We would like to thank A. Haraux and E. Zuazua for valuable references and comments.
Chen, G ., M. Delfour, A.M. Krall and G. Payre (1987). Modeling, stabilization and control of serially connected beams. SIAM .r J. Contr. Opt., 25(3), pp. 526-546. Conrad, F . and M. Pierre (1989). Stabilisation d'une poutre par feedback frontiere non lineaire (in preparation) . Haraux, A. (1988). Une remarque sur la stabilisation de certains systemes du deuxieme ordre en temps (to appear) . Ingham, A.E. (1936). Some trigonometrical inequalities with applications to the theory of series, Mathematische Zeitschrift, 41, pp . 367-379. Leblond, J . and J.P. Marmorat (1987) . Stabilization of a vibrating beam: a regularity result. Proceedings of the Workshop on Stabilization of flexible structures, COMCON, Optimization Software Inc., pp. 162183. Nakao, M . (1987). On solutions of the wave equation with a sublinear dissipative term. J. Differential Equations, 69, pp . 204215 . Pazy, A. (1982) . Initial value problems for nonlinear differential equations in Banach spaces. Proc . Sem. Coll. France, 84(5). Zuazua, E. (1988a). Stability and decay for a class of non linear hyperbolic problems . Asymptotic Analysis, 1, pp . 161-185. Zuazua, E. (1988b). Uniform stabilization of the wave equation by nonlinear boundary feedback (submitted for publication to SIAM J . Contr. Opt .).