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Stabilization and decay estimate for distributed bilinear systems
Systems & Control Letters 36 (1999) 167–171
Stabilization and decay estimate for distributed bilinear systems Larbi Berrahmoune DÃepartement de MathÃematiques, Ecole Normale SupÃerieure de Rabat, B.P. 5118, Rabat, Morocco Received 1 January 1997; received in revised form 11 May 1998
Abstract In this paper, we consider the problem of feedback stabilization for the distributed bilinear system y0 (t) = Ay(t)+u(t)By(t). Here A is the in nitesimal generator of a linear C 0 semigroup of contractions on a Hilbert space H and B : H → H is a linear bounded operator. A sucient condition for feedback stabilization is given and explicit decay estimate is established. c 1999 Elsevier Science B.V. All rights reserved. Applications to vibrating systems are presented. Keywords: Distributed bilinear system; Stabilization; Decay estimate
1. Introduction, preliminaries, main result
since then, formally
This paper considers the question of feedback stabilization of the bilinear system
ky(t)k2 − ky(0)k2 6−2
y0 (t) = Ay(t) + u(t)By(t);
y(0) = y0 ;
(1.1)
where A is the in nitesimal generator of a linear C 0 semigroup of contractions et A on a Hilbert space H with inner product h ; i and corresponding norm k k, so that A is dissipative, i.e. hA ; i60;
∀ ∈ D(A):
(1.2)
B is a linear bounded operator from H to H . The real valued function u(t) is a control, and we consider the problem of choosing it in such a way that all solutions y(t) of Eq. (1.1) converge to zero. A natural choice for u(t) is
Z
t 0
(hBy(s); y(s)i)2 ds; (1.3)
so that the function t 7→ ky(t)k2 does not increase. Therefore, we are led to consider the problem of proving that solutions of the autonomous equation y0 (t) = Ay(t) − hBy(t); y(t)i By(t); y(0) = y0 ∈ H;
(1.4)
decay to zero. Recall that, with an appropriate de nition of weak solution, a funciton y ∈ C([0; T ]; H ); T ¿0, is a weak solution of Eq. (1.4) if and only if y satis es the variation of constants formula [1]: y(t) = et A y0 −
u(t) = − hBy(t); y(t)i ; c 1999 Elsevier Science B.V. All rights reserved. 0167-6911/99/$ – see front matter PII: S 0 1 6 7 - 6 9 1 1 ( 9 8 ) 0 0 0 6 5 - 6
Z
t 0
hBy(s); y(s)i e(t−s)A By(s) ds: (1.5)
168
L. Berrahmoune / Systems & Control Letters 36 (1999) 167–171
A standard approximation argument shows that there exists a unique solution y of Eq. (1.5) de ned on [0; +∞) and satisfying Eq. (1.3) [2]. Let us mention the following results which can be easily deduced from existing ones. Theorem 1.1 (Ball and Slemrod [3]). Suppose that B is compact and also that if
tA ; et A = 0; ∀t¿0; Be then = 0. Then for every weak solution of Eq. (1.4), y(t) converges weakly to zero in H as t → ∞. Theorem 1.2. Let B be self-adjoint and satisfy one of the following conditions: hB ; i¿0;
∀ ∈ H;
(1.6)
hB ; i60;
∀ ∈ H:
(1.7)
Suppose also that the resolvent of A is compact and if
tA ; et A = 0; ∀t¿0; Be then = 0. Then for every weak solution of Eq. (1.4), y(t) converges strongly to zero in H as t → ∞. Theorem 1:2 can be deduced from standard theory of nonlinear contraction semigroups generated by maximal monotone operators by using LaSalle’s invariance principle [6]. Our main result is as follows: Theorem 1.3. Suppose that B is self-adjoint and satisÿes Eq. (1.6) and Z T0
tA Be ’0 ; et A ’0 dt; ∀’0 ∈ H; (1.8) k’0 k2 6 0
for some T0 ¿0. Then, as t → ∞, every solution of Eq. (1.4) satisÿes 1 : (1.9) ky(t)k2 = O t Note that Eq. (1.8) is equivalent to Z T0 kB1=2 et A ’0 k2 dt; ∀’0 ∈ H: k’0 k2 6 0
This amounts to saying that the following observed system ’0 (t) = A’(t);
(1.10)
z(t) = B1=2 ’(t);
(1.11)
is continuously observable on [0; T0 ], or, equivalently, that the following corresponding dual controlled system: 0 (t) = A∗ (t) + B1=2 (t);
(1.12)
is exactly controllable on [0; T0 ] [7]. The main contribution of the paper consists, by virtue of Eq. (1.9), on giving precisely the decrease of solutions to zero. To the author’s knowledge, no such results have been established for distributed bilinear systems. The proof technique is the same as the one introduced in [12] and used to obtain decay rates for solutions of a class of nonlinear, dissipative wave equations. The plan of the paper is as follows: Section 2 is devoted to the proof of Theorem 1.3. Illustrative examples derived from vibrating systems are considered in Section 3. 2. Proof of Theorem 1.3 For y solution of Eq. (1.4) and for every T ¿0, the following inequality can be deduced by a density argument [2]: Z T 2 (hBy(t); y(t)i)2 dt6ky(0)k2 : ky(T )k + 2 0
(2.1) Let us consider w the solution of the following equation: w0 (t) = Aw(t) − hBy(t); y(t)i By(t);
w(0) = 0:
For 06t6T , one can obtain by using the variation of constants formula and kesA k61 for every s¿0, Z T 1=2 | hBy(s); y(s)i |kB1=2 y(s)k ds kw(t)k 6 kB k 6 kB1=2 k
Z
0 T 0
kB1=2 y(s)k3 ds
from which we may deduce the obvious estimate 0
(1.8 )
2
kw(t)k 6T
1=2
kB
1=2 2
k
Z
T 0
kB
1=2
4
y(s)k ds
3=2 :
L. Berrahmoune / Systems & Control Letters 36 (1999) 167–171
From assumption (1:8) and the estimate above, we have 2
2
2
ky(0)k 6 2(kw(0)k + ky(0) − w(0)k ) ( Z 3=2 T0 1=2 4 kB y(s)k ds 6 c0 0
Z +
T0 0
kB1=2 y(s)k4 ds
T0
0
Z + Z +
T0 0
T0 0
kB
1=2
4
y(s)k ds
kB1=2 y(s)k4 ds :
(2.2)
This yields 2
k = 0; 1; 2; : : : :
For any t 0 we have t = k T0 + for some integer k¿0 and ∈ [0; T0 ). Since V (t) and S(t) are nonincreasing, we have t− V (0) V (t) 6 V (k T0 )6S T0 t − 1 V (0): 6S T0 On the other hand, the function p˜0 (s) := s − (I + p0 )−1 (s) satis es p˜0 (s + p0 (s)) = p0 (s);
2
2
ky(T0 )k 6c0 h(ky(0)k − ky(T0 )k );
(2.3)
where the function h is de ned by h(s) := s1=2 + s + s 3=2 :
(2.4)
Let p0 denote the inverse of c0 h. Then p0 is obviously a nonlinear increasing function on [0; +∞) and Eq. (2.3) gives ky(T0 )k2 + po (ky(T0 )k2 )6ky(0)k2 :
(2.5)
Furthermore, p0 has an asymptotic behavior at the origin proportional to s2 , i.e. p0 (s) ∼ C0 s2 (s → 0);
˜ (t)) = 0; X 0 (t) + p(X X (0) = ¿0:
V (kT0 )6S(k)V (0);
kB1=2 y(s)k4 ds 1=2
Then sk 6S(k)s0 where S(t) is the solution of
It follows from Eq. (2.7) and Lemma 3.1, applied to the sequence sk = V (kT0 ), that
1=2 )
for some positive generic constant c0 depending on T0 . Combining the above with inequality (2:1) gives ( Z 3=2 ky(T0 )k2 6 c0
169
C0 := c0−2 :
(2.6)
so that p˜0 (s) will have the same asymptotic behavior as p0 (s) as s → 0. Let us consider the function Z dt ; s 0: g(s) = p ˜ s 0 ( ) Then g is a decreasing function and g() = 0; g(0+) = +∞. Thus [0; +∞) is in the range of g and the solution of X 0 (t) + p˜0 (X (t)) = 0; X (0) = ; is given by
On the other hand, instead of the interval [0; T0 ], we could just as well work on the interval [kT0 ; (k + 1)T0 ]; k = 1; 2; : : : : Then if we set V (t) := ky(t)k2 , Eq. (2.5) would read
X (t) = g−1 (t);
p0 (V ((k + 1)T0 )) + V ((k + 1)T0 )6V (kT0 ):
t→∞
(2.7)
We now apply the following lemma from [12].
t¿0:
Since g(0+) = +∞, lim X (t) = lim g−1 (t) = 0: t→∞
Let 0; ≺ 1. There exists () 0 such that if 0 ≺ s ≺ (),
Lemma 3.1. Let p denote a positive increasing function such that p(0) = 0 and set p(s) ˜ = s−(I +p)−1 (s) where I denotes the identity function. Let {sk }∞ k=0 be a sequence of positive numbers such that
|p˜0 (s) − C0 s2 | ≺ C0 s2 :
p(sk+1 ) + sk+1 6sk ;
−p˜0 (X (t))6C0 ( − 1)(X (t))2 ;
k¿0:
Moreover, there exists t0 () 0 such that 0 ≺ X (t) ≺ () for t¿t0 (). Therefore, if t¿t0 (), we have
170
L. Berrahmoune / Systems & Control Letters 36 (1999) 167–171
x0 ∈ I RN such that ! is a neighborhood of the closure of the set
hence, X 0 (t) + C0 (1 − )(X (t))2 60;
t¿t0 ():
(x0 ) := {x ∈ =(x − x0 )(x) 0};
It follows that 1 ; X (t) = O t
(t → ∞):
This completes the proof of Theorem 1.3. 3. Applications In what follows, ⊂ I R N denotes a bounded open domain with suciently smooth boundary . Let ! be a non-empty open subset of . Then we introduce a function a ∈ L∞ ( ) de ned by a(x)¿0 ae on ;
a(x)¿c¿0 ae on !:
(3.1)
Example 3.1. Wave equation. Consider the system zt t = z + u(t)a(x)zt in (0; ∞) × ; z = 0 on (0; ∞) × :
(3.2)
This system has the form of Eq. (1.1) if we set 0 I 1 2 ; H = H0 ( ) × L ( ); A = P 0 0 B= ; I = identity operator; G
∀w ∈ L2 ( ):
The appropriate feedback is given by Z a(x)|zt |2 dx: u(t) = −
(3.3)
One can easily apply Theorem 1.3 provided that there exist constants C¿0 and T0 ¿0 such that Z {|t (0; x)|2 + |∇(0; x)|2 } dx
Z 6C
T0 0
Z !
|t (t; x)|2 dx dt;
(3.4)
where is the solution of the uncontrolled wave equation 00 = in (0; ∞) × ; = 0 on (0; ∞) × :
Remark 3.1. It is well known that when the domain
is of class C ∞ , a necessary and sucient condition for Eq. (3.4) to hold is that ! satis es the following “geometric control property”: every ray of geometric optics propagating in and being re ected on its boundary, enters the control region ! in a time less than T0 [4]. Example 3.2. Petrowsky system. In this example,
is assumed to be a parallelepipedical domain, say
= (0; d1 ) × · · · × (0; dn ). Let us consider the system zt t + 2 z = u(t)a(x)zt in (0; ∞) × ; z = z = 0 on (0; ∞) × :
with P = ; D(P) = H 2 ( ) ∩ H01 ( ); D(A) = D(P) × H01 ( ), and G de ned by (G w)(x) := a(x)w(x) ae on ;
where (x) denotes the unit outward normal at x ∈ [14]. This result has been exploited to obtain exponential decay for semilinear wave equations [15; 16]. Similar results concerning distributed bilinear systems with damping can be found in [5]. Furthermore, for the one-dimensional case = (0; l); (3:4) can be obtained for T0 ¿2l and ! arbitrary [8, 11].
(3.5)
Inequality (3:4) is established for T0 large enough and ! satisfying the following condition: there exists
(3.6)
This system has the form of Eq. (1.1) if we set 0 I ; H = H 2 ( ) ∩ H01 ( ) × L2 ( ); A = P 0 0 B= ; I = identity operator; G with P = −2 ;
D(P) = {w ∈ H 4 ( )=w= = w= = 0};
D(A) = D(P) × H 2 ( ) ∩ H01 ( ); and G de ned as in Example 3.1. Here, the space H 2 ( ) ∩ H01 ( ) is normed by kwkH 2 ( ) ∩ H01 ( ) := kwkL2 ( ) : The appropriate feedback is given by Eq. (3.3). In contrast to the wave equation, here ! and T0 can be chosen arbitrarily. Indeed, it is shown in [10] that there exists a constant C¿0 such that every solution of 00 + 2 = 0 in (0; ∞) × ; = = 0 on (0; ∞) ×
(3.7)
L. Berrahmoune / Systems & Control Letters 36 (1999) 167–171
satis es the inequality Z {|t (0; x)|2 + |(0; x)|2 } dx
T0Z
Z 6C
0
!
|t (t; x)|2 dx dt:
(3.8)
The application of Theorem 1:3 is then immediate. Remark 3.2. The multiplier techniques used in the proof of (3:4) have been used by various authors leading to inequalities of type (3:8) for di erent systems: plates, elasticity, thermoelasticity, : : : : The common features is that the set ! has to be a neighborhood of a subset of the boundary of the form (x0 ) (see, for instance [9]). Thus, this type of result also applies for Example 3.2 regardless of the boundary conditions. Remark 3.3. When is of class C ∞ , the “geometric control property” mentioned in Remark 3.1 above is sucient for Eq. (3.8) to hold for the plate equation [13]. Acknowledgements The author is grateful to the anonymous referees for very helpful suggestions and remarks. Also, he is indebted to an anonymous referee for suggesting the use of Lemma 3.1 that helped obtain a stronger result than the one stated in the rst version of the manuscript. References [1] J.M. Ball, Strong continuous semigroups, weak solutions, and the variation of constants formula, Proc. Amer. Math. Soc. 63 (1977) 370 –373.
171
[2] J.M. Ball, On the asymptotic behavior of generalized processes, with applications to nonlinear evolution equations, J. Di . Eq. 27 (1978) 244 –265. [3] J.M. Ball, M. Slemrod, Feedback stabilization of distributed semilinear control systems, Appl. Math. Optim. 5 (1979) 169–179. [4] C. Bardos, G. Lebeau, J. Rauch, Sharp sucient conditions for the observation, control and stabilization of waves from the boundary, SIAM J. Control Optim. 30 (1992) 1024 –1065. [5] L. Berrahmoune, Exponential decay for distributed bilinear control systems with damping, Rend. Circ. Mat. Palermo, in press. [6] H. Bounit, Contribution a la stabilisation et a la construction d’observateurs pour une classe de systemes a parametres distribues, These de doctorat, Universite de Lyon I, France, 1996. [7] R.F. Curtain, A.J. Pitchard, In nite Dimensional Linear Systems Theory, Lecture Notes in Control and Information Science, vol. 8, Springer, Berlin, 1978. [8] L.F. Ho, Exact controllability of the one-dimensional wave equation with locally distributed control, SIAM J. Control Optim. 28 (3) (1990) 733 –748. [9] J.U. Kim, Exact semi-internal control of an Euler-Bernoulli equation, SIAM J. Control Optim. 30 (5) (1992) 1001–1023. [10] V. Komornik, On the exact internal controllability of a Petrowsky system, J. Math. Pure Appl. 71 (1992) 331– 342. [11] J. Lagnese, Control of wave processes with distributed controls supported in a subregion, SIAM J. Control Optim. 21 (1) (1983) 68–85. [12] I. Lasiecka, D. Tataru, Uniform boundary stabilization of semilinear wave equation with nonlinear boundary conditions, J. Di . Integ. Eq. 6 (1993) 507–533. [13] G. Lebeau, Contrˆole de l’equation de Schrodinger, J. Math. Pure Appl. 71 (1992) 267–291. [14] J.-L. Lions, Contrˆolabilite Exacte, Perturbations et Stabilisation de Systemes Distribues, Tome 1, RMA No. 8, Masson, Paris, 1988. [15] E. Zuazua, Exponential decay for the semilinear wave equation with localized damping, Comm. Partial Di . Eq. 15 (2) (1990) 205–235. [16] E. Zuazua, Exponential decay for the semilinear wave equation with localized damping in unbounded domains, J. Math. Pure Appl. 70 (1992) 513–529.
Stabilization and decay estimate for distributed bilinear systems Larbi Berrahmoune DÃepartement de MathÃematiques, Ecole Normale SupÃerieure de Rabat, B.P. 5118, Rabat, Morocco Received 1 January 1997; received in revised form 11 May 1998
Abstract In this paper, we consider the problem of feedback stabilization for the distributed bilinear system y0 (t) = Ay(t)+u(t)By(t). Here A is the in nitesimal generator of a linear C 0 semigroup of contractions on a Hilbert space H and B : H → H is a linear bounded operator. A sucient condition for feedback stabilization is given and explicit decay estimate is established. c 1999 Elsevier Science B.V. All rights reserved. Applications to vibrating systems are presented. Keywords: Distributed bilinear system; Stabilization; Decay estimate
1. Introduction, preliminaries, main result
since then, formally
This paper considers the question of feedback stabilization of the bilinear system
ky(t)k2 − ky(0)k2 6−2
y0 (t) = Ay(t) + u(t)By(t);
y(0) = y0 ;
(1.1)
where A is the in nitesimal generator of a linear C 0 semigroup of contractions et A on a Hilbert space H with inner product h ; i and corresponding norm k k, so that A is dissipative, i.e. hA ; i60;
∀ ∈ D(A):
(1.2)
B is a linear bounded operator from H to H . The real valued function u(t) is a control, and we consider the problem of choosing it in such a way that all solutions y(t) of Eq. (1.1) converge to zero. A natural choice for u(t) is
Z
t 0
(hBy(s); y(s)i)2 ds; (1.3)
so that the function t 7→ ky(t)k2 does not increase. Therefore, we are led to consider the problem of proving that solutions of the autonomous equation y0 (t) = Ay(t) − hBy(t); y(t)i By(t); y(0) = y0 ∈ H;
(1.4)
decay to zero. Recall that, with an appropriate de nition of weak solution, a funciton y ∈ C([0; T ]; H ); T ¿0, is a weak solution of Eq. (1.4) if and only if y satis es the variation of constants formula [1]: y(t) = et A y0 −
u(t) = − hBy(t); y(t)i ; c 1999 Elsevier Science B.V. All rights reserved. 0167-6911/99/$ – see front matter PII: S 0 1 6 7 - 6 9 1 1 ( 9 8 ) 0 0 0 6 5 - 6
Z
t 0
hBy(s); y(s)i e(t−s)A By(s) ds: (1.5)
168
L. Berrahmoune / Systems & Control Letters 36 (1999) 167–171
A standard approximation argument shows that there exists a unique solution y of Eq. (1.5) de ned on [0; +∞) and satisfying Eq. (1.3) [2]. Let us mention the following results which can be easily deduced from existing ones. Theorem 1.1 (Ball and Slemrod [3]). Suppose that B is compact and also that if
tA ; et A = 0; ∀t¿0; Be then = 0. Then for every weak solution of Eq. (1.4), y(t) converges weakly to zero in H as t → ∞. Theorem 1.2. Let B be self-adjoint and satisfy one of the following conditions: hB ; i¿0;
∀ ∈ H;
(1.6)
hB ; i60;
∀ ∈ H:
(1.7)
Suppose also that the resolvent of A is compact and if
tA ; et A = 0; ∀t¿0; Be then = 0. Then for every weak solution of Eq. (1.4), y(t) converges strongly to zero in H as t → ∞. Theorem 1:2 can be deduced from standard theory of nonlinear contraction semigroups generated by maximal monotone operators by using LaSalle’s invariance principle [6]. Our main result is as follows: Theorem 1.3. Suppose that B is self-adjoint and satisÿes Eq. (1.6) and Z T0
tA Be ’0 ; et A ’0 dt; ∀’0 ∈ H; (1.8) k’0 k2 6 0
for some T0 ¿0. Then, as t → ∞, every solution of Eq. (1.4) satisÿes 1 : (1.9) ky(t)k2 = O t Note that Eq. (1.8) is equivalent to Z T0 kB1=2 et A ’0 k2 dt; ∀’0 ∈ H: k’0 k2 6 0
This amounts to saying that the following observed system ’0 (t) = A’(t);
(1.10)
z(t) = B1=2 ’(t);
(1.11)
is continuously observable on [0; T0 ], or, equivalently, that the following corresponding dual controlled system: 0 (t) = A∗ (t) + B1=2 (t);
(1.12)
is exactly controllable on [0; T0 ] [7]. The main contribution of the paper consists, by virtue of Eq. (1.9), on giving precisely the decrease of solutions to zero. To the author’s knowledge, no such results have been established for distributed bilinear systems. The proof technique is the same as the one introduced in [12] and used to obtain decay rates for solutions of a class of nonlinear, dissipative wave equations. The plan of the paper is as follows: Section 2 is devoted to the proof of Theorem 1.3. Illustrative examples derived from vibrating systems are considered in Section 3. 2. Proof of Theorem 1.3 For y solution of Eq. (1.4) and for every T ¿0, the following inequality can be deduced by a density argument [2]: Z T 2 (hBy(t); y(t)i)2 dt6ky(0)k2 : ky(T )k + 2 0
(2.1) Let us consider w the solution of the following equation: w0 (t) = Aw(t) − hBy(t); y(t)i By(t);
w(0) = 0:
For 06t6T , one can obtain by using the variation of constants formula and kesA k61 for every s¿0, Z T 1=2 | hBy(s); y(s)i |kB1=2 y(s)k ds kw(t)k 6 kB k 6 kB1=2 k
Z
0 T 0
kB1=2 y(s)k3 ds
from which we may deduce the obvious estimate 0
(1.8 )
2
kw(t)k 6T
1=2
kB
1=2 2
k
Z
T 0
kB
1=2
4
y(s)k ds
3=2 :
L. Berrahmoune / Systems & Control Letters 36 (1999) 167–171
From assumption (1:8) and the estimate above, we have 2
2
2
ky(0)k 6 2(kw(0)k + ky(0) − w(0)k ) ( Z 3=2 T0 1=2 4 kB y(s)k ds 6 c0 0
Z +
T0 0
kB1=2 y(s)k4 ds
T0
0
Z + Z +
T0 0
T0 0
kB
1=2
4
y(s)k ds
kB1=2 y(s)k4 ds :
(2.2)
This yields 2
k = 0; 1; 2; : : : :
For any t 0 we have t = k T0 + for some integer k¿0 and ∈ [0; T0 ). Since V (t) and S(t) are nonincreasing, we have t− V (0) V (t) 6 V (k T0 )6S T0 t − 1 V (0): 6S T0 On the other hand, the function p˜0 (s) := s − (I + p0 )−1 (s) satis es p˜0 (s + p0 (s)) = p0 (s);
2
2
ky(T0 )k 6c0 h(ky(0)k − ky(T0 )k );
(2.3)
where the function h is de ned by h(s) := s1=2 + s + s 3=2 :
(2.4)
Let p0 denote the inverse of c0 h. Then p0 is obviously a nonlinear increasing function on [0; +∞) and Eq. (2.3) gives ky(T0 )k2 + po (ky(T0 )k2 )6ky(0)k2 :
(2.5)
Furthermore, p0 has an asymptotic behavior at the origin proportional to s2 , i.e. p0 (s) ∼ C0 s2 (s → 0);
˜ (t)) = 0; X 0 (t) + p(X X (0) = ¿0:
V (kT0 )6S(k)V (0);
kB1=2 y(s)k4 ds 1=2
Then sk 6S(k)s0 where S(t) is the solution of
It follows from Eq. (2.7) and Lemma 3.1, applied to the sequence sk = V (kT0 ), that
1=2 )
for some positive generic constant c0 depending on T0 . Combining the above with inequality (2:1) gives ( Z 3=2 ky(T0 )k2 6 c0
169
C0 := c0−2 :
(2.6)
so that p˜0 (s) will have the same asymptotic behavior as p0 (s) as s → 0. Let us consider the function Z dt ; s 0: g(s) = p ˜ s 0 ( ) Then g is a decreasing function and g() = 0; g(0+) = +∞. Thus [0; +∞) is in the range of g and the solution of X 0 (t) + p˜0 (X (t)) = 0; X (0) = ; is given by
On the other hand, instead of the interval [0; T0 ], we could just as well work on the interval [kT0 ; (k + 1)T0 ]; k = 1; 2; : : : : Then if we set V (t) := ky(t)k2 , Eq. (2.5) would read
X (t) = g−1 (t);
p0 (V ((k + 1)T0 )) + V ((k + 1)T0 )6V (kT0 ):
t→∞
(2.7)
We now apply the following lemma from [12].
t¿0:
Since g(0+) = +∞, lim X (t) = lim g−1 (t) = 0: t→∞
Let 0; ≺ 1. There exists () 0 such that if 0 ≺ s ≺ (),
Lemma 3.1. Let p denote a positive increasing function such that p(0) = 0 and set p(s) ˜ = s−(I +p)−1 (s) where I denotes the identity function. Let {sk }∞ k=0 be a sequence of positive numbers such that
|p˜0 (s) − C0 s2 | ≺ C0 s2 :
p(sk+1 ) + sk+1 6sk ;
−p˜0 (X (t))6C0 ( − 1)(X (t))2 ;
k¿0:
Moreover, there exists t0 () 0 such that 0 ≺ X (t) ≺ () for t¿t0 (). Therefore, if t¿t0 (), we have
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L. Berrahmoune / Systems & Control Letters 36 (1999) 167–171
x0 ∈ I RN such that ! is a neighborhood of the closure of the set
hence, X 0 (t) + C0 (1 − )(X (t))2 60;
t¿t0 ():
(x0 ) := {x ∈ =(x − x0 )(x) 0};
It follows that 1 ; X (t) = O t
(t → ∞):
This completes the proof of Theorem 1.3. 3. Applications In what follows, ⊂ I R N denotes a bounded open domain with suciently smooth boundary . Let ! be a non-empty open subset of . Then we introduce a function a ∈ L∞ ( ) de ned by a(x)¿0 ae on ;
a(x)¿c¿0 ae on !:
(3.1)
Example 3.1. Wave equation. Consider the system zt t = z + u(t)a(x)zt in (0; ∞) × ; z = 0 on (0; ∞) × :
(3.2)
This system has the form of Eq. (1.1) if we set 0 I 1 2 ; H = H0 ( ) × L ( ); A = P 0 0 B= ; I = identity operator; G
∀w ∈ L2 ( ):
The appropriate feedback is given by Z a(x)|zt |2 dx: u(t) = −
(3.3)
One can easily apply Theorem 1.3 provided that there exist constants C¿0 and T0 ¿0 such that Z {|t (0; x)|2 + |∇(0; x)|2 } dx
Z 6C
T0 0
Z !
|t (t; x)|2 dx dt;
(3.4)
where is the solution of the uncontrolled wave equation 00 = in (0; ∞) × ; = 0 on (0; ∞) × :
Remark 3.1. It is well known that when the domain
is of class C ∞ , a necessary and sucient condition for Eq. (3.4) to hold is that ! satis es the following “geometric control property”: every ray of geometric optics propagating in and being re ected on its boundary, enters the control region ! in a time less than T0 [4]. Example 3.2. Petrowsky system. In this example,
is assumed to be a parallelepipedical domain, say
= (0; d1 ) × · · · × (0; dn ). Let us consider the system zt t + 2 z = u(t)a(x)zt in (0; ∞) × ; z = z = 0 on (0; ∞) × :
with P = ; D(P) = H 2 ( ) ∩ H01 ( ); D(A) = D(P) × H01 ( ), and G de ned by (G w)(x) := a(x)w(x) ae on ;
where (x) denotes the unit outward normal at x ∈ [14]. This result has been exploited to obtain exponential decay for semilinear wave equations [15; 16]. Similar results concerning distributed bilinear systems with damping can be found in [5]. Furthermore, for the one-dimensional case = (0; l); (3:4) can be obtained for T0 ¿2l and ! arbitrary [8, 11].
(3.5)
Inequality (3:4) is established for T0 large enough and ! satisfying the following condition: there exists
(3.6)
This system has the form of Eq. (1.1) if we set 0 I ; H = H 2 ( ) ∩ H01 ( ) × L2 ( ); A = P 0 0 B= ; I = identity operator; G with P = −2 ;
D(P) = {w ∈ H 4 ( )=w= = w= = 0};
D(A) = D(P) × H 2 ( ) ∩ H01 ( ); and G de ned as in Example 3.1. Here, the space H 2 ( ) ∩ H01 ( ) is normed by kwkH 2 ( ) ∩ H01 ( ) := kwkL2 ( ) : The appropriate feedback is given by Eq. (3.3). In contrast to the wave equation, here ! and T0 can be chosen arbitrarily. Indeed, it is shown in [10] that there exists a constant C¿0 such that every solution of 00 + 2 = 0 in (0; ∞) × ; = = 0 on (0; ∞) ×
(3.7)
L. Berrahmoune / Systems & Control Letters 36 (1999) 167–171
satis es the inequality Z {|t (0; x)|2 + |(0; x)|2 } dx
T0Z
Z 6C
0
!
|t (t; x)|2 dx dt:
(3.8)
The application of Theorem 1:3 is then immediate. Remark 3.2. The multiplier techniques used in the proof of (3:4) have been used by various authors leading to inequalities of type (3:8) for di erent systems: plates, elasticity, thermoelasticity, : : : : The common features is that the set ! has to be a neighborhood of a subset of the boundary of the form (x0 ) (see, for instance [9]). Thus, this type of result also applies for Example 3.2 regardless of the boundary conditions. Remark 3.3. When is of class C ∞ , the “geometric control property” mentioned in Remark 3.1 above is sucient for Eq. (3.8) to hold for the plate equation [13]. Acknowledgements The author is grateful to the anonymous referees for very helpful suggestions and remarks. Also, he is indebted to an anonymous referee for suggesting the use of Lemma 3.1 that helped obtain a stronger result than the one stated in the rst version of the manuscript. References [1] J.M. Ball, Strong continuous semigroups, weak solutions, and the variation of constants formula, Proc. Amer. Math. Soc. 63 (1977) 370 –373.
171
[2] J.M. Ball, On the asymptotic behavior of generalized processes, with applications to nonlinear evolution equations, J. Di . Eq. 27 (1978) 244 –265. [3] J.M. Ball, M. Slemrod, Feedback stabilization of distributed semilinear control systems, Appl. Math. Optim. 5 (1979) 169–179. [4] C. Bardos, G. Lebeau, J. Rauch, Sharp sucient conditions for the observation, control and stabilization of waves from the boundary, SIAM J. Control Optim. 30 (1992) 1024 –1065. [5] L. Berrahmoune, Exponential decay for distributed bilinear control systems with damping, Rend. Circ. Mat. Palermo, in press. [6] H. Bounit, Contribution a la stabilisation et a la construction d’observateurs pour une classe de systemes a parametres distribues, These de doctorat, Universite de Lyon I, France, 1996. [7] R.F. Curtain, A.J. Pitchard, In nite Dimensional Linear Systems Theory, Lecture Notes in Control and Information Science, vol. 8, Springer, Berlin, 1978. [8] L.F. Ho, Exact controllability of the one-dimensional wave equation with locally distributed control, SIAM J. Control Optim. 28 (3) (1990) 733 –748. [9] J.U. Kim, Exact semi-internal control of an Euler-Bernoulli equation, SIAM J. Control Optim. 30 (5) (1992) 1001–1023. [10] V. Komornik, On the exact internal controllability of a Petrowsky system, J. Math. Pure Appl. 71 (1992) 331– 342. [11] J. Lagnese, Control of wave processes with distributed controls supported in a subregion, SIAM J. Control Optim. 21 (1) (1983) 68–85. [12] I. Lasiecka, D. Tataru, Uniform boundary stabilization of semilinear wave equation with nonlinear boundary conditions, J. Di . Integ. Eq. 6 (1993) 507–533. [13] G. Lebeau, Contrˆole de l’equation de Schrodinger, J. Math. Pure Appl. 71 (1992) 267–291. [14] J.-L. Lions, Contrˆolabilite Exacte, Perturbations et Stabilisation de Systemes Distribues, Tome 1, RMA No. 8, Masson, Paris, 1988. [15] E. Zuazua, Exponential decay for the semilinear wave equation with localized damping, Comm. Partial Di . Eq. 15 (2) (1990) 205–235. [16] E. Zuazua, Exponential decay for the semilinear wave equation with localized damping in unbounded domains, J. Math. Pure Appl. 70 (1992) 513–529.