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Optimal shape and position of the actuators for the stabilization of a string
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Systems & Control Letters 48 (2003) 199 – 209 www.elsevier.com/locate/sysconle
Optimal shape and position of the actuators for the stabilization of a string Pascal H*ebrarda;1 , Antoine Henrotb;∗;2 b Ecole
a Institut Elie Cartan Nancy, BP 239, 54506 Vand uvre-l es-Nancy, France des Mines and Institut Elie Cartan Nancy, BP 239, 54506 Vand uvre-l es-Nancy, France
Received 11 October 2001; received in revised form 17 January 2002; accepted 3 April 2002 This paper is dedicated to J.L. Lions, father of French modern applied mathematics
Abstract The energy in a string subject to constant viscous damping k on a subset ! of length l ¿ 0 decays exponentially in time; we consider the problem of optimizing the decay rate for the ! which are the unions of at most N intervals. This rate is given by the spectral abscissa of the linear operator associated to the wave equation. We are interested in small values of k; therefore, we consider the derivative of the spectral abscissa at k = 0. We prove that, except for the case l = 12 , when the number of intervals is not 8xed a priori an optimal domain does not exist. We study numerically the case of one or two intervals using a genetic algorithm. These numerical results are not intuitive. In particular, the optimal position of one interval is never at the middle of the string. c 2002 Elsevier Science B.V. All rights reserved. Keywords: Damped wave equation; Optimal location; Spectral abscissa; Genetic algorithms
1. Introduction Let us consider a vibrating structure, say , that we want to stabilize with a damping acting only on a subdomain ! (the actuators). The question to determine the best location and shape of those actuators is very ∗
Corresponding author. E-mail addresses: [email protected] (P. H*ebrard), [email protected] (A. Henrot). URL: http://www.iecn.u-nancy/∼henrot 1 The work of this author was supported by CNRS and R* egion Lorraine. 2 The work of this author was supported by UMR 7502 CNRS and projet CORIDA, INRIA.
important in practice. In this paper, we are interested in that question for the simplest case of a 8xed string = [0; 1], when ! is a union of subintervals of [0; 1]. More precisely, we consider the following modelling. The displacement u of a string of unit length, 8xed at its ends, and in presence of viscous damping 2 ! (where ! denotes the characteristic function of the subdomain ! of positive length), satis8es utt (x; t) − uxx (x; t) + 2k ! (x)ut (x; t) = 0; x ∈ (0; 1); t ¿ 0; u(0; t) = u(1; t) = 0;
c 2002 Elsevier Science B.V. All rights reserved. 0167-6911/03/$ - see front matter PII: S 0 1 6 7 - 6 9 1 1 ( 0 2 ) 0 0 2 6 5 - 7
t ¿0
(1)
200
P. H,ebrard, A. Henrot / Systems & Control Letters 48 (2003) 199 – 209
upon being set in motion by the initial disturbance u(x; 0) = u0 (x);
ut (x; 0) = u1 (x);
∀x ∈ [0; 1]:
(2)
The energy of the string at time t 1 E(t) = [ux2 (x; t) + ut2 (x; t)] d x
n∈N
0
is known to obey E(t) 6 CE(0)e−2t
(3)
for some constants C ¿ 0 and ¿ 0 independent of the initial data. We de8ne the decay rate, as a function of k and !, to be the largest such :
(k; !) = sup{ : ∃C( ) ¿ 0 s:t: E(t) 6 CE(0)e−2 t ; for every solution of (1) and (2)}: Cox and Zuazua have shown in [7] that if ! is of bounded variation, i.e. ! is the union of a 8nite number of intervals, then (k; !) is equal to the opposite of the spectral abscissa of the operator A: (k; !) = − = −sup{Re : ∈ sp(A)}; where A denotes the linear operator associated to Eq. (1) 0 I A= ; d 2 =d x2 −2k ! (x) D(A) = (H 2 (0; 1) ∩ H01 (0; 1)) × H01 (0; 1):
In such a generality, looking for the maximizer of (k; !) → (k; !) is quite diMcult. In Section 2, we will explain how to simplify it by considering, instead of the decay rate, the quantity 1 ! (x) 2n (x) d x; (5) J (!) = inf ∗
(4)
Therefore, a natural question is to look for k and ! which minimize this spectral abscissa (or maximize (k; !)). For other related papers, with a damping given by a more general function a(x) instead of ! (x) in (1), we refer for example to [2,4], or [11] where the uniform stability property is proved even for non-positive a, and [3,6,10] where the question of the optimal a is studied. In [8] or [5], they discuss the optimization of a with another criterion involving the total energy (and therefore the norm of the solution of a Lyapunov equation). In a forthcoming work, see [12], we are going to study the two-dimensional version of the same question, of 8nding the optimal subdomain !. In this case, the decay rate does not depend only on the spectral abscissa of the operator, it also becomes a geometrical quantity involving trajectories of rays inside the domain, see e.g. [1]. Of course, the analysis becomes much more complicated.
0
where N∗ denotes the set of positive integers and ( n )n∈N∗ denote the normalized eigenfunctions for the √ problem without damping, i.e. n = 2 sin n x. Actually, when k is not too large, we have (k; !) kJ (!). We also study, in this section, the dependence k → (k; !) when the subdomain ! is 8xed. In Section 3, we study some general properties of the criterion J . It is easy to prove existence of a maximizer amongst the class of subdomains of total length l composed of at most N intervals (N 8xed). On the other hand, we prove the non-existence of a maximizer in the class of subdomains of total length l composed of a 8nite number of intervals (except for the particular cases l=0; 0:5; 1). This non-existence result would lead us to consider a relaxation of the problem, but we prove at the end of Section 3, that the natural extension of the functional J does not give a convenient relaxation of the problem. We study in Sections 4 and 5 the case of one or two intervals. We give numerical results showing what is the best location. These results are obtained, thanks to hybrid genetic algorithms because the functional we want to minimize has many local minima and is not diNerentiable at these minima. 2. Choice of the criterion We have decided, in this work, to get rid of the parameter k. There are two reasons which justify this choice. First of all, in practical cases, the value of the constant k could be small, so we may use the approximation (k) k(@=@k)k=0 . Moreover, as it is shown in Fig. 1, the dependence of on k seems to be quite linear even for relatively large values of k. So the above approximation holds for a large interval. Therefore, a good criterion to estimate the decay rate is (@=@k)k=0 = −(@=@k)k=0 . Theorem 2.1. Let ! be 8xed and let us consider the spectral abscissa (k; !) of the operator A de8ned in (4). The derivative of the spectral abscissa with
P. H,ebrard, A. Henrot / Systems & Control Letters 48 (2003) 199 – 209 1.5
201
1
Decay rate
Decay rate
1
0.5
0.5
0 0
-0.5
(a)
0
2
4
6
8
10
k
-0.5
(b)
0
2
4
6
8
10
k
Fig. 1. Decay rates for two intervals of length 0.3: (a) in the middle of the string and (b) at the left extremity of the string.
respect to k at k = 0 is given by 1 @ = − inf ∗ ! (x) 2n (x) d x; n∈N @k k=0 0
(6)
where ( n )n∈Z∗ denote the normalized eigenfunctions √ for the problem without damping, i.e. n = 2 sin n x. Proof. We are going to give the proof only when the above in8mum is achieved for a 8nite number of integers. As proved in Lemma 3.1 below, this always happens except for a very particular case: l = 12 and ! “antisymmetric” with respect to 12 (i.e. ! ∪ (1 − !) = [0; 1]). The result is also true in this particular case, but the proof is much longer, see [12]. The starting point is Theorem 3.4 in [11]: they prove that, if ! is 8xed, there exist a number k1 and a constant C0 such that all eigenvalues n (k) of A satisfy (Re(z) denotes the real part of the complex number z): 2C0 k ∀k ¡ k1 ; ∀n ∈ N∗ |Re(n (k)) + kl| 6 : (7) n We now require the 8rst derivative ˙n of a given eigenvalue with respect to the parameter k. We note that k → A is a holomorphic family of type (A) in the sense of Kato [13]. Since each eigenvalue n is simple for k = 0, it remains simple for small k, and we can apply classical results of [13] (see, e.g. Eq. (6.29), p. 422) to get: @n ˙n (k) = @k 1 n (k) 0 ! (x) 2n; k (x) d x =− : 1 1 n (k) 0 2n; k (x) d x+k 0 ! (x) 2n; k (x) d x (8)
Here n (k) denotes the (complex) eigenvalue close to in , for k small, and n; k is the corresponding eigenfunction. In particular, for k = 0, we get ( n; 0 = n ) 1 ! (x) 2n (x) d x; (9) ˙n (0) = − 0
which is a real negative quantity. According to Lemma 3.1 and our assumptions, we have 1 ! (x) 2n (x) d x inf ∗ n∈N
0
=
0
1
! (x) 2n0 (x) d x = l − ! ¡ l:
Now, by continuity of k → ˙n0 (k), there exists k2 ¿ 0 such that 0 6 k 6 k2 ⇒ Re(−˙n0 (k)) 6 l − !=2: Integrating the previous relation yields Re(n0 (k)) ¿ − (l − !=2)k: Now, if we choose n1 ¿ 6C0 = !, we have according to (7), ∀n ¿ n1 ; Re(n0 (k)) ¿ Re(n (k)) (note that, by construction n0 6 n1 ). Therefore, ∀k ¡ inf (k1 ; k2 ) (k) = sup Re(n (k)) = max Re(n (k)): n∈N∗
n6n1
Let us 8x a positive number ". By equicontinuity of the family {˙n (k); n 6 n1 } at 0, there exists k3 such that ∀n 6 n1 ; ∀k 6 k3
|˙n (k) − ˙n (0)| ¡ ";
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P. H,ebrard, A. Henrot / Systems & Control Letters 48 (2003) 199 – 209
which yields, by integration
self-adjoint operator $ + pa(x)I . That is
(˙n (0) − ")k 6 Re(n (k)) 6 (˙n (0) + ")k:
Sv + pa(x)v = v:
Taking the supremum for n 6 n1 gives
We are specially interested in 1 (p) the 8rst (i.e. the largest) eigenvalue. It is de8ned by the variational characterization
sup (˙n (0) − ")k 6 (k) 6 sup (˙n (0) + ")k;
n6n1
n6n1
which reads (k) lim = sup ˙n (0) = − inf ∗ n∈N k→0 k n6n1
1 (p)
0
1
! (x) 2n (x) d x:
It would be interesting to study more deeply the dependence k → (k; !). It is not our purpose here. We just point out here the phenomenon of overdamping (in some sense “more is not better”). This phenomenon is well-known, but we give here a proof for sake of completeness in a more general context. We also notice that numerical experiments seem to show that the function k → (k; !) seems to be 8rst decreasing then increasing. But we were not able to prove it. Theorem 2.2 (Overdamping). Let be a bounded open set in RN ; a(x) a 8xed function in L∞ () which is non-negative and positive on a set of positive measure and let us consider the spectral abscissa (k; a) of the operator A de8ned by 0 I ; A= $ −2ka(x) D(A) = (H 2 () ∩ H01 ()) × H01 ():
(10)
Then lim (k; a) = 0:
k→+∞
(11)
Proof. The proof uses the fact that for k large enough, there always exist (at least) two real eigenvalues and one of them converges to 0, when k → +∞. It follows a method suggested by Rauch and Taylor in [14] which was also used in [7] or [9]. Eigenvalues and eigenfunctions u of the operator A satisfy 2
Su = ( + 2ka(x))u:
(12)
Now, let us introduce like in [9] a parameter p and let us consider the eigenvalues and eigenfunctions of the
= sup
−
(13)
|∇ (x)|2 d x + p a(x) 2 (x) d x
∈H01 ()
2
(x) d x
:
(14) From (14), we see that the map p → 1 (p) is non-decreasing, convex (as supremum of aMne functions) and 1 (p) → +∞ when p → +∞ (according to the assumption satis8ed by a(x), it suMces to 8x in (14) a function 0 positive on the set {x; a(x) ¿ 0} and which vanishes outside). We will denote by (1 the curve of the map p → 1 (p). According to the properties of the function p → 1 (p) previously established, there exists k0 ¿ 0 such that for k ¿ k0 , the parabola = p2 =4k 2 intersects the curve (1 . Let pk be the abscissa of the lower intersection point and 1 (pk ) its ordinate. Setting = −pk =2k = − 1 (pk ), it is immediate to verify that is a real eigenvalue of (12) associated to the same eigenfunction than 1 (pk ). Now, when k → +∞, the parabola = p2 =4k 2 becomes more and more Tat and therefore 1 (pk ) tends to 0 and the theorem follows since = − 1 (pk ). 3. Existence and non-existence results We recall that we are looking for a domain ! of total length l, union of a 8nite number of intervals in [0; 1], which maximizes the criterion 1 J (!) = inf ∗ 2 ! (x) sin2 (n x) d x n∈N
0
= l − sup
n∈N∗
0
1
! (x) cos(2n x) d x:
When it is convenient, we will denote by In (!) the 1 integral 2 0 ! (x) sin2 (n x) d x. By symmetry of the sine function, it is clear that J (!) = J (1 − !) where 1 − ! denotes the symmetry of ! with respect to 12 . Of course, the criterion J being de8ned up to a set
P. H,ebrard, A. Henrot / Systems & Control Letters 48 (2003) 199 – 209
of zero measure, we can assume that the admissible domains ! are unions of closed intervals of positive measure. Then, we can precisely de8ne the number of connected components of such a set: De nition 3.1. For N ∈ N∗ and l ∈ [0; 1], we denote by ENl the set of subdomains of [0; 1] of total length l, union of at most N closed intervals of positive measure. Our 8rst result is a simple existence result in the class ENl previously de8ned. ∗
Theorem 3.1. For each N ∈ N and l ∈ [0; 1], there exists !0 ∈ ENl , such that !∈ENl
We will set JN (l) = max!∈ENl J (!). Proof. ! belongs to ENl if and only if there exists 2N real numbers (xi ) s.t. 0 6 x1 6 x2 6 · · · 6 x2N 6 1 2N
(x2i − x2i−1 ) = l
and (15)
and 2N
0 6 J (!) 6 l
[x2i−1 ; x2i ]:
i=1
Moreover, K = {(x1 ; x2 ; : : : ; x2N ) ∈ [0; 1]2N s:t: (15) holds} is a compact set, and, for each n ∈ N∗ , the map 1 from K into R: (xi )16i62N → 0 ! (x) 2n (x) d x is continuous. Hence, J is a u.s.c. function on K as it is an in8mum of continuous functions, and since K is compact, J achieves its supremum. Now, we are going to prove that if we remove the assumption that we are dealing with a 8xed number of intervals, we have no existence of a maximizer in general. More precisely, we will have existence only in the cases l = 0; l = 1 (which are trivial since we control either nothing or the whole string) and in the case l = 12 . Before, we need to prove the following lemma.
and therefore
JN (l) 6 l:
(16)
Moreover, we have the equality JN (l) = l only in the three cases: • l = 0, achieved for ! = ∅, • l = 1, achieved for ! = [0; 1], • l = 12 , achieved for ! such that ! and 1 − ! are a partition of [0,1]. Proof. When n goes to in8nity, sin2 n x goes to the weak-* topology in L∞ (0; 1). So, 1 1 2 lim 2 ! (x) sin n x d x = ! (x) = l 0
1 2
for
0
and J (!) 6 l. We want now to study the equality 1 cases. J (!) = l if and only if (because 0 ! (x) cos2n x d x → 0 when n → ∞) 1 sup ! (x) cos 2n x d x = 0 n∈N∗
0
⇔ ∀n ∈ N∗
i=1
!=
Lemma 3.1. Let l ∈ [0; 1]; N ∈ N∗ , and ! ∈ ENl . Then we have
n→∞
J (!0 ) = sup J (!):
203
0
1
! (x) cos 2n x d x 6 0:
Let us introduce D, the Dirichlet space of real-valued functions, 1-periodic, piecewise continuous and such that ∀x ∈ R; f(x+ ) + f(x− ) = 2f(x). Let us 8x a set ! in ENl such that J (!) = l. Now, we consider the function f de8ned by f(0) = 12 ( ! (0+ ) + ! (1− )); f(x) = 12 ( ! (x+ ) + ! (x− ));
∀x ∈ (0; 1);
and f 1-periodic. By construction f ∈ D and f = ! a.e. Let us introduce fe , the even part of f: fe (x) = 1 2 (f(x) + f(−x)). This function fe belongs to D and takes a.e. the values 0; 12 or 1. Its Fourier coeMcients are given by 1 A0 = 2l; An = 2 ! (x) cos 2n x d x; 0
∗
Bn = 0; n ∈ N :
Since fe ∈ D, the Fourier series n¿1 An cos 2n x converges for all x ∈ R and moreover ∞ ∀x ∈ R; fe (x) = l + An cos 2n x: n=1
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P. H,ebrard, A. Henrot / Systems & Control Letters 48 (2003) 199 – 209
Now, since we have assumed
that J (!) = l, for every n ¿ 1; An 6 0, the series n¿1 An
is absolutely convergent and the series of functions n¿1 An cos 2n x is normally convergent on R. Therefore, the function fe is continuous and since it can take only the values 0; 12 and 1, it has to be constant. We have three cases: • ∀x ∈ R; fe (x)=0, then ∀x ∈ [0; 1]; ! (x)=0; !=∅ and l = 0. • ∀x ∈ R; fe (x) = 1, then ∀x ∈ [0; 1]; ! (x) = 1; ! = [0; 1] and l = 1. • ∀x ∈ R; fe (x) = 12 , then ∀x ∈ [0; 1]; ! (x) + ! (1 − x) = ! (x) + 1−! (x) = 1: It exactly means that ! and 1 − ! make a partition of [0,1], and therefore l = 12 . The reverse assertion is clear by using the property ! + 1−! =1 to write 1 In (!) = 2 ! (x) sin2 n x d x 0
=2
0
1
(1 − ! (1 − x)) sin2 n x d x
= 1 − In (!) Then J (!) = In (!) =
1 2
= l.
Remark 3.1. The previous computation shows that if a set ! of length l satis8es J (!) = l, then ∀n ∈ N∗ In (!) = l. In some sense, this property can be generalized. Indeed, in [12] the following property is proved: ∀! ∈ ENl ; ∃n0 ∈ N∗ | J (!) = In0 (!): Moreover, if ! satis8es J (!) = JN (l) (i.e. ! achieves the maximum of J in the class ElN ) and if ! has K connected components, then there exists in general 2K distincts integers n1 ; n2 ; : : : ; n2K such that: J (!) = Ini (!);
1 6 i 6 2K:
Remark 3.2. We can also prove some simple properties for J and JN , for example (cf. [12]):
• |J (!1 ) − J (!2 )| ¡ |!1 !2 | where the right-hand side denotes the measure of the symmetric diNerence of the sets. • J is increasing for the inclusion: !1 ( !2 ⇒ J (!1 ) ¡ J (!2 ). • If l1 and l2 are two numbers in [0; 1] such that l1 ¿ l2 , then 0 ¡ JN (l1 ) − JN (l2 ) 6 l1 − l2 . In particular JN is 1-Lipschitzian. Now, we claim our main non-existence result (we recall that we proved existence of a maximizer in the particular cases l = 0; l = 12 ; l = 1 in Lemma 3.1): Theorem 3.2. Let l ∈ ]0; 12 [ ∪ ] 12 ; 1[, and N ∈ N∗ . Then there exists an integer P ¿ N such that JP (l) ¿ JN (l). In particular, the maximization problem max{J (!); ! union of a 8nite number of intervals} has no solutions when l = 0; 12 ; 1. 1 Proof. For ! ∈ ENl , we de8ne In (!) = 0 ! (x) cos × 2n x d x and J (!) = supn∈N∗ In (!). So, J (!) = |!| − J (!). For the proof we are going to use a quadrature method to compute In (!). We recall that if a function f is of class C 2 , the mid-point quadrature rule gives b a + b f(x) d x − (b − a)f a 2 6
(b − a)3 f ∞ : 24
Now, let ! ∈ ENl such that JN (l) = J (!). Since JN (l) ¡ l and In (!) → l when l → +∞, the set of integers n such that J (!) = In (!) = l − In (!) is 8nite. Let us denote it by N. In the sequel of the proof, the integer n belongs to N and then J (!) = In (!). Now, let us consider two subdivisions S1 and S2 of ! and !c : !=
K i=1
[ai ; bi ]
and
!c =
L j=1
[cj ; dj ]:
P. H,ebrard, A. Henrot / Systems & Control Letters 48 (2003) 199 – 209
The mid-point quadrature rule yields 1 ! (x) cos 2n x d x 0
For " ¿ 0 small enough, !" has N + K + L connected components and |!" | = |!| − "
K ai + bi − (bi − ai ) cos 2n 2
K
hi + "
i=1
i=1
= |!| − "(1 − l)
In (!" ) = In (!) −
i=1
1 c (x) cos 2n x d x 0 !
+
L
xi +hi "=2
xi −hi "=2
yj +lj "=2
yj −lj "=2
j=1
(dj − cj )
j=1
cos 2n x d x
cos 2n x d x:
Therefore L lj dIn (!" ) " cos 2n yj + lj (") = d" 2 2
L n2 2 (dj − cj )3 : 6
j=1
" + cos 2n yj − 2
j=1
Let us set xi = (ai + bi )=2; hi = (1 − l)(bi − ai ) for 1 6 i 6 K and yj = (cj + dj )=2; lj = l(dj − cj ) for 1 1 6 j 6 L. Since !c =1− ! ; 0 !c (x) cos 2n x d x= −In (!), it is possible to choose the subdivisions S1 and S2 such that, for all n ∈ N, K J (!) 1 hi cos 2n xi 6 ; (17) In (!) − 1−l 2 i=1
L J (!) In (!) + 1 l cos 2n y : j j 6 l 2 j=1
(18)
Now, we want to construct a new domain !" better than !. For that purpose, we make holes at the points xi and we add the removed material at the points yj K " " xi − hi ; xi + hi !" = ! − 2 2
j=1
K i=1
L cj + dj − (dj − cj ) cos 2n 2 j=1
L
(bi − ai ) + "l
L
Now
K n2 2 6 (bi − ai )3 ; 6
×∪
K
= |!| − "(1 − l)l + "l(1 − l) = |!|:
i=1
i=1
lj
i=1
K f ∞ 6 (bi − ai )3 24
L j=1
6
205
" " yj − lj ; yj + lj : 2 2
−
K hi i=1
2
" cos 2n xi + hi 2
" : + cos 2n xi − 2 In particular, L
K
j=1
i=1
dIn (!" ) lj cos 2n yj − hi cos 2n xi : (0) = d" According to (17) and (18), we have for all n ∈ N dIn (!" ) J (!) (0) 6 l − In (!) d" 2 J (!) − In (!) + (1 − l) 2 =−
J (!) ¡ 0: 2
Finally, there exists " ¿ 0, such that J (!" ) ¡ J (!), and JN +K+L (l) ¡ JN (l).
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P. H,ebrard, A. Henrot / Systems & Control Letters 48 (2003) 199 – 209
Remark 3.3 (Relaxation). Since we have no existence of a maximizer, it would be tempting to use some relaxation technique. Indeed the closure of the set of characteristic functions for the weak-* topology in L∞ (0; 1) is given by F l = 1 {a∈L1 (0; 1)|∀x∈[0; 1]; 06a(x)61and 0 a(x)dx=l} (when taking into account the volume constraint). Moreover, it seems very natural to extend J by considering the following functional de8ned on F l : 1 ˜ a(x) sin2 n x d x: J (a) = inf ∗ 2 n∈N
0
The maximization problem max J˜ (a); a ∈ F l , is now well posed and has many solutions. For example, the constant function a0 (x) = l is one of the solutions. Unfortunately, the extension J˜ is not the good one! Actually, it is strictly upper-semi continuous for the weak-* topology. For example, if we consider the sequence of domains de8ned by k−1 p p l ; !k = ; + k k k p=0
then we have obviously !k * a0 , but a straightforward computation yields sin m l cos m l : J (!k ) = inf ∗ l − m∈N m l That is J (!k ) does not depend on k and J (!k ) = J ([0; l]) ¡ l = J˜ (a0 ). 4. The case of one interval 4.1. Some preliminary results In this section, we consider the case where the control takes place only on one interval ! = [; − l=2; ; + l=2]. We de8ne un (;; l) = 1=n sin n l cos 2n ;; J (;; l)=supn∈N∗ un (;; l); J1 (l)=inf ;∈[l=2; 1=2] J (;; l) and J (;; l) = J ([; − l=2; ; + l=2]) = l − J (;; l). The problem is to 8nd the best location of ;, the center of the interval, that is for a 8xed length l, the maximum of the function ; → J (;; l) or the minimum of the function ; → J (;; l). Due to the symmetry property, it is enough to let ; vary in [l=2; 1=2] and according to Theorem 3.1, there exists at least one optimizer in this segment, which will be denoted by
;∗ (l). Moreover, according to Remark 3.1, there exist two distincts integers n1 (l) and n2 (l) such that J1 (l) = J (;∗ (l); l) = In1 (l) = In2 (l) . 4.2. Numerical results First of all, for a given l and ;, we need to compute the quantity J (;; l). Since |un (;; l)| 6 1=n , as soon as we have found a couple (n1 ; s1 ) which satis8es s1 = maxn6n1 un and n1 s1 ¿ 1, we know that supn∈N∗ =s1 . It follows from Lemma 3.1 that such a couple always exists except for the cases l = 0; 12 or 1. This remark allows us not to 8x a priori the number of eigenmodes of the system that we take into account. Fig. 2a shows the graph of the function ; → J (;; l) for l = 0:05 and the integers n for which the supremum is achieved. The second step consists in 8nding the global minimum of J . Since this function has many local minima and is not diNerentiable at these minima, we have chosen a genetic algorithm. This algorithm acts on a population with a 8xed number of individuals. Each individual has only one chromosome which is the Toating-point representation of ;. The initial population is randomly generated in the interval [l=2; 1=2] at the beginning of each run. The main loop is made up with three operators: • The crossover operation starts with two parents F and M chosen from the population by a 8tness-based selection method, and produces two sons S1 and S2 : S1 = t1 F + (1 − t1 )M; S2 = t2 F + (1 − t2 )M; where t1 and t2 are randomly chosen in [ − 21 ; 32 ] to avoid a too fast convergence of the population in the convex hull of the initial population. • The 8rst mutation operator replaces a randomly chosen vector by a new one also randomly chosen in the interval [l=2; 1=2]. The goal of this operator is to diversify the population. • The third operator starts with a randomly chosen individual and goes to the nearest local minimum. This operator is based on the remark that at a local minima there exists two distincts integer n1 and n2 s.t. J (;; l) = un1 (;; l) = un2 (;; l) and uses Newton’s algorithm.
P. H,ebrard, A. Henrot / Systems & Control Letters 48 (2003) 199 – 209 0.05 u
4
u
u
1
0.044
u
2
u
5
5
0.04
u6
1
J’ (l)
J’ (α,l)
0.046
0.06
u3
0.048
207
0.042
u
u
u7
7
7
0.04
0.02 u
u
8
0.038
8
0.036 0.1
0.2
(a)
α
0.3
0.4
0
0.5
α* optimal position of the control
0.8
J1 (l)
0.6
0.4
0.2
0
0.2
0.4 0.6 l length of the control
0.8
1
0.8
1
0.5
1
(c)
0
(b)
0
0.2
0.4
0.6
0.8
l length of the control
0.4
0.3
0.2
0.1
0
1
(d)
0
0.2
0.4
0.6
l length of the control
Fig. 2. (a) Graph of ; → J (;; l) for l = 0:05, (b) graph of J1 , (c) graph of J1 and (d) graph of ;∗ .
This hybrid algorithm—stochastic and determinist— is very robust and this gives a remarkably eMcient method in this case. We are now able to draw the graphs of the function J1 ; J1 and ;∗ shown, respectively, in Figs. 2b–d. First of all, we observe numerically the theoretical results for the case of equality J1 (l) = l. Secondly, we notice that the graphs of J1 and ;∗ have regular parts and points of discontinuity or angular points. In the neighborhood of a regular point, the function n1 (l) and n2 (l) are constants. It means that J1 and ;∗ are locally given by un1 (;∗ (l); l) = un2 (;∗ (l); l) = J1 (l) and that ; → J (;; l) have one valley deeper than others. A discontinuity point arises for a length l0 s.t. two picks of J take exactly the same value. In such a case, the optimal position ;∗ (l0 ) is not unique and there exists four distinct integers s.t. for l 6 l0 ; J1 and ;∗ are locally given by un1 (;∗ (l); l) = un2 (;∗ (l); l) = J1 (l) and for l ¿ l0 ; J1 and ;∗ are locally given by un3 (;∗ (l); l) = un4 (;∗ (l); l) = J1 (l) because we have jumped on the second pick. Finally, we notice that in the neighborhood of the equality cases (l = 0; 12 or 1) at least one of the integers n1 (l) and n2 (l) seems
to go to in8nity. It means that if the numbers of eigenmodes is 8xed, the results will not be accurate in these cases. Otherwise, n1 (l) and n2 (l) remain small. 5. The case of two intervals 5.1. Some preliminary results We are now interested in the case where the control takes place on two intervals: !=[;1 − l1 =2; ;1 + l1 =2] ∪ [;2 − l2 =2; ;2 + l2 =2] of total length l = l1 + l2 . Like for the case of one interval, we de8ne J (;1 ; ;2 ; l1 ; l2 ) = supn∈N∗ (1=n )(sin n l1 cos 2n ;1 + sin n l2 cos 2n ;2 ); J2 (l) = inf ;1 ;;2 ;l1 J (;1 ; ;2 ; l1 ; l − l1 ). We have now to 8nd, for a 8xed length l, the minimum of the function (;1 ; ;2 ; l1 ) → J (;1 ; ;2 ; l1 ; l−l1 ). Due to the symmetry property, we can assume that l1 6 l2 , the variables ;1 ; ;2 and l1 vary in an oblique prism whose base is a right isosceles triangle. According to Theorem 3.1 there exists at least one optimizer in that prism, which will be denoted by (;1∗ ; ;2∗ ; l∗1 ).
208
P. H,ebrard, A. Henrot / Systems & Control Letters 48 (2003) 199 – 209 0.06
0.3 1 interval 2 intervals
1 interval 2 intervals
0.25
0.04
Ji (l)
J’i (l)
0.2 0.15
0.02
0.1 0.05
0
0
(a)
0.2
0.4
0.6
0.8
l length of the control
0
1
0
0.05
(b)
0.1
0.15
0.2
0.25
0.3
l length of the control
Fig. 3. (a) Graph of J1 and J2 . (b) Graph of J1 and J2 for 0 6 l 6 0:3.
5.2. Numerical results As in the case of one interval, we do not 8x the number of eigenmodes a priori. The research of the global minima of J2 is also done with a genetic algorithm. But each individual has now three chromosomes which are the Toating-point representations of ;1 ; ;2 and l1 . The crossover operation still starts with two 8tness-selected parents to give two sons, but it is a little more complicated because of the new shape of the space of search. The third operator is also much more complicated because we have now to 8nd a point for which the supremum is achieved four times, but it is still based on Newton’s algorithm. Fig. 3a shows the graphs of the functions J1 and J2 . We notice that, as expected, we do not 8nd any new length l s.t. J2 (l) = 0. But except for the cases of equality, we obtain numerically that we always have J2 (l) ¡ J1 (l), which is, in a sense, an extension of Theorem 3.2. Fig. 3b shows the graphs of the functions J1 and J2 in the range where the diNerence is more signi8cant. Optimal positions ;1∗ and ;2∗ seem to be less regular than in the case of one interval. Nevertheless, we often meet the phenomena shown in Fig. 4. The optimal con8guration for the case of two intervals is made of one big interval which is very close to the optimal one for the case of one interval (l∗2 (l) l and ;2∗ (l) ;∗ (l)) and of one small interval, which gives however an important increase of the criterium J . For the case l = 0:3, the decrease of J is 24%, that is an increase of the criterium J of 2.3%.
1 interval, ’J1 = 0.02606, J1 = 0.2739 α* = 0.7661
2 intervals, ’ J = 0.01976, J = 0.2802 2
* l = 1
0.01627
α* = 0.4047 1
2
* l = 2
0.2837
α* = 0.7679 2
Fig. 4. Optimal positions in the case of one and two intervals for l = 0:3.
6. Conclusion The non-existence of a maximizer (except for the particular cases l=0; 12 ; 1) is somewhat not surprising. If we want to damp the string, the best way is probably to split the actuators into smaller and smaller parts. Therefore, the existence and characterization of maximizers for l = 12 is more interesting and unexpected. The numerical results are also interesting. They do not correspond to intuition. For example, in the case of one interval, the best location is never on the middle of the string nor on one extremity. References [1] C. Bardos, G. Lebeau, J. Rauch, Sharp suMcient conditions for the observation, control and stabilization of waves from the boundary, SIAM J. Control Optim. 30 (1992) 1024–1065.
P. H,ebrard, A. Henrot / Systems & Control Letters 48 (2003) 199 – 209 [2] A. Benaddi, B. Rao, Energy decay rate of wave equations with inde8nite damping, J. DiNerential Equations 161 (2) (2000) 337–357. [3] C. Castro, S. Cox, Achieving arbitrarily large decay in the damped wave equation, SIAM J. Control Optim. 39 (6) (2001) 1748–1755. [4] G. Chen, S.A. Fulling, F.J. Narcowich, S. Sun, Exponential decay of energy of evolution equations with locally distributed damping, SIAM J. Appl. Math. 51 (1) (1991) 266–301. [5] S. Cox, Designing for optimal energy absorption, 1: Lumped parameter systems, J. Vib. Acoustics 120 (1998) 339–345. [6] S. Cox, M. Overton, Perturbing the critically damped wave equation, SIAM J. Appl. Math. 56 (5) (1996) 1353–1362. [7] S. Cox, E. Zuazua, The rate at which energy decays in a damped string, Comm. Partial DiNerential Equations 19 (1994) 213–243. [8] F. Fahroo, K. Ito, Variational formulation of optimal damping designs, Contemp. Math. 209 (1997) 95–114.
209
[9] P. Freitas, On some eigenvalue problems related to the wave equation with inde8nite damping, J. DiNerential Equations 127 (1) (1996) 320–335. [10] P. Freitas, Optimizing the rate of decay of solutions of the wave equation using genetic algorithms: a counterexample to the constant damping conjecture, SIAM J. Control Optim. 37 (2) (1999) 376–387. [11] P. Freitas, E. Zuazua, Stability results for the wave equation with inde8nite damping, J. DiNerential Equations 132 (2) (1996) 338–352. [12] P. H*ebrard, Ph.D. Thesis, Universit*e Henri Poincar*e, November 2002. [13] T. Kato, Perturbation Theory for Linear Operators, 2nd Edition, Springer, Berlin, 1984. [14] J. Rauch, M. Taylor, Decay of solutions to nondissipative hyperbolic systems on compact manifolds, Comm. Pure Appl. Math. 28 (4) (1975) 501–523.
Systems & Control Letters 48 (2003) 199 – 209 www.elsevier.com/locate/sysconle
Optimal shape and position of the actuators for the stabilization of a string Pascal H*ebrarda;1 , Antoine Henrotb;∗;2 b Ecole
a Institut Elie Cartan Nancy, BP 239, 54506 Vand uvre-l es-Nancy, France des Mines and Institut Elie Cartan Nancy, BP 239, 54506 Vand uvre-l es-Nancy, France
Received 11 October 2001; received in revised form 17 January 2002; accepted 3 April 2002 This paper is dedicated to J.L. Lions, father of French modern applied mathematics
Abstract The energy in a string subject to constant viscous damping k on a subset ! of length l ¿ 0 decays exponentially in time; we consider the problem of optimizing the decay rate for the ! which are the unions of at most N intervals. This rate is given by the spectral abscissa of the linear operator associated to the wave equation. We are interested in small values of k; therefore, we consider the derivative of the spectral abscissa at k = 0. We prove that, except for the case l = 12 , when the number of intervals is not 8xed a priori an optimal domain does not exist. We study numerically the case of one or two intervals using a genetic algorithm. These numerical results are not intuitive. In particular, the optimal position of one interval is never at the middle of the string. c 2002 Elsevier Science B.V. All rights reserved. Keywords: Damped wave equation; Optimal location; Spectral abscissa; Genetic algorithms
1. Introduction Let us consider a vibrating structure, say , that we want to stabilize with a damping acting only on a subdomain ! (the actuators). The question to determine the best location and shape of those actuators is very ∗
Corresponding author. E-mail addresses: [email protected] (P. H*ebrard), [email protected] (A. Henrot). URL: http://www.iecn.u-nancy/∼henrot 1 The work of this author was supported by CNRS and R* egion Lorraine. 2 The work of this author was supported by UMR 7502 CNRS and projet CORIDA, INRIA.
important in practice. In this paper, we are interested in that question for the simplest case of a 8xed string = [0; 1], when ! is a union of subintervals of [0; 1]. More precisely, we consider the following modelling. The displacement u of a string of unit length, 8xed at its ends, and in presence of viscous damping 2 ! (where ! denotes the characteristic function of the subdomain ! of positive length), satis8es utt (x; t) − uxx (x; t) + 2k ! (x)ut (x; t) = 0; x ∈ (0; 1); t ¿ 0; u(0; t) = u(1; t) = 0;
c 2002 Elsevier Science B.V. All rights reserved. 0167-6911/03/$ - see front matter PII: S 0 1 6 7 - 6 9 1 1 ( 0 2 ) 0 0 2 6 5 - 7
t ¿0
(1)
200
P. H,ebrard, A. Henrot / Systems & Control Letters 48 (2003) 199 – 209
upon being set in motion by the initial disturbance u(x; 0) = u0 (x);
ut (x; 0) = u1 (x);
∀x ∈ [0; 1]:
(2)
The energy of the string at time t 1 E(t) = [ux2 (x; t) + ut2 (x; t)] d x
n∈N
0
is known to obey E(t) 6 CE(0)e−2t
(3)
for some constants C ¿ 0 and ¿ 0 independent of the initial data. We de8ne the decay rate, as a function of k and !, to be the largest such :
(k; !) = sup{ : ∃C( ) ¿ 0 s:t: E(t) 6 CE(0)e−2 t ; for every solution of (1) and (2)}: Cox and Zuazua have shown in [7] that if ! is of bounded variation, i.e. ! is the union of a 8nite number of intervals, then (k; !) is equal to the opposite of the spectral abscissa of the operator A: (k; !) = − = −sup{Re : ∈ sp(A)}; where A denotes the linear operator associated to Eq. (1) 0 I A= ; d 2 =d x2 −2k ! (x) D(A) = (H 2 (0; 1) ∩ H01 (0; 1)) × H01 (0; 1):
In such a generality, looking for the maximizer of (k; !) → (k; !) is quite diMcult. In Section 2, we will explain how to simplify it by considering, instead of the decay rate, the quantity 1 ! (x) 2n (x) d x; (5) J (!) = inf ∗
(4)
Therefore, a natural question is to look for k and ! which minimize this spectral abscissa (or maximize (k; !)). For other related papers, with a damping given by a more general function a(x) instead of ! (x) in (1), we refer for example to [2,4], or [11] where the uniform stability property is proved even for non-positive a, and [3,6,10] where the question of the optimal a is studied. In [8] or [5], they discuss the optimization of a with another criterion involving the total energy (and therefore the norm of the solution of a Lyapunov equation). In a forthcoming work, see [12], we are going to study the two-dimensional version of the same question, of 8nding the optimal subdomain !. In this case, the decay rate does not depend only on the spectral abscissa of the operator, it also becomes a geometrical quantity involving trajectories of rays inside the domain, see e.g. [1]. Of course, the analysis becomes much more complicated.
0
where N∗ denotes the set of positive integers and ( n )n∈N∗ denote the normalized eigenfunctions for the √ problem without damping, i.e. n = 2 sin n x. Actually, when k is not too large, we have (k; !) kJ (!). We also study, in this section, the dependence k → (k; !) when the subdomain ! is 8xed. In Section 3, we study some general properties of the criterion J . It is easy to prove existence of a maximizer amongst the class of subdomains of total length l composed of at most N intervals (N 8xed). On the other hand, we prove the non-existence of a maximizer in the class of subdomains of total length l composed of a 8nite number of intervals (except for the particular cases l=0; 0:5; 1). This non-existence result would lead us to consider a relaxation of the problem, but we prove at the end of Section 3, that the natural extension of the functional J does not give a convenient relaxation of the problem. We study in Sections 4 and 5 the case of one or two intervals. We give numerical results showing what is the best location. These results are obtained, thanks to hybrid genetic algorithms because the functional we want to minimize has many local minima and is not diNerentiable at these minima. 2. Choice of the criterion We have decided, in this work, to get rid of the parameter k. There are two reasons which justify this choice. First of all, in practical cases, the value of the constant k could be small, so we may use the approximation (k) k(@=@k)k=0 . Moreover, as it is shown in Fig. 1, the dependence of on k seems to be quite linear even for relatively large values of k. So the above approximation holds for a large interval. Therefore, a good criterion to estimate the decay rate is (@=@k)k=0 = −(@=@k)k=0 . Theorem 2.1. Let ! be 8xed and let us consider the spectral abscissa (k; !) of the operator A de8ned in (4). The derivative of the spectral abscissa with
P. H,ebrard, A. Henrot / Systems & Control Letters 48 (2003) 199 – 209 1.5
201
1
Decay rate
Decay rate
1
0.5
0.5
0 0
-0.5
(a)
0
2
4
6
8
10
k
-0.5
(b)
0
2
4
6
8
10
k
Fig. 1. Decay rates for two intervals of length 0.3: (a) in the middle of the string and (b) at the left extremity of the string.
respect to k at k = 0 is given by 1 @ = − inf ∗ ! (x) 2n (x) d x; n∈N @k k=0 0
(6)
where ( n )n∈Z∗ denote the normalized eigenfunctions √ for the problem without damping, i.e. n = 2 sin n x. Proof. We are going to give the proof only when the above in8mum is achieved for a 8nite number of integers. As proved in Lemma 3.1 below, this always happens except for a very particular case: l = 12 and ! “antisymmetric” with respect to 12 (i.e. ! ∪ (1 − !) = [0; 1]). The result is also true in this particular case, but the proof is much longer, see [12]. The starting point is Theorem 3.4 in [11]: they prove that, if ! is 8xed, there exist a number k1 and a constant C0 such that all eigenvalues n (k) of A satisfy (Re(z) denotes the real part of the complex number z): 2C0 k ∀k ¡ k1 ; ∀n ∈ N∗ |Re(n (k)) + kl| 6 : (7) n We now require the 8rst derivative ˙n of a given eigenvalue with respect to the parameter k. We note that k → A is a holomorphic family of type (A) in the sense of Kato [13]. Since each eigenvalue n is simple for k = 0, it remains simple for small k, and we can apply classical results of [13] (see, e.g. Eq. (6.29), p. 422) to get: @n ˙n (k) = @k 1 n (k) 0 ! (x) 2n; k (x) d x =− : 1 1 n (k) 0 2n; k (x) d x+k 0 ! (x) 2n; k (x) d x (8)
Here n (k) denotes the (complex) eigenvalue close to in , for k small, and n; k is the corresponding eigenfunction. In particular, for k = 0, we get ( n; 0 = n ) 1 ! (x) 2n (x) d x; (9) ˙n (0) = − 0
which is a real negative quantity. According to Lemma 3.1 and our assumptions, we have 1 ! (x) 2n (x) d x inf ∗ n∈N
0
=
0
1
! (x) 2n0 (x) d x = l − ! ¡ l:
Now, by continuity of k → ˙n0 (k), there exists k2 ¿ 0 such that 0 6 k 6 k2 ⇒ Re(−˙n0 (k)) 6 l − !=2: Integrating the previous relation yields Re(n0 (k)) ¿ − (l − !=2)k: Now, if we choose n1 ¿ 6C0 = !, we have according to (7), ∀n ¿ n1 ; Re(n0 (k)) ¿ Re(n (k)) (note that, by construction n0 6 n1 ). Therefore, ∀k ¡ inf (k1 ; k2 ) (k) = sup Re(n (k)) = max Re(n (k)): n∈N∗
n6n1
Let us 8x a positive number ". By equicontinuity of the family {˙n (k); n 6 n1 } at 0, there exists k3 such that ∀n 6 n1 ; ∀k 6 k3
|˙n (k) − ˙n (0)| ¡ ";
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P. H,ebrard, A. Henrot / Systems & Control Letters 48 (2003) 199 – 209
which yields, by integration
self-adjoint operator $ + pa(x)I . That is
(˙n (0) − ")k 6 Re(n (k)) 6 (˙n (0) + ")k:
Sv + pa(x)v = v:
Taking the supremum for n 6 n1 gives
We are specially interested in 1 (p) the 8rst (i.e. the largest) eigenvalue. It is de8ned by the variational characterization
sup (˙n (0) − ")k 6 (k) 6 sup (˙n (0) + ")k;
n6n1
n6n1
which reads (k) lim = sup ˙n (0) = − inf ∗ n∈N k→0 k n6n1
1 (p)
0
1
! (x) 2n (x) d x:
It would be interesting to study more deeply the dependence k → (k; !). It is not our purpose here. We just point out here the phenomenon of overdamping (in some sense “more is not better”). This phenomenon is well-known, but we give here a proof for sake of completeness in a more general context. We also notice that numerical experiments seem to show that the function k → (k; !) seems to be 8rst decreasing then increasing. But we were not able to prove it. Theorem 2.2 (Overdamping). Let be a bounded open set in RN ; a(x) a 8xed function in L∞ () which is non-negative and positive on a set of positive measure and let us consider the spectral abscissa (k; a) of the operator A de8ned by 0 I ; A= $ −2ka(x) D(A) = (H 2 () ∩ H01 ()) × H01 ():
(10)
Then lim (k; a) = 0:
k→+∞
(11)
Proof. The proof uses the fact that for k large enough, there always exist (at least) two real eigenvalues and one of them converges to 0, when k → +∞. It follows a method suggested by Rauch and Taylor in [14] which was also used in [7] or [9]. Eigenvalues and eigenfunctions u of the operator A satisfy 2
Su = ( + 2ka(x))u:
(12)
Now, let us introduce like in [9] a parameter p and let us consider the eigenvalues and eigenfunctions of the
= sup
−
(13)
|∇ (x)|2 d x + p a(x) 2 (x) d x
∈H01 ()
2
(x) d x
:
(14) From (14), we see that the map p → 1 (p) is non-decreasing, convex (as supremum of aMne functions) and 1 (p) → +∞ when p → +∞ (according to the assumption satis8ed by a(x), it suMces to 8x in (14) a function 0 positive on the set {x; a(x) ¿ 0} and which vanishes outside). We will denote by (1 the curve of the map p → 1 (p). According to the properties of the function p → 1 (p) previously established, there exists k0 ¿ 0 such that for k ¿ k0 , the parabola = p2 =4k 2 intersects the curve (1 . Let pk be the abscissa of the lower intersection point and 1 (pk ) its ordinate. Setting = −pk =2k = − 1 (pk ), it is immediate to verify that is a real eigenvalue of (12) associated to the same eigenfunction than 1 (pk ). Now, when k → +∞, the parabola = p2 =4k 2 becomes more and more Tat and therefore 1 (pk ) tends to 0 and the theorem follows since = − 1 (pk ). 3. Existence and non-existence results We recall that we are looking for a domain ! of total length l, union of a 8nite number of intervals in [0; 1], which maximizes the criterion 1 J (!) = inf ∗ 2 ! (x) sin2 (n x) d x n∈N
0
= l − sup
n∈N∗
0
1
! (x) cos(2n x) d x:
When it is convenient, we will denote by In (!) the 1 integral 2 0 ! (x) sin2 (n x) d x. By symmetry of the sine function, it is clear that J (!) = J (1 − !) where 1 − ! denotes the symmetry of ! with respect to 12 . Of course, the criterion J being de8ned up to a set
P. H,ebrard, A. Henrot / Systems & Control Letters 48 (2003) 199 – 209
of zero measure, we can assume that the admissible domains ! are unions of closed intervals of positive measure. Then, we can precisely de8ne the number of connected components of such a set: De nition 3.1. For N ∈ N∗ and l ∈ [0; 1], we denote by ENl the set of subdomains of [0; 1] of total length l, union of at most N closed intervals of positive measure. Our 8rst result is a simple existence result in the class ENl previously de8ned. ∗
Theorem 3.1. For each N ∈ N and l ∈ [0; 1], there exists !0 ∈ ENl , such that !∈ENl
We will set JN (l) = max!∈ENl J (!). Proof. ! belongs to ENl if and only if there exists 2N real numbers (xi ) s.t. 0 6 x1 6 x2 6 · · · 6 x2N 6 1 2N
(x2i − x2i−1 ) = l
and (15)
and 2N
0 6 J (!) 6 l
[x2i−1 ; x2i ]:
i=1
Moreover, K = {(x1 ; x2 ; : : : ; x2N ) ∈ [0; 1]2N s:t: (15) holds} is a compact set, and, for each n ∈ N∗ , the map 1 from K into R: (xi )16i62N → 0 ! (x) 2n (x) d x is continuous. Hence, J is a u.s.c. function on K as it is an in8mum of continuous functions, and since K is compact, J achieves its supremum. Now, we are going to prove that if we remove the assumption that we are dealing with a 8xed number of intervals, we have no existence of a maximizer in general. More precisely, we will have existence only in the cases l = 0; l = 1 (which are trivial since we control either nothing or the whole string) and in the case l = 12 . Before, we need to prove the following lemma.
and therefore
JN (l) 6 l:
(16)
Moreover, we have the equality JN (l) = l only in the three cases: • l = 0, achieved for ! = ∅, • l = 1, achieved for ! = [0; 1], • l = 12 , achieved for ! such that ! and 1 − ! are a partition of [0,1]. Proof. When n goes to in8nity, sin2 n x goes to the weak-* topology in L∞ (0; 1). So, 1 1 2 lim 2 ! (x) sin n x d x = ! (x) = l 0
1 2
for
0
and J (!) 6 l. We want now to study the equality 1 cases. J (!) = l if and only if (because 0 ! (x) cos2n x d x → 0 when n → ∞) 1 sup ! (x) cos 2n x d x = 0 n∈N∗
0
⇔ ∀n ∈ N∗
i=1
!=
Lemma 3.1. Let l ∈ [0; 1]; N ∈ N∗ , and ! ∈ ENl . Then we have
n→∞
J (!0 ) = sup J (!):
203
0
1
! (x) cos 2n x d x 6 0:
Let us introduce D, the Dirichlet space of real-valued functions, 1-periodic, piecewise continuous and such that ∀x ∈ R; f(x+ ) + f(x− ) = 2f(x). Let us 8x a set ! in ENl such that J (!) = l. Now, we consider the function f de8ned by f(0) = 12 ( ! (0+ ) + ! (1− )); f(x) = 12 ( ! (x+ ) + ! (x− ));
∀x ∈ (0; 1);
and f 1-periodic. By construction f ∈ D and f = ! a.e. Let us introduce fe , the even part of f: fe (x) = 1 2 (f(x) + f(−x)). This function fe belongs to D and takes a.e. the values 0; 12 or 1. Its Fourier coeMcients are given by 1 A0 = 2l; An = 2 ! (x) cos 2n x d x; 0
∗
Bn = 0; n ∈ N :
Since fe ∈ D, the Fourier series n¿1 An cos 2n x converges for all x ∈ R and moreover ∞ ∀x ∈ R; fe (x) = l + An cos 2n x: n=1
204
P. H,ebrard, A. Henrot / Systems & Control Letters 48 (2003) 199 – 209
Now, since we have assumed
that J (!) = l, for every n ¿ 1; An 6 0, the series n¿1 An
is absolutely convergent and the series of functions n¿1 An cos 2n x is normally convergent on R. Therefore, the function fe is continuous and since it can take only the values 0; 12 and 1, it has to be constant. We have three cases: • ∀x ∈ R; fe (x)=0, then ∀x ∈ [0; 1]; ! (x)=0; !=∅ and l = 0. • ∀x ∈ R; fe (x) = 1, then ∀x ∈ [0; 1]; ! (x) = 1; ! = [0; 1] and l = 1. • ∀x ∈ R; fe (x) = 12 , then ∀x ∈ [0; 1]; ! (x) + ! (1 − x) = ! (x) + 1−! (x) = 1: It exactly means that ! and 1 − ! make a partition of [0,1], and therefore l = 12 . The reverse assertion is clear by using the property ! + 1−! =1 to write 1 In (!) = 2 ! (x) sin2 n x d x 0
=2
0
1
(1 − ! (1 − x)) sin2 n x d x
= 1 − In (!) Then J (!) = In (!) =
1 2
= l.
Remark 3.1. The previous computation shows that if a set ! of length l satis8es J (!) = l, then ∀n ∈ N∗ In (!) = l. In some sense, this property can be generalized. Indeed, in [12] the following property is proved: ∀! ∈ ENl ; ∃n0 ∈ N∗ | J (!) = In0 (!): Moreover, if ! satis8es J (!) = JN (l) (i.e. ! achieves the maximum of J in the class ElN ) and if ! has K connected components, then there exists in general 2K distincts integers n1 ; n2 ; : : : ; n2K such that: J (!) = Ini (!);
1 6 i 6 2K:
Remark 3.2. We can also prove some simple properties for J and JN , for example (cf. [12]):
• |J (!1 ) − J (!2 )| ¡ |!1 !2 | where the right-hand side denotes the measure of the symmetric diNerence of the sets. • J is increasing for the inclusion: !1 ( !2 ⇒ J (!1 ) ¡ J (!2 ). • If l1 and l2 are two numbers in [0; 1] such that l1 ¿ l2 , then 0 ¡ JN (l1 ) − JN (l2 ) 6 l1 − l2 . In particular JN is 1-Lipschitzian. Now, we claim our main non-existence result (we recall that we proved existence of a maximizer in the particular cases l = 0; l = 12 ; l = 1 in Lemma 3.1): Theorem 3.2. Let l ∈ ]0; 12 [ ∪ ] 12 ; 1[, and N ∈ N∗ . Then there exists an integer P ¿ N such that JP (l) ¿ JN (l). In particular, the maximization problem max{J (!); ! union of a 8nite number of intervals} has no solutions when l = 0; 12 ; 1. 1 Proof. For ! ∈ ENl , we de8ne In (!) = 0 ! (x) cos × 2n x d x and J (!) = supn∈N∗ In (!). So, J (!) = |!| − J (!). For the proof we are going to use a quadrature method to compute In (!). We recall that if a function f is of class C 2 , the mid-point quadrature rule gives b a + b f(x) d x − (b − a)f a 2 6
(b − a)3 f ∞ : 24
Now, let ! ∈ ENl such that JN (l) = J (!). Since JN (l) ¡ l and In (!) → l when l → +∞, the set of integers n such that J (!) = In (!) = l − In (!) is 8nite. Let us denote it by N. In the sequel of the proof, the integer n belongs to N and then J (!) = In (!). Now, let us consider two subdivisions S1 and S2 of ! and !c : !=
K i=1
[ai ; bi ]
and
!c =
L j=1
[cj ; dj ]:
P. H,ebrard, A. Henrot / Systems & Control Letters 48 (2003) 199 – 209
The mid-point quadrature rule yields 1 ! (x) cos 2n x d x 0
For " ¿ 0 small enough, !" has N + K + L connected components and |!" | = |!| − "
K ai + bi − (bi − ai ) cos 2n 2
K
hi + "
i=1
i=1
= |!| − "(1 − l)
In (!" ) = In (!) −
i=1
1 c (x) cos 2n x d x 0 !
+
L
xi +hi "=2
xi −hi "=2
yj +lj "=2
yj −lj "=2
j=1
(dj − cj )
j=1
cos 2n x d x
cos 2n x d x:
Therefore L lj dIn (!" ) " cos 2n yj + lj (") = d" 2 2
L n2 2 (dj − cj )3 : 6
j=1
" + cos 2n yj − 2
j=1
Let us set xi = (ai + bi )=2; hi = (1 − l)(bi − ai ) for 1 6 i 6 K and yj = (cj + dj )=2; lj = l(dj − cj ) for 1 1 6 j 6 L. Since !c =1− ! ; 0 !c (x) cos 2n x d x= −In (!), it is possible to choose the subdivisions S1 and S2 such that, for all n ∈ N, K J (!) 1 hi cos 2n xi 6 ; (17) In (!) − 1−l 2 i=1
L J (!) In (!) + 1 l cos 2n y : j j 6 l 2 j=1
(18)
Now, we want to construct a new domain !" better than !. For that purpose, we make holes at the points xi and we add the removed material at the points yj K " " xi − hi ; xi + hi !" = ! − 2 2
j=1
K i=1
L cj + dj − (dj − cj ) cos 2n 2 j=1
L
(bi − ai ) + "l
L
Now
K n2 2 6 (bi − ai )3 ; 6
×∪
K
= |!| − "(1 − l)l + "l(1 − l) = |!|:
i=1
i=1
lj
i=1
K f ∞ 6 (bi − ai )3 24
L j=1
6
205
" " yj − lj ; yj + lj : 2 2
−
K hi i=1
2
" cos 2n xi + hi 2
" : + cos 2n xi − 2 In particular, L
K
j=1
i=1
dIn (!" ) lj cos 2n yj − hi cos 2n xi : (0) = d" According to (17) and (18), we have for all n ∈ N dIn (!" ) J (!) (0) 6 l − In (!) d" 2 J (!) − In (!) + (1 − l) 2 =−
J (!) ¡ 0: 2
Finally, there exists " ¿ 0, such that J (!" ) ¡ J (!), and JN +K+L (l) ¡ JN (l).
206
P. H,ebrard, A. Henrot / Systems & Control Letters 48 (2003) 199 – 209
Remark 3.3 (Relaxation). Since we have no existence of a maximizer, it would be tempting to use some relaxation technique. Indeed the closure of the set of characteristic functions for the weak-* topology in L∞ (0; 1) is given by F l = 1 {a∈L1 (0; 1)|∀x∈[0; 1]; 06a(x)61and 0 a(x)dx=l} (when taking into account the volume constraint). Moreover, it seems very natural to extend J by considering the following functional de8ned on F l : 1 ˜ a(x) sin2 n x d x: J (a) = inf ∗ 2 n∈N
0
The maximization problem max J˜ (a); a ∈ F l , is now well posed and has many solutions. For example, the constant function a0 (x) = l is one of the solutions. Unfortunately, the extension J˜ is not the good one! Actually, it is strictly upper-semi continuous for the weak-* topology. For example, if we consider the sequence of domains de8ned by k−1 p p l ; !k = ; + k k k p=0
then we have obviously !k * a0 , but a straightforward computation yields sin m l cos m l : J (!k ) = inf ∗ l − m∈N m l That is J (!k ) does not depend on k and J (!k ) = J ([0; l]) ¡ l = J˜ (a0 ). 4. The case of one interval 4.1. Some preliminary results In this section, we consider the case where the control takes place only on one interval ! = [; − l=2; ; + l=2]. We de8ne un (;; l) = 1=n sin n l cos 2n ;; J (;; l)=supn∈N∗ un (;; l); J1 (l)=inf ;∈[l=2; 1=2] J (;; l) and J (;; l) = J ([; − l=2; ; + l=2]) = l − J (;; l). The problem is to 8nd the best location of ;, the center of the interval, that is for a 8xed length l, the maximum of the function ; → J (;; l) or the minimum of the function ; → J (;; l). Due to the symmetry property, it is enough to let ; vary in [l=2; 1=2] and according to Theorem 3.1, there exists at least one optimizer in this segment, which will be denoted by
;∗ (l). Moreover, according to Remark 3.1, there exist two distincts integers n1 (l) and n2 (l) such that J1 (l) = J (;∗ (l); l) = In1 (l) = In2 (l) . 4.2. Numerical results First of all, for a given l and ;, we need to compute the quantity J (;; l). Since |un (;; l)| 6 1=n , as soon as we have found a couple (n1 ; s1 ) which satis8es s1 = maxn6n1 un and n1 s1 ¿ 1, we know that supn∈N∗ =s1 . It follows from Lemma 3.1 that such a couple always exists except for the cases l = 0; 12 or 1. This remark allows us not to 8x a priori the number of eigenmodes of the system that we take into account. Fig. 2a shows the graph of the function ; → J (;; l) for l = 0:05 and the integers n for which the supremum is achieved. The second step consists in 8nding the global minimum of J . Since this function has many local minima and is not diNerentiable at these minima, we have chosen a genetic algorithm. This algorithm acts on a population with a 8xed number of individuals. Each individual has only one chromosome which is the Toating-point representation of ;. The initial population is randomly generated in the interval [l=2; 1=2] at the beginning of each run. The main loop is made up with three operators: • The crossover operation starts with two parents F and M chosen from the population by a 8tness-based selection method, and produces two sons S1 and S2 : S1 = t1 F + (1 − t1 )M; S2 = t2 F + (1 − t2 )M; where t1 and t2 are randomly chosen in [ − 21 ; 32 ] to avoid a too fast convergence of the population in the convex hull of the initial population. • The 8rst mutation operator replaces a randomly chosen vector by a new one also randomly chosen in the interval [l=2; 1=2]. The goal of this operator is to diversify the population. • The third operator starts with a randomly chosen individual and goes to the nearest local minimum. This operator is based on the remark that at a local minima there exists two distincts integer n1 and n2 s.t. J (;; l) = un1 (;; l) = un2 (;; l) and uses Newton’s algorithm.
P. H,ebrard, A. Henrot / Systems & Control Letters 48 (2003) 199 – 209 0.05 u
4
u
u
1
0.044
u
2
u
5
5
0.04
u6
1
J’ (l)
J’ (α,l)
0.046
0.06
u3
0.048
207
0.042
u
u
u7
7
7
0.04
0.02 u
u
8
0.038
8
0.036 0.1
0.2
(a)
α
0.3
0.4
0
0.5
α* optimal position of the control
0.8
J1 (l)
0.6
0.4
0.2
0
0.2
0.4 0.6 l length of the control
0.8
1
0.8
1
0.5
1
(c)
0
(b)
0
0.2
0.4
0.6
0.8
l length of the control
0.4
0.3
0.2
0.1
0
1
(d)
0
0.2
0.4
0.6
l length of the control
Fig. 2. (a) Graph of ; → J (;; l) for l = 0:05, (b) graph of J1 , (c) graph of J1 and (d) graph of ;∗ .
This hybrid algorithm—stochastic and determinist— is very robust and this gives a remarkably eMcient method in this case. We are now able to draw the graphs of the function J1 ; J1 and ;∗ shown, respectively, in Figs. 2b–d. First of all, we observe numerically the theoretical results for the case of equality J1 (l) = l. Secondly, we notice that the graphs of J1 and ;∗ have regular parts and points of discontinuity or angular points. In the neighborhood of a regular point, the function n1 (l) and n2 (l) are constants. It means that J1 and ;∗ are locally given by un1 (;∗ (l); l) = un2 (;∗ (l); l) = J1 (l) and that ; → J (;; l) have one valley deeper than others. A discontinuity point arises for a length l0 s.t. two picks of J take exactly the same value. In such a case, the optimal position ;∗ (l0 ) is not unique and there exists four distinct integers s.t. for l 6 l0 ; J1 and ;∗ are locally given by un1 (;∗ (l); l) = un2 (;∗ (l); l) = J1 (l) and for l ¿ l0 ; J1 and ;∗ are locally given by un3 (;∗ (l); l) = un4 (;∗ (l); l) = J1 (l) because we have jumped on the second pick. Finally, we notice that in the neighborhood of the equality cases (l = 0; 12 or 1) at least one of the integers n1 (l) and n2 (l) seems
to go to in8nity. It means that if the numbers of eigenmodes is 8xed, the results will not be accurate in these cases. Otherwise, n1 (l) and n2 (l) remain small. 5. The case of two intervals 5.1. Some preliminary results We are now interested in the case where the control takes place on two intervals: !=[;1 − l1 =2; ;1 + l1 =2] ∪ [;2 − l2 =2; ;2 + l2 =2] of total length l = l1 + l2 . Like for the case of one interval, we de8ne J (;1 ; ;2 ; l1 ; l2 ) = supn∈N∗ (1=n )(sin n l1 cos 2n ;1 + sin n l2 cos 2n ;2 ); J2 (l) = inf ;1 ;;2 ;l1 J (;1 ; ;2 ; l1 ; l − l1 ). We have now to 8nd, for a 8xed length l, the minimum of the function (;1 ; ;2 ; l1 ) → J (;1 ; ;2 ; l1 ; l−l1 ). Due to the symmetry property, we can assume that l1 6 l2 , the variables ;1 ; ;2 and l1 vary in an oblique prism whose base is a right isosceles triangle. According to Theorem 3.1 there exists at least one optimizer in that prism, which will be denoted by (;1∗ ; ;2∗ ; l∗1 ).
208
P. H,ebrard, A. Henrot / Systems & Control Letters 48 (2003) 199 – 209 0.06
0.3 1 interval 2 intervals
1 interval 2 intervals
0.25
0.04
Ji (l)
J’i (l)
0.2 0.15
0.02
0.1 0.05
0
0
(a)
0.2
0.4
0.6
0.8
l length of the control
0
1
0
0.05
(b)
0.1
0.15
0.2
0.25
0.3
l length of the control
Fig. 3. (a) Graph of J1 and J2 . (b) Graph of J1 and J2 for 0 6 l 6 0:3.
5.2. Numerical results As in the case of one interval, we do not 8x the number of eigenmodes a priori. The research of the global minima of J2 is also done with a genetic algorithm. But each individual has now three chromosomes which are the Toating-point representations of ;1 ; ;2 and l1 . The crossover operation still starts with two 8tness-selected parents to give two sons, but it is a little more complicated because of the new shape of the space of search. The third operator is also much more complicated because we have now to 8nd a point for which the supremum is achieved four times, but it is still based on Newton’s algorithm. Fig. 3a shows the graphs of the functions J1 and J2 . We notice that, as expected, we do not 8nd any new length l s.t. J2 (l) = 0. But except for the cases of equality, we obtain numerically that we always have J2 (l) ¡ J1 (l), which is, in a sense, an extension of Theorem 3.2. Fig. 3b shows the graphs of the functions J1 and J2 in the range where the diNerence is more signi8cant. Optimal positions ;1∗ and ;2∗ seem to be less regular than in the case of one interval. Nevertheless, we often meet the phenomena shown in Fig. 4. The optimal con8guration for the case of two intervals is made of one big interval which is very close to the optimal one for the case of one interval (l∗2 (l) l and ;2∗ (l) ;∗ (l)) and of one small interval, which gives however an important increase of the criterium J . For the case l = 0:3, the decrease of J is 24%, that is an increase of the criterium J of 2.3%.
1 interval, ’J1 = 0.02606, J1 = 0.2739 α* = 0.7661
2 intervals, ’ J = 0.01976, J = 0.2802 2
* l = 1
0.01627
α* = 0.4047 1
2
* l = 2
0.2837
α* = 0.7679 2
Fig. 4. Optimal positions in the case of one and two intervals for l = 0:3.
6. Conclusion The non-existence of a maximizer (except for the particular cases l=0; 12 ; 1) is somewhat not surprising. If we want to damp the string, the best way is probably to split the actuators into smaller and smaller parts. Therefore, the existence and characterization of maximizers for l = 12 is more interesting and unexpected. The numerical results are also interesting. They do not correspond to intuition. For example, in the case of one interval, the best location is never on the middle of the string nor on one extremity. References [1] C. Bardos, G. Lebeau, J. Rauch, Sharp suMcient conditions for the observation, control and stabilization of waves from the boundary, SIAM J. Control Optim. 30 (1992) 1024–1065.
P. H,ebrard, A. Henrot / Systems & Control Letters 48 (2003) 199 – 209 [2] A. Benaddi, B. Rao, Energy decay rate of wave equations with inde8nite damping, J. DiNerential Equations 161 (2) (2000) 337–357. [3] C. Castro, S. Cox, Achieving arbitrarily large decay in the damped wave equation, SIAM J. Control Optim. 39 (6) (2001) 1748–1755. [4] G. Chen, S.A. Fulling, F.J. Narcowich, S. Sun, Exponential decay of energy of evolution equations with locally distributed damping, SIAM J. Appl. Math. 51 (1) (1991) 266–301. [5] S. Cox, Designing for optimal energy absorption, 1: Lumped parameter systems, J. Vib. Acoustics 120 (1998) 339–345. [6] S. Cox, M. Overton, Perturbing the critically damped wave equation, SIAM J. Appl. Math. 56 (5) (1996) 1353–1362. [7] S. Cox, E. Zuazua, The rate at which energy decays in a damped string, Comm. Partial DiNerential Equations 19 (1994) 213–243. [8] F. Fahroo, K. Ito, Variational formulation of optimal damping designs, Contemp. Math. 209 (1997) 95–114.
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[9] P. Freitas, On some eigenvalue problems related to the wave equation with inde8nite damping, J. DiNerential Equations 127 (1) (1996) 320–335. [10] P. Freitas, Optimizing the rate of decay of solutions of the wave equation using genetic algorithms: a counterexample to the constant damping conjecture, SIAM J. Control Optim. 37 (2) (1999) 376–387. [11] P. Freitas, E. Zuazua, Stability results for the wave equation with inde8nite damping, J. DiNerential Equations 132 (2) (1996) 338–352. [12] P. H*ebrard, Ph.D. Thesis, Universit*e Henri Poincar*e, November 2002. [13] T. Kato, Perturbation Theory for Linear Operators, 2nd Edition, Springer, Berlin, 1984. [14] J. Rauch, M. Taylor, Decay of solutions to nondissipative hyperbolic systems on compact manifolds, Comm. Pure Appl. Math. 28 (4) (1975) 501–523.