Optimal internal stabilization of the linear system of elasticity

XX Cong eso de Ecuaciones Di e enciales y Aplicaciones X Cong eso de Ma em´ a ica Aplicada Se illa, 24-28 sep iemb e 2007 (pp. 1–8) Op imal in e nal s abiliza ion o he linea sys em o elas ici y A. M¨ unch1, P. Ped egal2, F. Pe iago3 1Labo a oi e de Ma h´ema iques de Besan¸con, Uni e si ´e de F anche-Com ´e, UMR CNRS 6623, 16, ou e de G ay 25030 Besan¸con, F ance. E-mail: [email p o ec ed]. 2Dp o. de Ma em´a icas, ETSI Indus iales, Uni e sidad de Cas illa-La Mancha, 13071 Ciudad Real. E-mail: [email p o ec ed]. 3Dp o. de Ma em´a ica Aplicada y Es ad´ıs ica, ETSI Indus iales, Uni e sidad Poli ´ecnica de Ca agena, 30203 Ca agena. E-mail: [email p o ec ed]. Palab as cla e: Relaxa ion, s abiliza ion, sys em o elas ici y, nonlinea op imal design, g adien descen me hod. Resumen We conside he nonlinea op imal design p oblem which consis s in inding he bes posi ion and shape o he in e nal iscous damping se o he s abiliza ion o he linea sys em o elas ici y. Since non-exis ence o classical designs is usual in his con ex , a elaxa ion o he o iginal p oblem is p oposed. Then, he elaxed p oblem is sol ed nume ically. Finally, a penaliza ion echnique o eco e quasi-op imal classical designs om he elaxed ones and he o e -damping phenomenon a e analyzed in se e al nume ical expe imen s. 1. P oblem o mula ion Conside he ollowing nonlinea op imal design p oblem: (P) ´ın ω∈ΩL J(Xω) = 1 2ZT 0ZΩ³¯¯u0¯¯ 2+σ(u) : ε(u)´dxd (1) whe e o a ixed 0 < L < 1, ΩL={ω⊂Ω : |ω|=L|Ω|} , 1 A. M¨unch, P. Ped egal, F. Pe iago |ω|and |Ω|being he Lebesgue measu e o ωand Ω, espec i ely, and uis he solu ion o he elas ici y sys em        u00 − ∇x·σ+a(x)Xω(x)u0= 0 in (0, T)×Ω, u= 0 on (0, T)×Γ0, σ·n= 0 on (0, T)×Γ1, u(0,·) = u0,u0(0,·) = u1in Ω. (2) As usual, ε=ε(u) = 1 2³∇xu+ (∇xu)T´(3) is he linea ized s ain enso and σ=σ(u) = (σij =aijklεkl) (4) he s ess enso . The coe icien s aijkl ∈W1,∞(Ω), i, j, k, l = 1,· · · , N, a e such ha aijkl =aklij =ajikl and aijklεijεkl ≥αεijεij in Ω (5) o some ixed α > 0. Mo eo e , Xωis he cha ac e is ic unc ion o ω, ∇x·is he di e gence ope a o conside ed wi h espec o he spa ial a iable x, n= (n1,· · · , nN) is he ou wa d uni no mal ec o o Γ1,0< T ≤ ∞,and a=a(x)∈L∞(Ω) is a damping po en ial sa is ying a(x)≥a0>0 a. e. x∈ω. As o he physical meaning o sys em (2), he dissipa i e e m a(x)Xω(x)u0is usually e e ed in he li e a u e as a iscous damping because i is caused o he iscosi y o he medium in which he ib a ions o he sys em ake place. F om an enginee ing iewpoin , his e m may also be seen as a eedback con ol mechanism which measu es he eloci y o ib a ions, h ough use o senso s, and ac s on he sys em acco dingly o hese measu es by means o ac ua o s. In his sense, Xωindica es he place and shape o senso s and ac ua o s. I is hen na u al and e y impo an in p ac ise o analyze he ques ion o de e mining he bes posi ion and shape o senso s and ac ua o s ha minimize he ene gy o he sys em o e a ime in e al. This is ou p oblem (P). 2. Relaxa ion As shown ecen ly by he au ho s o he case o he wa e equa ion [4, 5] (see also [2, 3] o some ela ed wo ks), (P) is usually ill-posed in he sense ha he e is no minimize in he class o cha ac e is ic unc ions. Then, a ull elaxa ion o he o iginal p oblem was ca ied ou by using a sui able ep esen a ion o di e gence- ee ec o ields which enables o ans o m he o iginal p oblem in o a non-con ex, ec o a ia ional one. The aim o his wo k is o ex end he esul s in [4, 5] o he case o he sys em o linea elas ici y. Howe e , ou app oach he e does no equi e he in oduc ion o auxilia y po en- ials associa ed wi h di e gence- ee ec o ields. Ins ead o hose, we use di -cu l Young measu es as gi en by [7]. This makes he ea men much mo e di ec and dimension- independen . P ecisely, conside he elaxed p oblem (RP) ´ın s∈L∞(Ω) J(s) = ZT 0ZΩ³¯¯u0¯¯ 2+σ(u) : ε(u)´dxd (6) 2 Op imal in e nal s abiliza ion o he elas ici y sys em whe e usol es he new sys em        u00 − ∇x·σ+a(x)s(x)u0= 0 in (0, T)×Ω, u= 0 on (0, T)×Γ0, σ·n= 0 on (0, T)×Γ1, u(0,·) = u0,u0(0,·) = u1in Ω, (7) and now he compe ing unc ions ssa is y he poin -wise and olume cons ain s 0≤s(x)≤1 and ZΩ s(x)dx =L|Ω|.(8) Ou main heo e ical esul ollows. Theo em 1 Assume ha he ini ial da a o sys em (2) ha e he egula i y (u0,u1)∈³¡H2(Ω)¢N∩V0´×V0.(9) Then (RP)is a ull elaxa ion o (P)in he sense: (i) The e a e op imal solu ions o (RP ). (ii) The minimum o (RP )equals he in imum o (P). (iii) Minimizing sequences o (P)a e eco e ed by i s -o de lamina es wi h any no mal in space and independen o ime. Fo a ull p oo o his esul we e e o [6]. Ne e heless, a ew commen s on i a e now in o de . Fi s , he egula i y condi ion (9) on he ini ial da a is a su icien condi ion in o de o a oid concen a ion o ene gy phenomena and he e o e his enables us o use he Young measu e heo y o compu e he cos limi o a minimizing sequence o p oblem (P). Second, we will show la e on ha o some alues o he damping po en ial a he e is a nume ical e idence ha p oblem (P) is ill-posed. This jus i ies he elaxa ion s a ed in poin s (i) and (ii) abo e. Finally, in wha conce ns poin (iii), i ells us how he mic os uc u e o he op imal damping designs looks like. This in o ma ion is codi ied by he op imal Young measu e associa ed wi h he elaxed p oblem (RP). In ac , om his i can be p o ed ha i Xωjis a minimizing sequence o (P), hen he associa ed displacemen s ujcon e ge o he op imal displacemen u o he elaxed p oblem (RP) in a s ong sense. 3. Nume ical esolu ion o p oblems (P)and (RP ) 3.1. Algo i hm o minimiza ion We p opose a i s -o de g adien descen me hod o sol e (RP). P ecisely, we de ine he descen di ec ion s1(x) = −µa(x)ZT 0 u0( , x)·p( , x)d +γ¶,∀x∈Ω,(10) 3 A. M¨unch, P. Ped egal, F. Pe iago whe e he mul iplie γis de e mined so ha o any unc ion η∈L∞(Ω,R+), wi h η6= 0, and ||s+ηs1||L1(Ω) =L|Ω|we ha e Jγ(s+ηs1)≤Jγ(s) o Jγ(s) = J(s) + γ||s||L1(Ω).(11) This leads o ake γ=(RΩs(x)dx −L|Ω|)−RΩη(x)a(x)RT 0u0( , x)·p( , x)d dx RΩη(x)dx .(12) As o p, his is he solu ion o he adjoin p oblem        p00 − ∇x·σ(p)−a(x)s(x)p0=u00 +∇x·σ(u),in (0, T)×Ω, p= 0,on (0, T)×Γ0, p·n= 0,on (0, T)×Γ1, p(T, ·) = 0,p0(T, ·) = u0(T, ·) in Ω. (13) A las , he unc ion ηis chosen so ha s(x) + η(x)s1(x)∈[0,1], o all x∈Ω. A simple and e icien choice consis s in aking η(x) = ²s(x)(1 −s(x)) o all x∈Ω wi h ²a small eal posi i e. Consequen ly, he descen algo i hm o sol e nume ically he elaxed p oblem (RP) may be s uc u ed as ollows : le Ω ⊂RN, (u0,u1)∈((H2(Ω))N∩V0)×V0,L∈(0,1), T > 0, and ² < 1, ²1<< 1 be gi en : Ini ializa ion o he densi y unc ion s0∈L∞(Ω; ]0,1[); Fo k≥0, i e a ion un il con e gence (i.e. |J(sk+1)−J(sk)| ≤ ²1|J(s0)|) as ollows : •Compu a ion o he solu ion usko (7) and hen he solu ion psko (13), bo h co esponding o s=sk. •Compu a ion o he descen di ec ion sk 1de ined by (10) whe e he mul iplie γkis de ined by (12). •Upda e he densi y unc ion in Ω: sk+1 =sk+²sk(1 −sk)sk 1(14) wi h ε∈R+small enough in o de o ensu e he dec ease o he cos unc ion and sk+1 ∈L∞(Ω,[0,1]). 3.2. Nume ical expe imen s Nex , we p esen some nume ical simula ions o N= 2 and he uni squa e Ω = (0,1)2. Fo simplici y, we conside he case Γ0=∂Ω and assume ha Ω is composed o an iso opic homogeneous ma e ial o which aijkl =λδijδkl +µ(δikδjl +δilδjk). λ > 0 and µ > 0 a e he Lam´e coe icien s and δdesigna es he K onecke symbol. The s ess enso becomes simply: σ(u) = λ (∇x·u)IN×N+ 2µε(u). 4 Op imal in e nal s abiliza ion o he elas ici y sys em Sys ems (7) and (13) a e sol ed in space using a C0- ini e elemen me hod and he ime disc e iza ion is pe o med in a s anda d way using cen e ed ini e di e ences o o de wo. Wi hou loss o gene ali y, we conside a cons an damping unc ion a(x) = aXΩ(x) in Ω since he dependence in xis con ained in he densi y s. We ea he simple condi ions : u0= (sin(πx1) sin(πx2),sin(πx1) sin(πx2)),u1= (0,0).(15) Resul s a e ob ained wi h h= 10−2,²1= 10−5,L= 10−1,T= 1, s0(x) = Lon Ω and ²= 10−2(see he algo i hm). 3.2.1. In luence o he damping cons an alue a Nume ical simula ions exhibi a bi u ca ion phenomenon wi h espec o he alue o he damping cons an a. When his alue is small enough, he op imal densi y is always a cha ac e is ic unc ion which sugges s ha he o iginal p oblem (P) is well-posed. On he o he hand, o ala ge enough i appea s ha he op imal densi y akes alues s ic ly in (0,1). In his case, he ill-posedness is ela ed o he o e -damping phenomenon : when m´ınωa(x)goes o in ini y, he damping e m a(x)Xωac s as penaliza ion e m and en o ces he solu ion u o be cons an in ime in ω: a he limi , he e is no mo e dissipa ion in ω (and so in Ω) and he ene gy is cons an (see also [1]). In o de o a oid his phenomenon (which hus appea s i as(x) is oo la ge), he densi y smus ake (a leas locally) alues lowe han 1 in o de o compensa e a. As a esul , (P) is no mo e well-posed. This jus i ies he in oduc ion o he elaxed p oblem (RP ). Fo (λ, µ) = (1/2,1), Figu e 1 depic s he iso- alues o he op imal densi y sop - ob ained a he con e gence o he algo i hm - o se e al alues o a. 3.2.2. F om op imal elaxed designs o quasi-op imal classical designs In he case whe e he op imal densi y sop is no in L∞((0, T)×Ω; {0,1}), one may associa e o sop a cha ac e is ic unc ion spen ∈L∞((0, T )×Ω; {0,1}) whose cos J(spen) is a bi a ily nea o J(sop ). Following [5], one may p oceed as ollows: we i s decompose he domain (0,1)×(0,1) in o M×Ncells such ha Ω = ∪i=1,M [xi, xi+1]×∪j=1,N [yj, yj+1] whe e {xi}(i=1,M+1) and {yj}(j=1,N+1) designa e wo uni o m subdi isions o he in e al (0,1). Then, we associa e o each cell he mean alue mi,j ∈[0,1] de ined by mi,j =1 (xi+1 −xi)(yj+1 −yj)Zxi+1 xiZyj+1 yj sop (x, y)dx dy (16) A las , we de ine he unc ion spen M,N in L∞(Ω,{0,1}) by spen M,N (x, y) = M X i=1 N X j=1 X[xi,(1−√mi,j )xi+√mi,j xi+1]×[yj,(1−√mi,j )yj+√mi,j yj+1](x, y).(17) We easily check ha ||spen M,N ||L1(Ω) =||sop ||L1(Ω), o all M, N > 0. Thus, he cha ac e is ic unc ion spen M,N akes ad an age o he in o ma ion codi ied in he densi y sop . Le us illus a e his poin wi h he op imal densi y ob ained o (λ, µ) = (1/2,1) and a= 50 (see Figu e 1 bo om igh ). The co esponding alue o he cos unc ion is J(sop )≈2,0883. Table 1 collec s he alue o J(spen M,N ) o se e al alues o M=Nand sugges s he con e gence o J(spen M,N ) owa d J(sop )≈2,0883 as Ninc eases. 5 A. M¨unch, P. Ped egal, F. Pe iago 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x1 x2 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x1 x2 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x1 x2 0.25 0.2 0.15 0.1 0.05 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x1 x2 Figu a 1: T= 1, (λ, µ) = (1/2,1), a(x) = aXΩ(x) - Iso- alue o he op imal densi y sop on Ω o a= 5 ( op le ), a= 10 ( op igh ), a= 25 (bo om le ) and a= 50 (bo om igh ). 6 Op imal in e nal s abiliza ion o he elas ici y sys em 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x1 x2 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x1 x2 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x1 x2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 1 2 3 4 5 6 7 8 9 E( ) N=1 N=2 N=5 N=16 Figu a 2: T= 1, (λ, µ) = (1/2,1), a(x) = 50XΩ(x) - Iso- alue o he penalized densi y o N= 2, N= 5 and N= 16 - Bo om igh :E( ) s. associa ed o spen N,N . 7 A. M¨unch, P. Ped egal, F. Pe iago N1 2 3 4 5 6 7 16 N→ ∞ J(spen N,N ) 3.824 3.261 2.446 2.238 2.161 2.137 2.109 2.096 2.0883 Tabla 1: (λ, µ) = (1/2,1), T= 1 - a= 50- Value o he cos unc ion o he penalized cha ac e is ic densi y. Ag adecimien os The second au ho is suppo ed by p ojec s MTM2004-07114 om Minis e io de Edu- caci´on y Ciencia (Spain), and PAI05-029 om JCCM (Cas illa-La Mancha). The hi d au ho is suppo ed by p ojec s MTM2004-07114 om Minis e io de Educaci´on y Ciencia (Spain) and 00675/PI/04 om Fundaci´on S´eneca (Gobie no Regional de Mu cia). Re e encias [1] C. Cas o and S.J. Cox, Achie ing a bi a ily la ge decay in he damped wa e equa ion, SIAM J. Con ol Op im. 39(6), (2001) 1748-1755 [2] S.J. Cox, Designing o op imal ene gy abso p ion II, The damped wa e equa ion, In e na ional se ies o nume ical ma hema ics 126, (1998) 103–109. [3] P. Heb a d and A. Hen o , Op imal shape and posi ion o he ac ua o s o he s abiliza ion o a s ing, Sys ems and con ol le e s 48, (2003) 199–209. [4] A. M¨unch, P. Ped egal and F. Pe iago, A a ia ional app oach o a shape design p oblem o he wa e equa ion, C. R. Acad. Sci. Pa is, Se . I 343 (2006) 371–376. [5] A. M¨unch, P. Ped egal and F. Pe iago, Op imal design o he damping se o he s abiliza ion o he wa e equa ion, J. Di e en ial Equa ions 231 (2006) 331–358. [6] A. M¨unch, P. Ped egal and F. Pe iago, Op imal in e nal s abiliza ion o he linea sys em o elas ici y. P ep in a ailable a h p://ma ema icas.uclm.es/ome a . [7] P. Ped egal, Di -Cu l Young measu es and op imal design in any dimension, Re is a Ma . Complu- ense 20 (2007), no1, 239–255. 8