Boundary-to-Displacement Asymptotic Gains for Wave Systems With Kelvin-Voigt Damping

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1 Boudary-o-Displaceme Asympoic Gais for Wave Sysems Wih Kelvi-Voig Dampig Iasso Karafyllis *, Maria Kooriaki ** ad Miroslav Krsic *** * Dep. of Mahemaics, Naioal Techical Uiversiy of Ahes, Zografou Campus, 578, Ahes, Greece, iasokar@ceral.ua.gr ** Deparme of Saisics ad Operaios Research, Uiversiy of Mala, Tal-Qroqq Campus, Msida, 8, Mala, koorimaria@gmail.com *** Dep. of Mechaical ad Aerospace Eg., Uiversiy of Califoria, Sa Diego, La Jolla, CA 993-4, U.S.A., krsic@ucsd.edu Absrac We provide esimaes for he asympoic gais of he displaceme of a vibraig srig wih edpoi forcig, modeled by he wave equaio wih Kelvi-Voig ad viscous dampig ad a boudary disurbace. Two asympoic gais are sudied: he gai i he L spaial orm ad he gai i he spaial sup orm. I is show ha he asympoic gai propery holds i he L orm of he displaceme wihou ay assumpio for he dampig coefficies. The derivaio of he upper bouds for he asympoic gais is performed by eiher employig a eigefucio expasio mehodology or by meas of a small-gai argume, whereas a ovel frequecy aalysis mehodology is employed for he derivaio of he lower bouds for he asympoic gais. The graphical illusraio of he upper ad lower bouds for he gais shows ha ha he asympoic gai i he L orm is esimaed much more accuraely ha he asympoic gai i he sup orm. Keywords: Wave equaio, ISS, dampig, boudary disurbaces.. Iroducio Asympoic gai properies for fiie-dimesioal sysems were iroduced i [,3,4,]. More specifically, i [] i was show he Ipu-o-Sae Sabiliy (ISS) superposiio heorem, which was exeded i [] for he case of he oupu asympoic gai propery. Recely, he asympoic gai propery has bee used i ime-delay sysems (see [9]) ad absrac ifiie-dimesioal sysems (see [4]). For liear sysems, where he asympoic gai fucio is liear, he asympoic gai (coefficie) may be used as a measure of he sesiiviy of he sysem wih respec o exeral disurbaces. The wave equaio wih viscous ad Kelvi-Voig dampig is he prooype Parial Differeial Equaio (PDE) for he descripio of vibraios i elasic media wih eergy dissipaio bu may also arise i differe physical problems (e.g., moveme of chemicals udergroud; see [5,]). The sudy of he dyamics of he wave equaio wih viscous ad Kelvi-Voig dampig has araced he ieres of may researchers (see [,4,6,3,5,6,]). Recely, he corol of he wave equaio wih Kelvi-

2 Voig dampig was sudied i [,], while he corol of he wave equaio wih viscous dampig was sudied i [7,8,9]. The log-ime behavior of he wave equaio wih viscous ad Kelvi-Voig dampig uder a boudary disurbace was sudied i [], wihi he heoreical framework of he ISS propery (see [7,]). Moreover, a upper boud of he maximum displaceme ha ca be caused by a ui boudary disurbace (Ipu-o-Oupu Sabiliy gai i he sup orm) was give uder a specific assumpio for he dampig coefficies. The aim of he prese work is he exesio of he resul i [] o various direcios by employig he asympoic gai propery for he wave equaio wih viscous ad Kelvi-Voig dampig give by 3 u (, x) u (, x) u (, x) u (, x) x x u(,) d( ) u (,), for ( x, ) (, ) (,), (), for, () where, are cosas. This is he mahemaical model of a vibraig srig wih ieral ad exeral dampig ad u(, x ) deoes he displaceme of he srig a ime ad posiio x [,]. Oe ed of he srig (a x ) is pied dow while exeral forcig acs o he oher ed of he srig (a x ). The effec of he exeral forcig is described by he boudary disurbace d () ad we show ha he asympoic gai propery holds i he L orm of he displaceme wihou ay assumpio for he dampig coefficies (Theorem 3.), we provide upper ad lower bouds for he asympoic gais i he sup-orm ad L orm of he displaceme (Theorems.3 ad.4 ad Theorems 3. ad 3.). This is he firs ime ha a sysemaic sudy for he asympoic gais of a PDE wih a boudary disurbace is beig performed. For example, Fig. ad Fig. depic he lower ad upper bouds of he asympoic gai, i he sup-orm ( L ad U ) ad L orm ( L ad U ), respecively, for ad for a wide rage of values for. Fig. : The lower (red) ad upper (blue) bouds for he asympoic gai i he spaial sup-orm for. The grey area depics possible values for he asympoic gai i he sup-orm.

3 The derivaio of he upper bouds for he asympoic gais is performed by eiher employig a eigefucio expasio mehodology (see also [8,]) or by meas of a small-gai argume (as i [,]). The derivaio of he lower bouds for he asympoic gais is performed by meas of a ovel mehodology, which has ever used before for boudary disurbaces i PDEs: a frequecy aalysis mehodology. The mehodology is coveie ad ca be used for oher PDEs. The srucure of he prese work is as follows: Secio iroduces he reader o he oio of he asympoic gai ad describes he frequecy aalysis mehodology. All mai resuls i he sup-orm are saed i Secio, while all mai resuls i he L orm are give i Secio 3. The proofs of all mai resuls are provided i Secio 4. Secio 5 provides a graphical illusraio of he mai resuls provided i Secios ad 3. Fially, he cocludig remarks of he prese work are coaied i Secio 6. Fig. : The lower (red) ad upper (blue) bouds for he asympoic gai i he spaial L orm for. The grey area depics possible values for he asympoic gai i he L orm. Noaio. Throughou his paper, we adop he followig oaio. : [, ). For every x, x deoes he ieger par of x. Le A be a ope se ad le U, be ses wih A U A. By C U (or C U; ), k we deoe he class of coiuous mappigs o U (which ake values i ). By C U (or k C U; ), where k, we deoe he class of coiuous fucios o U, which have coiuous derivaives of order k o U (ad also ake values i ). L (,) deoes he equivalece class of measurable fucios f :[,] for which f / ( ) f x dx. (,) which f : ess sup f ( x) x(,) L deoes he equivalece class of measurable fucios f :[,] for. Le u : [,] be give. We use he oaio u [] o deoe he profile a cerai, i.e. ( u[ ])( x) u(, x), for all x [,]. Whe u (, x) is differeiable wih respec o x [, ], we use he oaio u (, x) for he derivaive of u wih respec o [, ] u(, x) u x (, x). x, i.e., k For a ieger k, H (, ) deoes he Sobolev space of fucios i L (, ) wih all is weak derivaives up o order k i L (, ). 3

4 . Asympoic Gai i he Sup Norm.. Backgroud m Le (P) be a specific -D PDE corol sysem wih domai [,] ad le d () be a disurbace ipu ha eers he sysem. Le X be he sae space of he sysem ( X is a ormed liear fucioal space of fucios defied o (,) ) ad for each ( u, w) X le ( u, w) deoe he se of bouded disurbaces d : for which he sysem has a uique soluio ( u[ ], w[ ]) X for (a coiuous m mappig ( u[ ], w[ ]) X wih ( u[], w[]) ( u, w) for which he classical semigroup propery holds). We assume ha here exiss a leas oe iiial sae ( u, w) X ad a o-zero bouded disurbace d ( u, w) for which he sysem has a uique soluio ( u[ ], w[ ]) X for (because oherwise he oly allowable disurbace would be he zero ipu). Suppose ha (P) saisfies a oupu asympoic gai propery wih liear gai fucio, i.e., suppose ha here exiss a cosa such ha for every iiial sae ( u, w) X ad for every allowable disurbace d ( u, w) he followig iequaliy holds for he soluio ( u[ ], w[ ]) X : u d limsup [ ] sup ( ), (3) where u [] is a orm of a fucioal space of fucios defied o (,). Two higs should be emphasized a his poi: ) Propery (3) is a oupu asympoic gai propery. Noice ha he sae may have addiioal compoes, deoed here by w []. The oupu map is he map X ( u, w) u S, where S is a ormed liear fucioal space (of fucios defied o (,) ) wih orm e. We assume ha he map X ( u, w) u S is coiuous. Noice ha eve i he case where he oly compoe of he sae is u [], he oupu map is o he ideiy mappig whe orm e does o coicide wih he orm of X. ) Propery (3) is a oupu asympoic gai propery wih liear gai fucio. The fucio g sup d ( ) sup d ( ) ha appears o he righ had side of (3) is liear. This feaure allows us o use he ame d -o- u asympoic gai i he orm of S for he quaiy limsup u [ ] : sup :( u, w ) X, d ( u, w ), d, (4) sup d ( ) where ( u[ ], w[ ]) X deoes he soluio wih ( u[], w[]) ( u, w), correspodig o d ( u, w). I should be oiced ha, i.e., he d -o- u asympoic gai i he orm of S defied by (4), is he smalles cosa for which (3) holds for every iiial sae ( u, w) X ad for every bouded disurbace d : for which he sysem has a uique soluio ( u[ ], w[ ]) X for m. 4

5 Remark.: Due o he semigroup propery, he oupu asympoic gai propery wih liear gai fucio (3) holds if ad oly if he followig iequaliy holds for every iiial sae ( u, w) X ad for every d ( u, w) : limsup u[ ] limsup d( ). (5) Therefore, he d -o- u asympoic gai i he orm of S defied by (4) saisfies he followig equaio limsup u [ ] sup :( u, w ) X, d ( u, w ), limsup d( ) limsup d ( ) provided ha here exiss a leas oe iiial sae ( u, w) X ad a disurbace d ( u, w) wih limsup d ( ). If for every ( u, w) X he allowable disurbace se ( u, w) coais oly disurbaces d : m wih d.. Frequecy Aalysis limsup ( ), he defiiio (4) ad iequaliy (5) imply ha. The frequecy aalysis of he d -o- u asympoic gai i he orm of S cosiss of wo seps. m Sep : Le d : be a parameerized family of o-zero ipus, wih parameer, which are periodic wih period T, i.e., d d ( ) for all. For each, fid a periodic soluio ( u[ ], w[ ]) X of (P) wih period T ha correspods o he ipu : m d. Sep : If Sep ca be accomplished, he esimae he d -o- u asympoic gai i he orm of S, by meas of he followig iequaliy max u [ ] / sup sup d ( ). (6) / Iequaliy (6) is a direc cosequece of (4), he fac ha ( u[ ], w[ ]) X is a periodic soluio of (P) m wih period T ha correspods o he o-zero ipu : ad he fac ha he mappigs X ( u, w) u S, ( u[ ], w[ ]) X for each ). d are coiuous mappigs (ad herefore max u [ ] / exiss.3. Wave Equaio wih Kelvi-Voig ad Viscous Dampig Cosider he wave equaio wih Kelvi-Voig ad viscous dampig (), (), where, are cosas. Wihou loss of geeraliy, he esio, i.e., he coefficie of u x (, x) i () has bee assumed o be equal o ; his ca always be achieved wih appropriae ime scalig. I order o obai a exisece/uiqueess resul for he wave equaio wih Kelvi-Voig ad viscous dampig we firs eed o move he disurbace from he boudary o he domai ad make he boudary codiios homogeeous. However, his process should be doe wih cauio because we would also like he o-homogeeous erm ha will appear i he PDE o be expressed by a sufficiely fas coverge Fourier series. This may be doe i he followig way: if we assume ha dc 4 he we may defie he fucios 5

6 s i i i i, for, ( ) ( ) ( ) exp ( s) ds i,, (7) wih ( ) d( ) d( ), for, (8) ad perform he followig rasformaio 3 s u(, x) w(, x) d( )( x) x 3x x exp ( s) ds s 8x x 5x 3x exp 4 ( s) ds 36 which rasforms problem (), () o he followig problem w w w w 8x x 5x 3x (, x) (, x) (, x) (, x) 6 x x 36 ( ), for ( x, ) (, ) (,), () w(,) w(,), for. () For every pair of fucios f H 6 (,), 4 (4) (4) g H (,) wih f () f () f (), f () f () f (), g() g(), g() g(), a soluio wc [,] wih w [ ] C [,] for all of he iiial-boudary value problem (), () wih w w[] f, [] g () ca be expressed by he formula where a () for,,... wih are he soluios of w(, x) a ( )si( x), for, x [,] (3) ( ) a ( ) a ( ) a ( ) 8x x 5x 3x si( x) dx 6 (4) 36 a () f ( x)si( x) dx, (9) a () g( x)si( x) dx, for,,.... (5) 6 Ideed, sice a () O ( ) (a cosequece of he fac ha 6 (4) f H (,) wih f () f () f (), (4) 4 f () f () f () ), a () O ( ) (a cosequece of he fac ha g H 4 (,) wih g() g(), 6 ) ad sice c : 36 g() g() x x 5 x 3 x si( x ) dx O ( ), i ca be show 6 ha he soluio of (4), (5) saisfies a ( ) O ( ) for every fixed. This follows from he fac ha for sufficiely large (so ha ), we have for : a () a ( ) ( k r )exp( ( k r ) ) ( r k )exp( ( k r ) ) r a() c exp( ( ) ) exp( ( ) ) k r k r exp( ( k r )( s )) exp( ( k r )( s )) ( s ) ds r r where k ( ), 7 r 4 (oice ha c r O( ) ad ha here exiss a cosa L so ha k r L for all sufficiely large ). Therefore, he fucio w defied by 6

7 (3) is of class C [,] wih w [ ] C [,] for all ad saisfies (), () for all x [,]. Usig he rasformaio (9), we are i a posiio o give he followig proposiio. 4 Proposiio.: Suppose ha (4) (4) f (), f () f () f (), g() d() dc ad f H 6 (,), g H 4 (,) wih f() d(), g () / d() d(),, f (), g() g(). The he iiial-boudary value problem (), () wih u u[] f, [] g (6) has a uique soluio uc [,] wih u [ ] C [,] for all, which saisfies (), () for all, x [,]. Proof: A sraighforward applicaio of rasformaio (9) ad formulas (7), (8). Uiqueess follows by cosiderig he eergy fucioal u v u v E( ) (, x) (, x) (, x) (, x) dx x x where u, v C [,] wih u [ ] C [,], v [ ] C [,] for all, are wo arbirary soluios he iiial-boudary value problem (), (), (6). Sice E ( ) for all ad sice E(), we obai ha E ( ) for all. This implies ha u x(, x) v x(, x) for all, x [,], which gives u v. The proof is complee. Le X be he liear space of all ( f, g) C ([,]) C ([,]) for which here exiss a bouded ipu dc such ha he iiial-boudary value problem (), (), (6) has a uique soluio uc [,] wih u [ ] C [,] for all. Noice ha Proposiio. guaraees ha D X, where 6 4 (4) (4) (, ) (,) (,): () () () () () () (). D f g H H f f f f f g g For every ( f, g) X, le ( f, g) deoe he se of bouded disurbaces dc for which he iiial-boudary value problem (), (), (6) has a uique soluio uc [,] wih u [ ] C [,] for all..4. Asympoic Gai i he sup orm We assume ha he followig assumpio holds. (H) There exiss a cosa such ha for every bouded disurbace dc ( ) for which (), () has a uique soluio uc [,] wih u [ ] C [,] for all, he followig iequaliy holds: limsup u[ ] sup d( ). (7) I was show i [] ha Assumpio (H) holds whe. Whe Assumpio (H) holds, we are i a posiio o defie he asympoic gai of he displaceme i he sup orm by meas of he formula 7

8 limsup u [ ] : sup :( f, g) X, d ( f, g), d sup d ( ), (8) where uc [,] wih u [ ] C [,] for all is he soluio of he iiial-boudary value problem (), (), (6). The resuls i [] show ha whe, he followig iequaliy holds g, (9) where g s : if si( ) p( s, ) : s s ad p( s, ) : s s ( ). Our firs mai resul i he prese work cocerig he sup orm is a sharper resul ha he resul give i []. Is proof is provided i Secio 4. Theorem.3: Cosider he wave equaio wih Kelvi-Voig ad viscous dampig (), (), where, are cosas wih. The Assumpio (H) holds ad iequaliy (9) holds wih g s : if si( ) p( s, ) : s s p( s, ) : s s ( ). ad Our secod mai resul i he prese work cocerig he sup orm is give below ad i is proved by meas of he frequecy aalysis procedure ha was described above. Is proof is provided i Secio 4. Theorem.4: Cosider he wave equaio wih Kelvi-Voig ad viscous dampig (), (), where, are cosas. Suppose ha Assumpio (H) holds. Defie for each : r : a: rcos( / ) b: rsi( / ), () () where (, ) is he uique agle ha saisfies he equaios The he followig iequaliy holds: cos( ), si( ). () r r max cosh( ax) cos( bx) x[,] sup cosh( a) cos( b). (3) Noicig ha he righ had side of (3) is always greaer or equal o, Theorem.3 allows us o obai he followig corollary for he sup orm. 8

9 Corollary.5: Cosider he wave equaio wih Kelvi-Voig ad viscous dampig (), (), where, are cosas. Suppose ha. The. 3. Asympoic Gai i he L Norm 3.. Exisece of Asympoic Gai i he L Norm I is o clear wheher Assumpio (H) holds for all, for he wave equaio wih Kelvi- Voig ad viscous dampig. However, he aalogue of Assumpio (H) for he L orm holds for all,. This is a cosequece of he followig heorem. Theorem 3.: Cosider he wave equaio wih Kelvi-Voig ad viscous dampig (), (), where, are cosas. There exiss a cosa such ha for every bouded disurbace dc ( ) for which (), () has a uique soluio uc [,] wih u [ ] C [,] for all, he followig iequaliy holds: limsup u [ ] sup d ( ). (4) Moreover, he cosa saisfies he iequaliy G(, ), (5) where if 3 G(, ) : (6) A if ad A A exp A, : if, 4, (7) A, if ad, (8) exp, if, (9) arccos 4 exp 4, if. (3) 9

10 The proof of Theorem 3. ca be foud i he followig secio. A direc cosequece of Theorem 3. is he fac ha for every,, we ca defie he asympoic gai of he displaceme i he L orm for he wave equaio wih Kelvi-Voig ad viscous dampig i he followig way: limsup u [ ] : sup :( f, g) X, d ( f, g), d, (3) sup d ( ) where uc [,] wih u [ ] C [,] for all is he soluio of he iiial-boudary value problem (), (), (6). Moreover, i follows from Theorem 3. ha he followig iequaliy holds: G(, ), (3) where he fucio G(, ) is defied by (6). Fially, if Assumpio (H) holds he he followig iequaliy is a direc cosequece of he fac ha u for all u C ([,]) : u. (33) 3.. Lower Boud of Asympoic Gai i he L Norm Our secod mai resul i he prese work cocerig he L orm is give below ad i is proved by meas of he frequecy aalysis procedure ha was described above. Is proof is provided i Secio 4. Theorem 3.: Cosider he wave equaio wih Kelvi-Voig ad viscous dampig (), (), where, are cosas. Defie r, a, b by meas of (), () for each, where (, ) is he uique agle ha saisfies equaios (). The he followig iequaliy holds: where Q( ) : p M wih sup Q( ), (34) 4 a b 4 b si( b) cosh( a) a sih( a) cos( b) sih ( a) si ( b) M :, (35) 6a b cosh( a) cos( b) bsih( a) asi( b) p 4ab cosh( a) cos( b). (36) Combiig Theorem 3. ad Theorem 3., we ge he followig corollary, which is proved i he followig secio. Corollary 3.3: Cosider he wave equaio wih Kelvi-Voig ad viscous dampig (), (), where, are cosas. Suppose ha. The 3.

11 4. Proofs of Mai Resuls Proof of Theorem.3: The fac ha Assumpio (H) holds whe is a direc cosequece of Theorem. i []. We ex oice ha every soluio uc [,] wih u [ ] C [,] for all, of (), () is a soluio of he sysem of he iegro-differeial equaio u u s (, x) (, x) u(, x) exp u( s, x) ds x (37) u u exp (, x) (, x) u(, x) x Le (, ), [, ) be a pair of agles wih (such a pair of agles exiss due o he fac ha ). Defie he posiive fucio ( x) : si( x) for x [,] ad he orm u : max u( x) ( x). Noice ha Assumpios (H), (H), (H3), (H4) i [] hold for he PDE problem, x u u (, x) (, x) u(, x) f (, x) x wih u(,) d( ) u(,). Applyig Corollary 4.3 i [] o (37) wih s u u f (, x) exp u( s, x) ds exp (, x) (, x) u(, x) x ad usig he semigroup propery ad he fac ha limsup f [ ], sup u[ ], u, d u, limsup [ ] limsup ( ) limsup [ ] si( ), we ge I follows from he above iequaliy ad he fac ha u u for all u C ([,]) ha, s s si( ) for all (, ), [, ) wih s s every (, ) wih s s ad s :. Therefore, for, i holds ha s s ( ) si( ) ad p( s, ) : s s ( ) holds wih complee. g s : if si( ) p( s, ) : s s. Thus, iequaliy (9). The proof is Proof of Theorem.4: We apply he frequecy aalysis mehodology for he parameerized family of ipus d ( ) si( ) wih parameer. A periodic soluio u [] of (), () ha correspods o he ipu d ( ) si( ) is give by he formula where hg, are soluios of he boudary-value problem u (, x) si( ) h( x) cos( ) g( x), for, x [,] (38)

12 g ( x) g( x) h( x) h( x), (39) g( x) g( x) h ( x) h( x) I ca be verified ha he fucios h(), h() g() g(). (4) cosh( a( x)) cos( b)cosh( ax) cos( bx) si( b)si( bx)cosh( ax) hx ( ), cosh( a) cos( b) (4) si( b)cos( bx)sih( ax) sih( a( x)) cos( b)sih( ax) si( bx) gx ( ), cosh( a) cos( b) where ab, are defied by (), are soluios of he boudary-value problem (39), (4). Usig (6) ad (38), we obai he iequaliy Sice / x sup max max si( ) h( x) cos( ) g( x). (4) max max si( ) h( x) cos( ) g( x) max max si( ) h( x) cos( ) g( x) max h ( x) g ( x) (43) / x x / x ad sice (4) implies ha cosh( a( x)) cos( b( x)) h ( x) g ( x), for x [,], cosh( a) cos( b) (44) we obai from (4), (43) ad (44) he desired iequaliy (3). The proof is complee. Proof of Theorem 3.: I suffices o show ha here exiss a cosa such ha for every bouded disurbace dc ( ) for which (), () has a uique soluio uc [,] wih u [ ] C [,] for all, wih he followig propery: For every here exiss T such ha u[ ] sup d( s) for all T. Le a arbirary bouded disurbace dc ( ) for which (), () has a uique soluio [,] wih u [ ] C [,] for all. Defie: u C y ( ) u (, x )si( x ) dx s, for,,.... (45) Defiiio (45) i cojucio wih (), () implies ha he followig differeial equaios hold for all ad,,... : Whe he we may defie y ( ) y ( ) y ( ) d( ) d( ) (46) k :, r 4 : (47)

13 ad express he soluio of (46) by he followig formula: exp( ( k r) ) y ( ) k r ( k r )exp( r ) y () y () exp( r ) r exp( ( k r ) ) exp( r ) d() g ( ), r g ( ) : ( k r )exp( ( k r )( )) ( r k )exp( ( k r )( )) d( ) d. (49) r Whe (or equivalely whe ) he he soluio of (46) is give by he followig formulas: (48), (5) y ( ) k y ()exp( k ) y () exp( k ) exp( k ) d() g ( ) Whe he we may defie:. (5) g ( ) ( k )( s) exp( k ( s)) d( s) ds : 4 (5) ad i his case he soluio of (46) is give by he followig formulas: k y() y ( ) y () si( ) cos( ) exp( k) si( )exp( k) si( )exp( k) d() g ( ), (53) g ( ) cos( ( s)) ( k )si( ( s)) exp( k ( s)) d( s) ds. (54) We ex show ha for every here exiss T such ha for all T i holds ha: y ( ) g ( ) for all T,,,... (55) Defiiios (47) imply ha here exis posiive cosas,, c so ha he followig iequaliies hold for all wih : k, r r c. I follows from (48) ad (56) ha he followig esimae holds for all ad wih : exp( ) y ( ) g ( ) ( k r ) y () y () exp( ) d() r c (56). (57) Defiiio (45) ad () imply ha y d u x x dx. Therefore, we ge: ( ) ( ) (, )cos( ) () ( ) d y u [ ], for all,,,... (58) Similarly, defiiio (3) implies ha: 3

14 ( ) u y [ ], for all ad. (59) Usig (56), (57), (58) ad (59), we obai for all ad wih : exp( ) u y ( ) g( ) ( r ) d() u[] [] exp( ) d(). r c (6) I follows from (56) ad (6) ha he followig iequaliy holds for all ad wih : exp( ) u y( ) g( ) d() u[] [] d(). (6) c c c holds for all iegers he (55) is a direc cosequece of (6). We ex cosider If he case ha here exis iegers for which. Le N be he ieger defied i he followig way: N if N ( ) if. Noice ha for all N. Wihou loss of geeraliy, we may assume ha he posiive cosa ad for which (56) holds, also saisfies he iequaliy. I he follows from (47), (5) ad (53) ha here exiss a cosa G (depedig o he iiial codiios ad d () ) such ha he followig iequaliy holds for all ad,..., N wih : The above iequaliy implies ha y ( ) g ( ) G exp( ). N y( ) g( ) G exp( ). (6) Therefore, if here exis iegers for which, he (55) is a direc cosequece of (6) ad (6). Thus, we have show ha (55) holds. For all wih, defiiios (47), (49) imply he followig esimaes for all : g ( ) d ( k r )exp( ( k r )( )) ( r k )exp( ( k r )( )) d r r d k r k r if or (equivalely if k r ) (63) g ( ) d ( k r )exp( ( k r )( )) ( r k )exp( ( k r )( )) d r k k r r d k r k r r k r k r k r if ad (equivalely if k r ), (64) 4

15 where d : sup s d( s). Noice ha (which implies for all ), defiiios (47), (5) ad esimae (63) give: g () d, for all ad. (65) Usig (63), (64), (5), (54), (47), (5) ad defiiios (7), (8), (3), we obai he followig esimaes for he case : g() A d, for all ad. (66) We are ready o fiish he proof. Le arbirary be give. Sice lim A (a cosequece of defiiio (7)), i follows ha he sequece { A } is bouded. Le A be a upper boud for he sequece { A }. Pick 6 3 d A. The here exiss sufficiely small so ha T so ha (55) holds. Usig he riagle iequaliy, we obai from (55) for all ad T: y ( ) g ( ) g ( ). (67) Defiiio (45) ad he fac ha he se of fucios si( x):,,... is a orhoormal basis of L (,) implies ha Parseval s ideiy holds, i.e., (67) for all T: Usig (68) ad he fac u[ ] y ( ). If he we ge from (66), u[] d A d A 6, we ge for all T:. (68) where A is he upper boud for he sequece { } 3d A u[ ] G(, ) d d A 6 3, (69) A. I follows from (69) ad he fac ha 6 ha u[ ] G(, ) d for all T. The aalysis is similar for he case (simply se A i he above aalysis). The proof is complee. Proof of Theorem 3.: We apply he frequecy aalysis mehodology for he parameerized family of ipus d ( ) si( ) wih parameer. A periodic soluio u [] of (), () ha correspods o he ipu d ( ) si( ) is give by (38), where hg, are give by (4). Usig (38), we ge for all : where u[ ] p q cos( ) q si( ), (7) p : h ( x) g ( x) dx, (7) q : g ( x) h ( x) dx, (7) q : h( x) g( x) dx. (73) 5

16 Usig (4), (44), (7), (7), (73), we obai (36) as well as he followig formulas: cosh( a) cos( b) bsi( b) a sih( a) q cosh( a) cos( b) 4a b cosh( a) cos( b) Equaio (7) implies ha q sih( a) si( b) a si( b) b sih( a) cosh( ) cos( ) a b 4a b cosh( a) cos( b) max u[ ] : p q q (74) (75), (76) Equaio (4.4) combied wih (74), (75) ad (6) gives (34) wih Q( ) : p M ad M give by (35). The proof is complee. Proof of Corollary 3.3: I suffices o show ha Q lim ( ) 3, where Q( ) : p M wih pm, defied by (35), (36). Ideed, defiiios (35), (36) imply ha p 6 ad M 36 as ( ab, ) (,). Moreover, defiiios (), () imply ha a ad b as. The proof is complee. 5. Graphical Illusraio of he Theorem Saemes This secio is devoed o he graphical illusraio of he heorems saemes for he upper ad lower bouds of he asympoic gais. To his purpose, we defie: U : g where, for, wih, (77) ad p( s, ) : s s ( ) g s : if si( ) p( s, ) : s s ad ab, are beig defied by (), where A( ) : L x[,] : sup A( ),, (78) max cosh( ax) cos( bx) L cosh( a) cos( b), (79) Q, (8) : sup ( ) Q( ) : p M, (8) ad pm, are beig defied by (35), (36) ad U : G(, ), (8) where G(, ) is defied by (6). Noice ha he resuls of he previous secios guaraee ha L U, for all, wih, (83), for all, for which Assumpio (H) holds, (84) 6

17 L U, for all,. (85) Fig. ad Fig. 3 depic he lower ad upper bouds of he asympoic gai i he sup orm for ad.5, respecively, ad for a wide rage of values for. Fig. 3: The lower (red) ad upper (blue) bouds for he asympoic gai i he sup orm for.5. The grey area depics possible values for he asympoic gai i he sup orm. Fig. 4, Fig. 5 ad Fig. 6 are Bode-like plos for he logarihm of A( ) defied by (79) for.,. ad., respecively, ad for four differe values of.these plos idicae ha for small ad, A( ) preses spikes a frequecies which differ by. O he oher had, for sufficiely large ad, A( ) ideically equal o. Fig. ad Fig. 7 depic he lower ad upper bouds of he asympoic gai i he L orm, for ad.5, respecively, ad for a wide rage of values. Noice ha for, we ge L U / 3. Comparig Fig. ad Fig. 7 wih Fig. ad Fig. 3, we coclude ha he asympoic gai i he L orm is esimaed much more accuraely ha he asympoic gai i he sup orm. Fig. 4: Bode-like plo of l( A( )) for.. 7

18 Fig. 5: Bode-like plo of l( A( )) for.. Fig. 6: Bode-like plo of l( A( )) for.. Fig. 7: The lower (red) ad upper (blue) bouds for he asympoic gai i he L orm for.5. The grey area depics possible values for he asympoic gai i he L orm. 8

19 Fig. 8, Fig. 9, Fig. ad Fig. are Bode-like plos for he logarihm of Q( ) defied by (8) for.,.,. ad.5, respecively, ad for four differe values of. As i he sup orm, hese plos idicae ha for small ad, Q( ) preses spikes a frequecies which differ by. These are dagerous frequecies: he ampliude of he oscillaio of he boudary of a srig becomes magified i is domai. Fig. 8: Bode-like plo of l(q( )) for.. Fig. 9: Bode-like plo of l(q( )) for.. Fig. : Bode-like plo of l(q( )) for.. 9

20 Fig. : Bode-like plo of l(q( )) for Cocludig Remarks We have provided esimaes for he asympoic gais of he displaceme of a vibraig srig wih exeral forcig. By cosiderig a exeral boudary disurbace for he wave equaio wih Kelvi- Voig ad viscous dampig, we have show ha he asympoic gai propery holds i he spaial L orm of he displaceme wihou ay assumpio for he dampig coefficies. We have also provided upper ad lower bouds for he asympoic gais i he spaial sup-orm ad he spaial L orm of he displaceme. As oed i he iroducio, his is he firs sysemaic sudy of he asympoic gais for a PDE wih a boudary disurbace. Alhough here we sudied he wave equaio wih Kelvi-Voig ad viscous dampig, i is expeced ha similar sudies will be performed for may oher impora PDEs. Refereces [] Ageli, D., B. Igalls, E. D. Soag ad Y. Wag, Separaio Priciples for Ipu-Oupu ad Iegral-Ipu o Sae Sabiliy, SIAM Joural o Corol ad Opimizaio, 43, 4, [] Che, S., K. Liu ad Z. Liu, Specrum ad Sabiliy for Elasic Sysems wih Global or Local Kelvi-Voig Dampig, SIAM Joural o Applied Mahemaics, 59(), 998, [3] Coro, J. M., L. Praly ad A. Teel, Feedback Sabilizaio of Noliear Sysems: Sufficie Codiios ad Lyapuov ad Ipu-Oupu Techiques, i Treds i Corol, A. Isidori, ed., Spriger- Verlag, Lodo, 995. [4] Gerbi, S., ad B. Said-Houari, Exisece ad Expoeial Sabiliy of a Damped Wave Equaio Wih Dyamic Boudary Codiios ad a Delay Term, Applied Mahemaics ad Compuaio, 8(4),, 9-9. [5] Gueher, R. B., ad J. W. Lee, Parial Differeial Equaios of Mahemaical Physics ad Iegral Equaios, Dover, 996. [6] Guo, B.-Z., J.-M. Wag ad G.-D. Zhag, Specral Aalysis of a Wave Equaio wih Kelvi- Voig Dampig, Zeischrif für Agewade Mahemaik ud Mechaik, 9(4),, [7] Karafyllis, I. ad Z.-P. Jiag, Sabiliy ad Sabilizaio of Noliear Sysems, Spriger-Verlag Lodo (Series: Commuicaios ad Corol Egieerig),.

21 [8] Karafyllis, I. ad M. Krsic, ISS Wih Respec o Boudary Disurbaces for -D Parabolic PDEs, IEEE Trasacios o Auomaic Corol, 6(), 6, [9] Karafyllis, I. ad M. Krsic, Predicor Feedback for Delay Sysems: Implemeaios ad Approximaios, Birkhäuser, Boso (Series: Mahemaics, Sysems & Corol: Foudaios & Applicaios), 7. [] Karafyllis, I. ad M. Krsic, Small-Gai Sabiliy Aalysis of Cerai Hyperbolic-Parabolic PDE Loops, Sysems ad Corol Leers, 8, 8, 5-6. [] Karafyllis, I. ad M. Krsic, Ipu-o-Sae Sabiliy for PDEs, Spriger-Verlag, Lodo (Series: Commuicaios ad Corol Egieerig), 8. [] Krsic, M. ad A. Smyshlyaev, Boudary Corol of PDEs: A Course o Backseppig Desigs, SIAM, 8. [3] Liu, K. ad B. Rao, Expoeial Sabiliy for he Wave Equaios Wih Local Kelvi Voig Dampig, Zeischrif für Agewade Mahemaik ud Physik, 57, 6, [4] Mirocheko, A. ad F. Wirh, Characerizaios of Ipu-o-Sae Sabiliy for Ifiie- Dimesioal Sysems, IEEE Trasacios o Auomaic Corol, 63(6), 8, [5] Pellicer, M., Large Time Dyamics of a Noliear Sprig-Mass-Damper Model, Noliear Aalysis, 69(), 8, [6] Pellicer, M. ad J. Sola-Morales, Aalysis of a Viscoelasic Sprig-Mass Model, Joural of Mahemaical Aalysis ad Applicaios, 94(), 4, [7] Roma, C., D. Bresch-Pieri, C. Prieur ad O. Seame, Robusess of a Adapive Oupu Feedback for a Ai-Damped Boudary Wave PDE i Presece of I-Domai Viscous Dampig, Proceedigs of he America Corol Coferece, 6, Boso, U.S.A.. [8] Roma, C., D. Bresch-Pieri, E. Cerpa, C. Prieur, ad O. Seame, Backseppig Observer Based- Corol for a Ai-Damped Boudary Wave PDE i Presece of I-Domai Viscous Dampig, Proceedigs of he 55 h IEEE Coferece o Decisio ad Corol, 6, [9] Roma, C., D. Bresch-Pieri, E. Cerpa, C. Prieur, ad O. Seame, Backseppig Corol of a Wave PDE Wih Usable Source Terms ad Dyamic Boudary, IEEE Corol Sysems Leers,, 8, [] Siraosia, A., M. Krsic, A. Smyshlyaev, ad M. Beme, Moio Plaig ad Trackig for Tip Displaceme ad Deflecio Agle for Flexible Beams, Joural of Dyamic Sysems, Measureme, ad Corol, vol 3, 9. [] Soag, E. D. ad Y. Wag, New Characerizaios of Ipu o Sae Sabiliy, IEEE Trasacios o Auomaic Corol, 4, 996, [] Zhag, Q., Expoeial Sabiliy of a Elasic Srig wih Local Kelvi Voig Dampig, Zeischrif für Agewade Mahemaik ud Physik, 6(6),, 9 5.