D DAVID PUBLISHING. Simultaneous Exact Controllability of Wave Sound Propagations in Multilayer Media. 1. Introduction. v t q t.

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1 Jornal of Maheaics and Syse Science 6 (6-9 doi:.765/59-59/6.8. D DAVID PUBLISHING Silaneos Exac Conrollabiliy of Wave Sond Propagaions in Mlilayer Media Alexis Rodrigez Carranza, Lis. J. Cacha, Obidio Rbio Mercedes Naional niversiy of rjillo, Per. Received: Janary, 6 / Acceped: Febrary 6, 6 / Pblished: Ags 5, 6. Absrac: In his wor we sdy one proble of aheaical ineres for heir applicaions in several opics in Applied Science. We sdy silaneos conrollabiliy of a pair of syses which odel he evolion of sond in a copressible flow considered as a ransission proble. We show he well posed of he proble. Frherore provided appropriae condiions in he geoery of he doain are valid and siable asspions on he flid, is possible o condce he pair of syses o he eqilibri in a silaneos way sing only one conrol. Key words: Conrollabiliy, wave sond, H.U.M. Inrodcion In his wor, we considered an eqaions syse o describe an evolion of he wave sond or copressible flids. A linear odel well now is given by a syse [7] + α p=, in Ω (, p + div( =, in Ω (, (. η = Q, in S (, p=, in S (, x (, = ( x, px (, = p( x where p= px (, is acosic pressre, = (,, and j = j( x, are flid velociy field, α > is he densiy of eqilibri and > is he copressibiliy facor of flid. Here Ω is an open sbse of IR wih reglariy bondary condiions S S = Ω and S S =. o solve he silaneos conrollabiliy we considered a syse given by Corresponding ahor: Alexis Rodrigez Carranza, Naional niversiy of rjillo, Per. v + γ q=, in Ω (, q + τdiv( v =, in Ω (, ( q= P, in S (, q=, in S (, vx (, = v( x, qx (, = q( x where γ > and τ >. Q and P in ( and (, respecively; hese are conrol fncions. In 986, D.L. Rssell [] and J.L.Lions [8] proposed o solve a exac conrollabiliy proble for an evolion odel, sing only one conrol fncion. hey called ha proble as silaneos conrollabiliy. he absences of dissipaive effecs as in ( and (, he proble presen difficlies for he solion, see he exaples [4], [5] and [8], where hey perrbed he lipliers sed for he conrollabiliy. he proble of silaneos conrollabiliy for he syses ( and ( is o ae a conrol for boh of syse sing only one conrol fncion, i.e., given > any iniial condiion, (, p, v, q, and final (, p, v, q in appropriae fncional space, find Px (, and Qx (, sch ha A solion {, pvq,, } of ( and ( saisfied in

2 4 Silaneos Exac Conrollabiliy of Wave Sond Propagaions in Mlilayer Media ( (.,, p(.,, v(.,, q(., = (, p, v, q he conrol fncion, Px (,, for ( was given in ers of Qx (,. A ehod o solve he conrollabiliy proble is Hilber Uniqeness Mehod (H.U.M proposed by J.L.Lions, i is a consrcion of an appropriae srcre for he Hilber space in he iniial condiions space. hese srcre are conneced by niqeness properies. An iporan conribion o he conrollabiliy probles ( and ( were ade by Kapionov e. G. Perla Menzala [5]. In [5] he ahor answered posiively for a silaneos conrol and hey showed ha he conrol P = γ Q cold be se o solve a proble. In his wor we sdy a conrollabiliy proble of hese syses wih a perspecive for applicaions as a proble of ransission; his is described below. > and =,,,. For each, loo a b an open and conneced sbse wih reglar bondary and sch ha σ b σ, b b +. place ω = b σ, Ω = b+ b, =,,, e ω = σ b. Given σ and σ open liied sbse and conneced in ir, wih σ σ. Also ω = σ σ, we denoed σ = s, σ = s. And fixed an ineger > and =,,,. For each, b is an open sbse and conneced, wih reglariy in he bondary sch ha, σ b σ, b b +. We p ω = b σ, ω = b+ b, =,,, and ω = σ b. and, ω = ω, for i j, we ae ω i ω j = j= j and ω = s s. Exaples for his decoposiion is showed in Fig. We need a solion defined by par on each sb doain; for ha, we considered he syses ( and ( rewrie on sb doains Ω, and + α p =, in Ω (, p div( in ( + =, Ω, ( x, = ( x, p ( x, = p( x ( Fig. Case = and =.

3 Silaneos Exac Conrollabiliy of Wave Sond Propagaions in Mlilayer Media 5 v + γ q =, in Ω (, q div( v in ( + τ =, Ω, v ( x, = v( x, q ( x, = q( x =,,,,. (4 wih bondary condiions ( and (. he inerfaces of ransission condiions Γ = Ω, given by α p = α p (. η = (. η =,,, ( x, Γ (, (5 γ q = γ q ( τ v. η = τ ( v. η =,,, ( x, Γ (, (6 for he syses ( and (4, respecively. he fncions α,, γ and τ are he resricion for he fncions αγτ,,, on he syses ( and (, we assed ha hose fncions were consan by pars, sricly posiive and we los he coniniy only in Γ, =,,,. he objecive in his secion is o ge he esiaion of + α ( p ( dx Ω = + τ v + γ ( q (7 h C [ αp τ( v. η ] ds S d. For soe >, C > and >. he ineqaliy (7 is naed fro an ineqaliy of observaion which is in he heore assing geoerical properies on doain Ω and in he inerfaces Γ. Sch ha, o prove (7 we assed onooniciy condiions in he coefficiens of he syses ( and (4. he reqireen necessary were fond by Lions[8] in his sdy of ransission proble. Lagnese[4] sed he sae hypohesis o prove he resl of conrollabiliy for a hyperbolic proble.. Fncional Spaces Given he Hilber space X = L ( Ω L ( Ω, associae o (. We define an scalar prodc in X, given by (, p,(, p X, hen: X = { α p } (, p,(, p =. + p dx Ω (8 Conseqenly, we considered X = L ( Ω L ( Ω associae o (4. We define a scalar prodc in X, as ( vq, (, vq, X, hen: X = { τ v γ q } ( v, q(, v, q = v. + q dx Ω (9 We have considered a oal energy o he proble (, (4, (5, (6 and he bondary condiions in (, (, as + α ( p + E( = dx ( Ω = τ v + γ q ( Maing a rigorosly way for he inerfaces condiions, we can see a lea ; for ore deails see Perla e al.[5]. Lea. Given Ω bonded region in IR, wih reglariy in he bondary Ω. he applicaion C ( Ω C ( Ω = (,, η. where η = η( x is as exerior ni noral vecor in

4 6 Silaneos Exac Conrollabiliy of Wave Sond Propagaions in Mlilayer Media x Ω. We can exend by coniniy applicaion H H / ( Ω and where H = { L } ( Ω, div( L ( Ω H / ( Ω is dal space of H / ( Ω o siplify he noaion we wrie as, hen, Given ( α A(, p = p, div( ( vq, DA ( ( γ τ A( vq, = q, div( v as, he sae way for all sybols in he region Ω. by he lea is clearly ha he spaces H (, p X, = X ( α p,div( X ( vq, X, H= q v X ( γ,τdiv( we can define he sb spaces: (, p H, Z = α p = α p (. η = (. η, inγ.. η = S, p = S and, ( vq, H, Z = γ q = γ q ( τ v. η = τ ( v. η, inγ, q =, in Ω = S S Observe ha C ( Ω Z j, j =,. Also Z and Z are dense in X and X, respecively. Considering he bonded operaor defined as Given hen, A: Z = DA ( X X j j j j j ( p, DA ( X Perla e al. [5] showed ha operaor A is sew-adjoin, i.e, A = A, he sae resl was proved for A. Using he Sone s heore, we have proved ha A and A generae infiniesially a grop of srongly coninos ni operaors { U j( } IR, in X and X, respecively. Moreover, U j( w j is srongly differeniable in relaion o and for any wj DA ( j, d U j( w j = AUw j ( j d. Ineqaliy of Observabiliy Using he liplier s heory (see Koorni[], we ae he proof. he liplier was odified o ge a good esiaes in he bondary. hese liplier were sed in several wors. he invarian of he syses ( and (, in relaions o dilaaions grops in all variables, see [5]. Given h: C( Ω C ( Ω IR an axiliary fncion, i will be chosen in he nex seps; and, given ( p, V DA ( a solion of he syse (. Considering he liplier given by: ( α α M = p. h + p( x, s ds = ( M p h M = and (, p solion of (, we have he ideniies { } { α } = M p + div + M. + p + { ( s α } M. + p ds

5 Silaneos Exac Conrollabiliy of Wave Sond Propagaions in Mlilayer Media 7 he expression above, we ae rewrie as A = div( B J ( where ( ( ( ( A = + αp p. h + αp p x, s ds x,. x, s ds ( B = αp + α p h h + (. h α p( x, s ds h J = ( α( h p i j i, j= xi xj We choosea fncion as : hx ( = x x + δ Φ ( x ( where x σ and Φ saisfy Φ = en Ω Φ Vol( Ω =, in, S area( S Φ Vol( Ω =, in, S area( S Is easy show ha µ ( Ω. η =. η e Γ = η e Γ = l ( x e Γ l H ( Ω e h= e S η ( (4 Now, considered he hypohesis in he doain Ω. Given δ > sch ha, soe x σ, we have δ ( µ ( Ω <, Vol( Ω ( x x. η δ, for x S area( S Vol( Ω (5 ( x x. η δ, for x S area( S Φ x x. η + δ, x Γ, =,,, η = Lea. aing Φ as in (, he geoery properies (5, and he hypohesis of onoony of coefficiens (6 and he hypohesis (4; hese were ade for he iniial condiion. hen, C5 >, wihindependence of,,, p, sch ha δ ( µ ( Ω C 5 Ω + α ( p dx h α p ds S d. he sae anner, we obain he ineqaliy of observabiliy for he syse (-(4 wih heir inerface condiions and he onooniciy of he coefficiens, given by: γ τ γ (7 τ heore. Assing Φ as in (, he onooniciy for he coefficiens (7 and he hypohesis of he heore wih hx ( = x x + δφ ( x and ( v, q V DA (, v =, wih H ( Ω, = in S. hen, here is a consan 6 v,, q sch ha C >, wih independence of α α (6

6 8 Silaneos Exac Conrollabiliy of Wave Sond Propagaions in Mlilayer Media [ δ ( µ ( ] τ v γ ( q dx Ω = Ω + C6 τ v + γ q dx Ω = ( h τ v. η ds S d. heore. Assing he hypohesis of he heore,. Moreover, we sppose ha : α = γτ τ= τ. (8 hen, here is a posiive consan C > sch ha + α ( p ( dx Ω = + τ v + γ ( q h C [ αp τ( v. η ] ds S d. C5+ C6 { ( ( Ω } > ax,. δ µ 4. Exac Conrollabiliy Given G = ( w, and G = (, l arbirary eleens of Y. We denoe by ( wx (,, ( x, = U( ( w, ( x (,, l( x, = U ( (, l consider he following fncions ( ( ( Q= α x, τx,. η P= Q γ (9 and given (, p and ( vq, he solion of ( and (4, hese are nll in he insan, ( > and he conor condiions (9. Considering he ap : Y Y ( GG, ( GG, = ( pvq,,, = Using (? in = and sbsiing P and Q given by (9, we have ( G, G (,, p, v, q Γ Γ [ αqp γτ Pv η ] = +. = ( γ τ. η( αp τv. η = ( G, G (,, p, v, q X Y ( of (, we can conclde ha is an isoorphis of Y in Y. We proved following heore heore. Assing he hypohesis of heore. If given > and iniial condiion (, p, v, q Y he proble (, (, (5, (5. hen, here is exis an conrol Qx (, C(, ; L( Γ sch ha he corresponding solion (, pvq,, wih he bondary condiion and wih P η. = Q, in S (, p =, in S (, x (, = ( x, px (, = p( x q = P, in S (, q =, in S (, vx (, = v( x, qx (, = q( x = γ, saisfy, for x Ω Q Acnowledgen ( x, = p( x, = vx (, = q( x, = ( ( ( We han o professors G. Perla Menzala for he sggesions in his aricle. References [] R. A. Adas, Sobolev Spaces, Acadeic Press, New Yor, 975. [] J. M. Coron, Conrol and nonlineariy, Maheaical Srvey and Monographs 9.

7 Silaneos Exac Conrollabiliy of Wave Sond Propagaions in Mlilayer Media 9 [] V. Koorni, Exac Conrollabiliy and Sabilizaion. he liplier ehod, RAM:Research in Applied Maheaics. Masson, Paris; Jhon Wiley and Sons, Chicheser, 994. [4] J. E. Lagnese, Bondary conrollabiliy in probles of ransission for a class of second order hyperbolic syses, ESAIM Conrol Opi. Calc Var., (997, 4-57 [5] B.V. Kapionov, and G. Perla, Silaneos exac conrollabiliy for a pair of syses of evolion of sond in a copressible flid, Jornal of Maheaics and Syse Science ( [6] B.V. Kapionov, and G. Perla, Bondary Sabilizaion and a proble of ransission for a syse of propagaion of sond, Fncialaj Evacioj, 49 (6, 7-. [7] L.D. Landa and E.M. Lifshiz, Flid Mechanics Vole 6 of Corse of heorical Physics Pergaon Press, Oxford, New Yor, Paris (96 [8] J. L. Lions, Exac conrollabiliy, Sabilizaion and perrbaions for disribed syse, SIAM Review (988, -68. [9] A. Pazy, Seigrop of Linear Operaor and Applicaion o Parial Differenial eqaions, Springer-Verlag, New Yor, Berlin, 98. [] D. L. Rssell, A nified bondary conrollabiliy heory for hyperbolic and parabolic parial differenial eqaions, Sdies in Appl. Mah. 5 (97, 89-. [] D. L. Rssell, Conrollabiliy and Sabilizabiliy heory for linear parial differenial eqaions: recen progress an open qesions, SIAM Review (4 (978,