Min-max Game Theory for Elastic and Visco-Elastic Fluid Structure Interactions

Transcription

1 Send Order or Reprint to The Open Applied Mathematic Journal, 3, 7, -7 Open Acce Min-max Game Theory or Elatic and Vico-Elatic Fluid Structure Interaction Irena Laiecka,*, Roberto Triggiani and Jing Zhang Department o Mathematic, Univerity o Virginia, Charlotteville, VA 9, USA Department o Mathematic and Computer Science, Virginia State Univerity, St Peterburg, VA, USA Abtract: We preent the alient eature o a min-max game theory developed in the context o coupled PDE with an interace. Canonical application include linear luid-tructure interaction problem modeled by Oeen equation coupled with elatic wave. We hall conider two model or the tructure: elatic and vico-elatic. Control and diturbance are allowed to act at the interace between the two media. The ought-ater addle olution are expreed in a pointwie eedback orm, which involve a Riccati operator; that i, an operator atiying a uitable non-tandard Riccati dierential equation. Motivation, application a well a a brie hitorical account are alo provided. Keyword: Elaticity, luid-tructure interaction, min-max game problem, Riccati operator, vico-elaticity.. INTRODUCTION.. General Comment We conider a min-max game problem a ormulated or coupled PDE control ytem ariing in luid tructure interaction model. We hall conider two dratically dierent model or the tructure: elatic and vicoelatic. While the irt prevail in engineering application when the olid i immered into a luid, the econd vicoelatic model i oten ued in biological-medical application where the tructure ha ome vicoelatic repone. The purpoe o thi article i to preent general abtract reult in the area o game theory and the aociated eedback ynthei via a non-tandard Riccati equation along with their application to the luid tructure interaction mentioned above. One hould tre at the outet that while game theory wa etablihed by John von Neumann [] and later developed by J. Nah, Iaac and other [-7, 5], it wa a work o Baar and Bernhard [8] that ha had inluence on thi theory developed or dynamic governed by abtract ODE and PDE. An extenive, yet not complete lit i given in [9, ]. It uice to ay that the theory ha been coniderably developed in the context o linear emigroup with bounded control action and bounded diturbance. However, the preence o unbounded control-diturbance provide or a main novelty and challenge o the problem under conideration in thi article. On the other hand, application uch a they arie in the context o coupled PDE ytem, where the controlditurbance i typically active at *Addre correpondence to thi author at the Department o Mathematic, Univerity o Virginia, Charlotteville, VA 9, USA; Tel: ; ilv@virginia.edu the interace hence on a maniold o lower dimenion, -lead invariably to unbounded control -diturbance ormulation o the problem. Thee application have provided the impetu or recent development in abtract variational analyi. Thi now require an arenal o new tool enabling one to deal with unbounded actuation-diturbance control problem. It i the purpoe o thi article to give an overview o recent abtract reult and put thee in the context o phyical application. We hall how how to apply abtract theory, how to veriy theoretical aumption, impoed by that theory, on a peciic cla o model o interet. Finally, we hall interpret game theory reult in the context o boundary controlled luid tructure interaction. It turn out that type o elatic or vicoelatic repone o the tructural material ha igniicant implication on the inal game theoretic reult claimed. The main aim o thi article i to preent a complete mini-max theory in the context o model coniting o coupled parabolic - parabolic dynamic with control action occurring on the interace between two dierent environment luid and tructure. A we hall ee, thi etup bring orward new mathematical challenge... Game Theory or a Fluid-Structure Interaction Model: Engineering and Biological Motivation The luid-tructure interaction model i well etablihed in both mathematical and engineering literature [, ] and the application range rom naval and aeropace engineering to cell biology and biomedical engineering [-7]. In order to motivate a reader we hall decribe ome o them. It i important to note that depending on the application conidered the model o tructure or olid may be purely elatic or vico-elatic Bentham Open

2 The Open Applied Mathematic Journal, 3, Volume 7 Laiecka et al. Fig. 3. An elatic body immered inide a luid. Fig.. Submarine immered in water. Fig. 4. A luid urrounded by an elatic wall. Fig.. Blood cell urrounded by blood plama.. One intereting application o thi model i the meaurement o the doage o antibiotic precribed to a patient. The reearch carried out in thi area ha raied wide attention a precription drug doage ha become a at growing problem recent year. It i documented that in 8, a total o 36,45 death were attributed to drug overdoe, and thi number i till increaing. Our PDE model provide a mathematical approach or determining the optimal doage or the antibiotic. Here the ``optimal doage i deined a the doage that give the deired eect with minimum ide eect. We work under the aumption that doctor alway want to minimize the antibiotic doage precribed to patient o a to minimize the ide eect. While on the other hand, i the doage i too low, the antibiotic can not kill the bacteria; wore till, bacteria can even get antibioticreitant. In thi cenario, the antibiotic and the bacteria become two player in a noncooperative game. Our reearch target the goal o inding the minimum antibiotic doage that kill the wort bacteria in the patient, thu cure the bacteria inection completely. Thi i a typical minimax algorithm in game theory. In the PDE model, the eect o the antibiotic and the bacteria to the patient i decribed by the control and the diturbance repectively. Our objective i to develop the eedback optimal doage or the antibiotic baed on the olution the eedback operator o a certain Riccati equation. It i known that the Riccati theory i a very powerul tool in deigning and computing eedback control in the linear quadratic regulator problem. In our tudy, we extend the claic Riccati equation to o called non-tandard Riccati equation. The dierence between a non-tandard Riccati equation and the uual Riccati equation i an extra term correponding to the preent diturbance in the PDE ytem, which, however, occur with a ``bad negativeign in ront. The contruction o the non-tandard Riccati equation aociated to the PDE model provide u with a eedback algorithm or computing the optimal antibiotic doage or a particular patient due to hi phyical condition, which can be monitored by hi blood tet, uch a the white blood cell count, etc; wherea our PDE model decribe the blood tatu by the interaction between the blood cell modeled by the equation o the ytem o elaticity that account or vicoelatic eect and the blood plama modeled by the linearized Navier-Stoke equation Fig... Another medical application involve tomography and ultraound technique where the goal i a detection o a olid within the cavity illed with body luid. The actuation i in a orm o a orce laer orce applied to the interace. 3. A claical example o engineering application i a olid -elatic body ubmered in a water ubmarine Fig.. The goal here i to control vibration o the elatic body or ele minimizing ignature o the olid. Thi i done on the boundary o the olid by applying rictional or vicou damper layered material to the urace o the body. In thi cae, the body i modeled by the dynamic ytem o elaticity that may ignore vico-elatic eect.. THE FLUID-STRUCTURE INTERACTION - PDE MODEL.. Decription o the Model In what ollow we hall conider luid-tructure interaction model correponding to the ituation where the elatic body i immered inide the luid Fig. 3. Other coniguration that correpond to a luid urrounded by elatic wall blood aorta Fig. can alo be conidered in the ame manner. In thi latter cae the luid contitute the internal domain while the tructure-wall are on the exterior. See Fig. 4 below. We hall ocu in thi paper on the irt

3 Min-max Game Theory or Elatic and Vico-Elatic Fluid The Open Applied Mathematic Journal, 3, Volume 7 3 coniguration. The aumption made i that the ocillation o the body are rapid but mall. Thi allow to neglect the dynamic o the interace []. Let R n, coniting o an interior region n =,3 be an open bounded domain and an exterior region, ee picture below. We denote by the outer boundary o the domain while i the boundary o region which alo border the exterior region and where the interaction o the two ytem take place. Thu i the interace between the two media. Let u be an n - dimenional vector unction deined on repreenting the velocity o the luid, while the calar unction p repreent the preure. Additionally, let v and v be the n- dimenional diplacement and velocity unction o the olid. We alo denote by the unit normal vector outward with repect to the domain. The boundaryinterace control i repreented by g L,T;L n and i active on the boundary. For implicity but it alo eem a phyically reaonable ituation, we irt aume that the determinitic diturbance w = { w, w} act with one component w in and one component w in. We work under the aumption o mall but rapid ocillation o the olid body, o that the interace may be aumed to be tatic, ee [] or more modeling detail. Additional reerence to the non-linear model o the Navier-Stoke equation include [], p6, which, in turn, make reerence to a biological model [8]. The correct model o a moving tructure immered in a Navier-Stoke luid ay a boat that drit under the action o the luid; not a el-propelled tructure appear to be till unreolved. A dicuion leading to an arbitrary Lagrange-Euler ormulation i given in [9], equation 8.6, p9. A more recent eort i in []... Boundary Control with BoundaryInterior Diturbance PDE Model We hall conider a model reulting rom the coupling o Oeen operator luid with dynamic ytem o elaticity. The coupling i on interace between the two dierent environment. Given the boundary control g = g,g L,T;U with U = H n L n at the interace, and the interior determinitic diturbance w = {w,w,w 3 }L,T;V where V = L n L n L n, we conider the ollowing parabolic-hyperbolic coupled ytem o partial dierential equation which couple Oeen equation luid with dynamic ytem o elaticity rep. vicoelaticity. The unknown are{u,v,v t, p}, decribing luid velocity u, olid diplacement v, olid velocity v t and the preure p. t. u t u L u p = G w in Q $,T ] divu = in Q $,T ] v tt divv divv t = G w in Q $,T ] v t = u g in $ *,T ] u = in $ *,T ] v v t, = -u, p, g G 3 w 3 in $ *,T ] u, = u in v, = v, v t, = v in.a.b.c.d.e..g.h The elatic train tenor and the tre tenor, are n n ymmetric matrice repectively given by ij u = u i x j u j x i = ji u. and n u = trui µu = ij u i, j= n.3a ij u = $ kk u ij µ$ ij u = ji u.3b k= where Obviouly we have, > and > µ are the Lamé contant. u u ; $u max{,µ} u max{,µ} u.4 The boundary condition in. repreent balance o the tree, while condition.d repreent matching o the velocitie between the luid and the olid. The contant play an important role. When > the model o the tructure account or vico-elatic eect - a typical ituation when modeling blood paing through the arterie [3, 4]. The cae o a pure elatic olid uch a ubmarine immered in the water correpond to >. From the mathematical point o view, the two model are vatly dierent. The irt one > correpond to a parabolicanalytic coupling o dynamic, while the econd cae = correpond to parabolic-hyperbolic coupling. We hall ee later that thi act ha implication on the reult obtained or each model. The term L u i a linearization o the convective term o the Navier-Stoke term uu and i deined a L u = u y e y e u, div y e =, y e =.5 where y e i a teady-tate or equilibrium olution orced by an external orce L ; that i, a time independent mooth vector unction in [H ] n with the propertie: div y e =, y e =. The diturbance operator G,G,G 3 are aumed linear. They decribe location and intenity o the internal diturbance aecting the luid G w and the olid G w. The third diturbance G 3 w 3 L i located at the interace between the two region.

4 4 The Open Applied Mathematic Journal, 3, Volume 7 Laiecka et al. The control unction g and g act on the boundary - interace between the two region. Our goal i to elect uitable boundary-control mechanim g, g, expreed in the eedback orm a a unction o the tate variable ut,vt,v t t, that will provide optimal repone to the ytem in the preence o diturbance..3. Mini-max Optimization Problem Let U rep. V be given a above. With T > preaigned, control unction g = g,g L,T;U = L U and diturbance w = w, w,w 3 L V conider the quadratic cot unctional T J y, g, w = gt Ryt U Z dt T wt dt.6 V Here the tate variable yt = ut,vt,v t t H atiie. or a given control g and diturbance w. Hilbert pace H repreent tandard inite energy pace aociated with the phyical model repreenting luid and wave to be introduced later and the obervation R LH Z with a given output Hilbert pace Z. The min-max game problem correponding to.6 i up in J y, g, w = J y *, g *, w *.7 wl V gl U Our aim i to ind tate-eedback repreentation or both control and diturbance g * t = Kty * t, w * t = Dty * t,t,t.8 where the eedback operator K t and D t are bounded. Thi i to ay: Kt LH U, Dt LH V, t [,T ].9 We note that the requirement that the tate eedback be bounded very practical engineering requirement provide or mathematical challenge o the problem. It i well known that the unbounded control-diturbance -uch a point or boundary control - while till yielding well deined open loop control, lead to unbounded tate eedback repreentation in the cloed loop ormulation, ee [-3] and reerence therein. Thu, in order to have bounded tate eedback repreentation in the context o interace control ome pecial tructure need to be dicovered. It i the aim o thi work to reveal rather pecial tructure o the phyical ytem ingular etimate property which i reponible or well behaving tate eedback to be later contructed by Riccati theory. While mini-max game theory problem ha been conidered in the pat literature-both abtractly and alo concretely in the context o luid tructure interaction [4, 5], thi i the irt time when vicoelatic tructure i conidered and Dirichlet type boundary control unction g are employed. In act, while Neumann boundary control, due to higher regularity and variational nature o the ormulation [] are more amenable to the mathematical treatment, phyically attractive Dirichlet control invariably lead to much more challenging analyi they do not reide in a natural energy pace. Dirichlet boundary control problem or luid tructure model, even in the context o claical Riccati Equation, ha been open and conidered diicult or ome time. Thu, thi i the irt manucript where thi open problem i being addreed and olved. The outline o the paper i a ollow. In ection 3 we recall general mini-max theory reult that were recently obtained or abtract ytem with unbounded controlditurbance. In ection 4 we hall repreent luid-tructure interaction model a an abtract control ytem. The main reult i tated in Theorem 3., Section 4. Section 5 i devoted to the proo. Here the main point i to veriy the abtract aumption impoed by the general theory o ection 3. Thi will allow or application o uch uniied treatment. The paper conclude with concluion and open quetion. 3. MIN-MAX GAME THEORY - GENERAL THEORY 3.. General Setup and Overview To tudy the min-max game theory or parabolichyperbolic type coupled PDE, our main diiculty i the contruction o the aociated non-tandard Riccati equation in the context o boundary controlboundary diturbance. The latter yield a mathematically challenging analyi in the etting o highly unbounded operator, where the tandard approach within the etting o ``bounded operator ail. Thi i due to the act that the eedback operator along with the aociated coeicent in the non-tandard Riccati equation may not be well deined. The intrinic reaon or thi i the lack o uicient regularity in the PDE ytem. The theoretical ramework o contructing the Riccati equation aociated with ytem generating analytic emigroup ha indeed been etablihed [9,, 6], Chapter 6; wherea or each o thee parabolic-hyperbolic coupled PDE model, the overall dynamic may not be mooth enough due to the preence o the hyperbolic component. Neverthele, the parabolic component doe impoe a partial moothing on the overall dynamical evolution, which i captured in the ``ingular etimate property. Thi property i critical or the development and validity o mini-max Riccati theory in the cae o unbounded action or unbounded coeicient. We hall report below the main reult which are pertinent to thi tudy. Paper [4] conider the ollowing abtract dynamic: yt = Ayt Bgt Gwt on [D A * ] Abtract Model : $ y = y Y 3. where {yt, gt, wt} repreent the tate, control and diturbance repectively, A i the generator o a trongly continuou emigroup on the Hilbert tate pace Y, B i a linear, highly unbounded operator over the control pace U, uch that B :U [D A * ] or A B LU;Y, imilarly or G over the diturbance pace V. Moreover, both atiy the ollowing ``ingular etimate at the origin: e At Bg Y C T t g, e At Gw U Y C T t w, V < <, < t T 3.

5 Min-max Game Theory or Elatic and Vico-Elatic Fluid The Open Applied Mathematic Journal, 3, Volume 7 5 Remark 3. We note that the ingular etimate in 3. i automatically atiied when the control operator i At bounded U Y. Alo, when emigroup e i analytic and the operator B i α bounded ie R, A B :U Y i bounded with <, the ingular etimate i jut a conequence o analyticity. It will be hown later how one can arrive at uch etimate. It i important to notice that our treatment doe not aume a-priori analyticity o the reulting emigroup. In act, in typical application to luid tructure interaction, the emigroup i not analytic due to the preence o hyperbolic component aociated with the dynamic o the wave equation modeling the elatic olid. However, the trongly damped wave > lead to analytic emigroup that decribe the model. Thi highly non-trivial act will be hown later in Section 4. The min-max game theory problem or the dynamic 3.atiying 3. i a ollow: For any y Y, ind a olution {y * t, y,g * t, y,w * t, y } on,t, uch that it olve the ollowing problem: up wl V in { Ryt gt Z U $ wt V }dt 3.3 gl U T where R i a linear bounded obervation operator rom Y to another Hilbert pace Z, and i a poitive contant. Thu, qualitatively, the min-max game problem conit in inding the minimal cot o the ytem under the wort diturbance. For thi problem, we obtain a critical value or, and prove the exitence and uniquene o the olution under the condition > c or all initial condition through a variational approach, while or < c there i no olution. The key reult i that the olution pair control g *, diturbance orm o the dynamic * w can be expreed in pointwie eedback * y : g * t, y = B * Pty * t, y ; w * t, y = G * Pty * t, y t T c 3.4 where Ptatiie an operator Riccati dierential equation. Becaue the abtract ytem 3. i ininite dimenional with unbounded generator A, and highly unbounded control and diturbance operator B and G, a noted beore, the wellpoedne o all quantitie a well a o the aociated Riccati dierential equation i a undamental technical iue ee Theorem 3.. The preence o the ``ingular etimate - which ubtitute or the lack o regularity o emigroup - turn out through a technical tour-de-orce to guarantee that the eedback operator B * Pt and G * Pt are both bounded and, in act, coincide with the ought ater operator Kt and Dt in the ormula.8. Remark 3. The correponding cae, under ingular etimate 3. or T < i tudied in [7]. Intead, [8] tudie the min-max problem in the abtract hyperbolic cae or T <. The theory reerred to above i baed on the abtract model 3., and the reult i optimal and complete in howing the validity or the contruction o the aociated non-tandard Riccati equation. Thi allow one to include everal phyically relevent illutration uch a tructural acoutic chamber, thermoelatic plate, compoite beam, etc, or thi abtract theory. However, the luid-tructure interaction model we are intereted in happen to be more pathological. It tudy require olving additional technical and conceptual obtacle becaue uch model ail the critical aumption o our original theory, the``ingular etimate 3. rom the controlditurbance pace to the tate pace Y. In act, in the preent cae, a weaker ``ingular etimate hold true, not in the original tate pace, but in a pace lightly larger. Thi i due, intrinically, to the act that there i a mimatch between the regularity o the hyperbolic component and the regularity o the parabolic component at the interace. A a conequence, the term `` e At Bu and `` e At Gw no longer belong to the tate pace Y, and thu 3. ail in the preent model. Our treatment in thi part i to introduce a weaker and more generalingular etimate condition, the Output Singular Etimate the original ingular etimate i then called the State Singular Etimate correpondingly. Then we extend our original theory to thi new Output Singular Etimate PDE ytem. 3.. Abtract Min-Max Game Theorem or the Output Singular Etimate PDE Sytem In thi ection, we develop the general min-max game theory in the context o abtract dynamic, with unbounded control and diturbance action. Abtract Dynamic. Let U control, Y the tate, Z obervation and V diturbance be given Hilbert pace. We conider the dynamic governed by the tate equation y t = Ay Bg Gw; on [D A * ] ; y = y Y, 3.5 and ubject to the ollowing aumption, to be maintained throughout the paper. Hypothee. Let U control, Y tate, Z obervation and V diturbance be given Hilbert pace. H. A :Y D A Y i the ininiteimal generator o a At trongly continuou.c.emigroup e on Y, t. H. B i a linear operator on U = DB [D A * ], atiying the condition R, AB LU;Y, or ome A, where R, A i the reolvent o A and A i the reolvent et o the A. Thi aumption i automatically atiied i B LU,Y i.e. B i bounded a in the tandard cae H.3 G i a linear operator on V = DG [D A * ], atiying the condition R, AG LV ;Y, or ome A. Thi aumption i automatically atiied i G LV,Y i.e. G i bounded a in the tandard cae H.4 Let R :DR Z be a linear operator, uch that DR {e At BU, < t T}{e At GV, < t T}Y 3.6

6 6 The Open Applied Mathematic Journal, 3, Volume 7 Laiecka et al. H.5 The triple {A, B, R} atiie the ollowing Output Singular Etimate Condition: There exit < < and a contant C T > uch that Re At B LU ;Z = B * e A*t R * LZ;U C T t, < t T 3.7a or Re At Bg C[,T ];Z, g U 3.7b where Bg, y Y = g, B * y U, g U, y DB * D A *. The unction pace C[,T ]; i deined in below. H.6 With R a in H.4, the triple { A, G, R} atiie the ollowing Output Singular Etimate Condition: Re At G LV ;Z = G * e A*t R * LZ;V C T t, < t T 3.8a or Re At Gw C[,T ];Z, w V 3.8b where Gw, y Y = w,g * y V, w V. For notational impliication, we take the ame contant in 5.3 and 5.4. H.7 With R a in H.4, R LY;Z Remark 3.3 I B, G and R are all bounded operator, then aumption H.-H.7 are automatically atiied. More iginiicantly, we hall how that the above abtract aumption H.-H.7 are a-ortiori atiied in the cae o luid-tructure interaction problem. with appropriately deined input-output pace. Actually, aumption H.6 accommodate alo the cae where the determinitic diturbance act at the interace. Thu, model. where however now both control g and diturbance w act on, ay, two ditinct portion o the interace i included in the preent abtract etting. Thi i explicitly given at the end o Section 4. Remark 3.4 Note that the Hypothei H.5 require ome balance between ingularity o the control operator B and the obervation R. The obervation i R i allowed to mitigate trong ingularity o B. Similar phenomenon applie to the diturbance G and it relation to obervation R in the hypothei H.6. Notation For urther reerence, we next deine the Banach pace r C[,T ]; X, where < r <. Thi pace meaure the ingularity on the let point. The topology on C[,T ], X i deined a [, 6] p 3: r C[,T ]; X = { C,T ]; X : r upt $ r t r C[,T ];X X } <tt 3.9 Min-max game theory problem over inite time interval [,T ] For a ixed < T < and ixed >, we aociate with 3.5 the cot unctional Jg, w; y Jg, w; yg, w; y = T [ Ryt gt Z U $ wt V ]dt where yt = yt; y i the olution o 3.5 due to gt and wt. What we are going to tudy i the ollowing game theory problem up in Jg, w; y 3. wl V gl U where the inimum i taken over all g L,T;U or w L,T;V ixed, and the upremum i then taken over all w L,T;V. Abtract Theory Equipped with the above aumption and propertie, we have [4] the ollowing reult or the abtract dynamical model 3.5. Theorem 3. Aume H.-H.7. Then there exit a critical value, uch that: c. I c > and < < c, then taking the upremum in w lead to or all initial condition y Y ; that i, there i no inite olution o the game theory problem 3.. I > c, then: there exit a unique olution {g * ; y, w * ; y, y * ; y } o the game theory problem Moreover there exit a bounded non-negative eladjoint operator Pt = P * t LY, t T, deined explicitly in term o the problem data, uch that: Ptcontinuou :Y C[,T ];Y 4. B * Ptx U G * Ptx V C T x Y, t T; < 5. Pt deine the cot unctional in 4.7b o the olution o the min-max game problem initiating at the point x Y and at the time t, over the interval [t,t ], or all t [,T ] : Ptx, x Y up in Jg, w;x 3. wl t,t ;V gl t,t ;U 6. Ptatiie a non-tandard dierential Riccati equation: or all x, y D A, we have PT = Ptx, y Y = Rx, Ry Y Ptx, Ay Y PtAx, y Y $ 3. B * Ptx, B * Pty U G * Ptx,G * Pty V 7. The ollowing pointwie eedback relation hold true or the min-max game theory olution: g * t;x = B * Pty * t;x C[,T ];U, x Y 3.3 w * t;x = G * Pty * t;x C[,T ];V, x Y The operator Rt,x Ry * t,;x C[,T ];Z, x Y i an evolution type operator, which i trongly continuou in each variable t and eparately, holding the other ixed. Moreover, or all x D A, t,x i dierentiable with repect to t in the dual ene on [D A * ] ; that i d dt t,x = [A BB* Pt GG * Pt]t,x $C[,T];Y Rt,B and Rt,G atiy the ame ingular etimate a Re At B and Re At G in H.5, H.6; that i:

7 Min-max Game Theory or Elatic and Vico-Elatic Fluid The Open Applied Mathematic Journal, 3, Volume 7 7 Rt,Bg, Rt,Gw C[,T ];$, 3.6. Finally, or all x D A, Rt,x i dierentiable with repect to in the pace C[,t];Z, and d [Rt,x] = Rt, d [A BB * P GG * P]x $ C[,t]; Z 3.7 Remark 3.5 I B, G, R are bounded, Then Theorem 3. hold true under only one aumption H.. Proo. Indeed, ince the operator B i bounded, then R, AB LU,Y, a a compoition o two bounded operator. Thu H. i atiied automatically. The ame applie to the diturbance G in Hypothei H.3. When R i bounded then DR coincide with the entire pace, o that H.4 i an empty aumption. Singular etimate Hypothee in H.5 and H.6 are automatically atiied with α =. There i no ingularity at the origin. Aumption H.7 hold on the trength o trong continuity o the emigroup combined with the boundedne o R. 4. ELASTIC AND VISCO-ELASTIC FLUID STRUCTURE GAME THEORY PROBLEM Having at our dipoal the abtract theory we are back to concrete application in the context o luid-tructure interaction. 4.. The Model The aim o thi ection i to put the original control problem within the abtract ramework. For thi we need to identiy uitable operator decribing the abtract etup. Boundary ControlBoundary and Interior Diturbance model: In the preent model, both control g, g and diturbance w 3 act at the interace between the two media. The diturbance w and w aect the luid and the elatic body. To decribe the model, we introduce the Cauchy Polya tenor dicribing the luid velocity and the preure: Tu, p u pi, u $u $T u 4. We hall alo introduce unction localizing the eect o diturbance to ome ubregion o the peciied domain. To x thi end, let denote a mooth unction that i upported in a open et. Similarly x will localize in a mooth manner to a ubet. The PDE model under conideration i the ollowing: - u t divtu, p L u = w in Q $,T ] divu = in Q $,T ] v tt divv v t = w in Q $,T ] 4.. v t = u g in $ *,T ] u = in $ *,T ] v v t, = Tu, p, g in $ *,T ] v v t, = Tu, p, w 3 in $ *,T ] with the initial data u, = u in,v, = v, v t, = v in Here =, = Ø. That i, the control g and diturbance w act on two ditinct portion o the interace. The linear operator L u i a irt order dierential operator. In practical application to Navier Stoke equation, thi operator reult rom linearization o Navier Stoke operator and i given a in.5: where y i an equilibrium point correponding to a tatic but orced verion o N-S equation divt y e, p y e y e =, div y e =, y e = on 4.3 with ome orce L. The orce can alo be een a a ixed diturbance or the model. 4.. Specialization o Abtract Model to the Fluid- Structure Interaction Model In thi ection, we adapt the abtract theory we have developed to the luid tructure interaction model. The abtract pace o Section 4 aociated with the coupled PDE model 4. are given by: H H H n L n or {u,v,v t }; U H n L n or g = g, g ; V L n L n L $ n or w = w, w,w where H {u L n :divu =, u $ = } 4.5 We moreover deine the pace E {u H n :divu =, u $ = } 4.6 Notation: Henceorth, we hall drop the ymbol e n, n =,3 or all Sobolev pace H and L pace pertaining to u and v, or the ake o implicity o notation. Accordingly, ay on the domain and correponding boundary, the L -inner product are denoted by v,v = v v d ; $u,u = u u d 4.7a On, we ue u,u = u u d. Moreover the pace E i topologized with repect to the inner product u,u E = u,u, $u $u d 4.7b We alo denote the induced norm by, which i equivalent to the uual H norm via Korn inequality and Poincaré inequality [9]: u, = u d 4.8

8 8 The Open Applied Mathematic Journal, 3, Volume 7 Laiecka et al. The pace H i topologized with repect to the inner product given by v, z, v$zd v$zd 4.9a o that the i norm induced by the inner product above, v, = v$vd v, 4.9b Thi again equivalent to the uual H norm by Korn inequality [9]. Sobolev norm o order are denoted by u, = u H, u, = u. We hall alo ue the abbreviated notation H u = u, With T > preaigned, the quadratic cot unctional correponding to 4. i then T Ju,v, g, w = gt R U ut L R vt H T R 3 v t t L dt wt dt 4. V where g = g,g and R LL, R LL H, R 3 LL The min-max game problem correponding to 4. i the ollowing: up in Ju,v, g, w 4. wl V gl U where u, v atiy with given g, w. We next tranorm the PDE Sytem 4. into the abtract tate equation and then ue the abtract theory in Section 3 to maxi-minimize the cot unctional 4.. The literature already contain two treatment leading to the abtract model correponding to thi PDE ytem. One approach - introduced in [3] and purued in [3, 3] - eliminate the preure by introducing two uitable Green map o appropriate elliptic problem in ; one with Dirichlet datum on and homogeneou Neumann datum on ; the econd, the other way around: homogeneou Dirichlet datum in and Neumann datum on. Thi approach wa actually tudied or the cae o interet in thee reerence, where g in. The econd approach, intead, i variational: it wa preented in [33-36]. We hall ue thi approach here. We irt deine the ollowing operator: A : E E, uch that Au, = Tu, p,$ = $u,$, E 4. A i linear, bounded rom E to E and coercive. Thi allow u to conider operator A a acting on H with domain D A = {u E : u, $ C E }. 4.3 Thu, A i negative, el-adjoint and generate an analytic At emigroup e on H. In addition, we deine the Neumann map N : L L H, given by N $ $ = { * H :,, =. -,,.,,,, The Neumann map 4.4 N i linear bounded rom H H to H [33, 35]. We are in a poition to deine the dynamic o the overall coupled ytem. Let A = A L AN $ AN $ I div div * 4.5 By [36], becaue L i relatively bounded by Lemma 5. rom D A to H, the pertubation A L till generate an analytic emigroup. It wa hown in [33]. A with domain DA H H deined a ollow DA = {u,v, z H:u E, Au Nv z$ H, z H, divv z L $, z = u in H } 4.6 t will generate a trongly continuou emigroup e A on H. Thi emigroup will be hown analytic when > Main Reult-Min-Max Game or Fluid-Structure Interaction Model In what ollow we hall reer to: the generator A given by 4.5, control operator B in 5.4 and diturbance operator G given by 5.37 phae pace H = H H L, control pace V = H L, diturbance pace V = L L L. The adjoint o B :U H and G: V H are given by B * u,v,z = [u, $ z] or u,v, z DA * where i a poitive irt order tangential operator on : and u L = u H G * u,v,z = [ u, z,u ]. Theorem 4. Let aume that the tatic diturbance i in a bounded et in L. In reerence to model 4. with > and the min-max game problem 4., there

9 Min-max Game Theory or Elatic and Vico-Elatic Fluid The Open Applied Mathematic Journal, 3, Volume 7 9 exit a critical >, or each initial condition in H, that c i, y = u,v,v H H L = H,. Nonexitence. i < < c, Ju,v,g,w a we take upremum over w = w, w, w 3 L V. Thu there i not inite olution or 4. or any initial condition y H.. Exitence. i > c, then or each initial condition y H there exit a unique control g * = g *, g * C[,T ],, H L, a unique diturbance w *,w *, w 3 * C[,T ], L, L L L $ and the correponding optimal tate y * t = u * t,v * t,v t * t C[,T ],H H L uch that J y *, g *, w * = up in J yg, w, g, w wl V gl U Furthermore, or all t [,T ] a g * t g * L t C y H H, b w * t w L t w L t 3 L C y. H * v t * v t * u t C y H t L H H c Feedback Synthei. There exit a poitive eladjoint n n operator matrix P t on H, let Ptv * t,v * t t,u * t = p t, p t, p 3 t with p t, p t and p 3 t being n -dimenional vector unction, uch that the control i given by g * t = B * Pty * t = p t,$ p 3 t 4.7 w * t = G * Pty * t = [ $ p t, $ p 3 t, p t ] 4.8 d Boundedne o the Gain. The eedback operator B * Pt LH,U or all t T with the etimate B * Pty U C y H, G * Pty L V C y H 4.9 e Riccati Equation. Pt i the unique olution to the ollowing non-tandard Riccati equation: PT = and Ptx, y H = Rx,Ry H Ptx,Ay H PtAx, y H $ B * Ptx,B * Pty U G * Ptx,G * Pty V where R = R, R, R 3 with R LL, R LH, R 3 LL. Remark 4. The main novelty and interet in Theorem 4. i the act that the control action i unbounded ie B i unbounded, yet we obtain ully atiactory wellpoedne o Riccati equation along with eedback ynthei that i point-wiely in time deined via gain operator that are bounded. Indeed Kt B * Pt, Dt G * Pt. Thi exhibit certain moothing mechanim diplayed by. olution o Riccati equation. The latter relect the eect o ingular etimate. Remark 4. The theory preented above with > doe not require any moothing with repect to the tate pace property impoed on the obervation operator R. In act, one can alo conider unbounded obervation eg irt order dierential operator in the Neumann cae. Thi i in contrat with other treatment where uch moothing wa required [5, 35, 36]. When =, in which cae only g i active, the ame reult i valid with one dierence that the obervation hould atiy incremental moothing property R = R, R, R 3 with R LL, R LH $ H $. 5. PROOF OF THEOREM Preliminarie The ollowing Lemma i critical or the development. Lemma 5. The operator L u i a irt order dierential operator in u bounded on L. Thi i to ay L u L C L u L 5. Proo. By elliptic regularity [8], one ha y e H H, div y e = 5. and the ollowing etimate i valid: y e C 5.3 H L By Sobolev embedding n < 3 y e L C y e H C L 5.4 Thi give y e u L $ C L u L 5.5 A or the econd term ollow u in u ye L, we etimate a u y e L $ u L 6 y e L Since y e H and H L 6, we iner y e L 6. Thi give u y e L $ C u H y e L6 C u y H e u C 5.7 H H L And the inal concluion reult rom Poincare inequality. 5.. Generation o the Semigroup In what ollow y = u, v, vt = u, v, z and y u, v,. = z

10 The Open Applied Mathematic Journal, 3, Volume 7 Laiecka et al. Theorem 5. The operator A generate a trongly continuou emigroup on H or all. When L = the correponding emigroup i contractive. Proo. In the cae when = the proo o thi reult i given in [33]. Similar argument can be applied alo to the cae >. To thi end it uice to note the ollowing relation ReAy, y H = z,$z $u, $u L u,u, y DA 5.8 The above equality i obtained by integration by part with the ue o cancellation occurring in the boundary condition. By Korn inequality one ha z,z $ z, Thi along with the Lemma 5. - implie ReAy, y H c[ u, H boundedne o $ z, ] c u 5.9 L u -ee 5. where c = when L =. The relation in 5. prove diipativity o A. The proo o maximality ollow now the ame argument a in [3, 33]. In the vico-elatic cae, when > a much tronger concluion hold. Theorem. For > the operator A generate an analytic emigroup o contraction on H. Proo. Here the argument i dierent and it boil down to obtaining uitable etimate or the reolvent operator. Our aim i to prove that there exit contant C > and > uch that R,A LH C, Re > $ 5. To proceed with the etimate, we introduce the ollowing change o variable. v v t 5. Then ee 5. with g =, w = t = v t v tt = v [div$ ] 5.3 With the above notation we conider the ollowing PDE ytem in the variable u, v,, p : u t divtu, p L u =, divu = v t v = t div$ = v accompanied by the boundary condition on the interace $ = Tu, p$, on = v u, on,u = on * Thi amount, ater elimination o the preure, to the conideration o the ollowing matrix operator A L AN $ A = I I I div I * DA = {u,v, H:u E, Au N $ H, H, 5.6 div L $, = [v u] in H } 5.7 We are looking or a olution u,v, H H $ H $ L which atiie u,v, = A u,v, The above topological etup in H i equivalent to the original pace H with the variable u,v,v t. We note that 5.4 i a irt order ytem o three equation ater elimination o the preure in the three variable. The irt and the third equation are ``parabolic like and the econd one i an abtract ODE. Parabolicity o equation involved i a good predictor or analytic behavior o the aociated emigroup. However, the boundary condition provide or ubtantial and unbounded coupling -thu potentially detroying analyticity o the uncoupled ytem. Our main eort goe into howing that thee boundary trace can be handled and eventually lead to an overall analytic dynamic generated by A. We note that the reolvent etimate or A i equivalent to that or A when > We hall etimate the reolvent operator correponding to the ytem 5.4. Thi amount to olving u,v, = R,A h,h,h 3 where u divtu, p L u = h, divu = v v $ = h $ div$ $ v = h 3 Where we take 5.8 h,h,h 3 H = H H L 5.9 and the boundary condition are the ame a in 5.5. Adopting the notation rom the previou ection and neglecting the lower order term, the latter have no impact on analyticity, we are let to conider the ollowing ytem u Au AN$ = h v v $ = h * $ div$ = h 3 $ = v u, on, 5. Multiplying the irt equation by u and the third by, integrating by part and recalling N * Au = u yield u u $,u = h,u 5.

11 Min-max Game Theory or Elatic and Vico-Elatic Fluid The Open Applied Mathematic Journal, 3, Volume 7 $, $, * = h 3, 5. Uing boundary condition = v u on, u u $ $,$ $,v* = h,u h 3,$ 5.3 The econd equation in 5. i treated via product. Thi give Re v $ v = $,v h,v H inner Re v C $ Re C v,h give a very ueul etimate relecting upon the act that v i an ODE: Re v C Re $ C v,h 5.5 In order to eliminate boundary term in the equation 5.3, we multiply equation by v and integrate by part,v $,v $,v* = h 3,v 5.6 which give upon uing 5., $,v =,v,v * h 3,v = v h, $,v h 3,v C $,v $ v h,$ h 3,v 5.7 By Young inequality,v $ C $ v 5.8 On the other hand, the econd equation in 5. or v with 5.4 give along with 5.5 v Hence, C Re v,h $ v 5.9,v $, C Re $, v, C $ Re v,h v C Re $ C Re v, h 5.3 Ater taking Re uiciently large which i caled to, Re > c.,v $[, Thi give, by uing 5.7 v, ] C $ v,h 5.3 $,v [, v, C[ v v,h v,h 3,h ] 5.3 u, The above etimate along with 5.3 lead to, Re$ [ u ] C [ v u,h v,h,h v,h 3,h 3 ] 5.33 Taking Re large enough, recalling 5.9 along with the econd equation in 5. yield to u,, v, C [ u, h v,h $, h v, h 3 $, h 3 ] 5.34 where the etimate i uniorm or large with Re > >. Recall that DA = {u,v,z H,u E,v H,z H,u = z on }. Hence u H, v H, H and uch that = v u, on i equivalent to the memberhip o u,v, DA. In view o thi, 5.34 can be written a A u,v, H C [ u,h $v,$h,h v,h 3,h 3 ] 5.35 Thu, by duality, the olution to the reolvent equation u,v, = R,A h,h,h 3 atiie A u,v, H C A h,h,h 3 H or A R, AF H C A F H 5.36 or equivalently to AR,AF H C F H,Re > Where the etimate i uniorm with repect to uch that Re >. The above, via the reolvent identity, prove 5., hence the the analyticity o the emigroup Control Problem Setting We introduce the abtract control and diturbance operator. Firt, when g = the control Neumann operator and the diturbance operator have the orm

12 The Open Applied Mathematic Journal, 3, Volume 7 Laiecka et al. B = AN $ P AN $, G = 5.37 Here N g = Ng,, N g = N, g. When g Dirichlet it i convenient to provide a dierent repreentation or the operator A. Our aim i to incorporate Dirichlet action into the tructure o the operator. Indeed, let D : H H denote the uual harmonic Dirichlet extenion o the boundary data into the interior. We hall denote by A the correponding elatic operator D div with the zero Dirichlet data. The trace operator denote the retriction to the boundary: v. Then the operator A ha an equivalent repreentation A = v A L AN$ AN$ *, I, A D D A D A D D A, D DA = {u,v, z H:u E, Au Nv z$ H, 5.38 z H,divv $z L, z = u in H } 5.39 o aumption impoed on the control and obervation operator Veriication o the Abtract Aumption Control Operator B. The operator B deined in 5.4 i unbounded rom L to H a the operator AN i unbounded rom L to H. Likewie the operator A D D i unbounded. To apply Theorem 3. to 5.4, it i critical to veriy the validity o Aumption H. or the operator B. Propoition 3. The reolvent R,Aatiie R,AB LU H or all in the reolvent et o A. Proo. When = it ha been already hown in [36] that R,AB LU H, > 5.4 We hall then conider now the cae. Without lo o generality we may take L =. Writing R,ABg, g =,v,z yield: AN g = A Nv Nz $ z = v A D Dg = A D v z D v D $z 5.43 which i equivalent to A Nv Nz $ N g = and the operator B i given by $ Bg, g = $ $ AN g A D Dg 5.4 A D v z D v Dg D = $z = $ v 5.44 Thi amount to olving the ollowing ytem A Nv $v = N g v v A D v D$ v = Dg 5.45 In order to jutiy a new tructure o A, it uice to recall 4.5, deinition o Dirichlet map and write div v = div v div D v = div v D v = A D v D v Similarly, ater uing compatibility condition on the boundary z = u on or u,v,z DA div z = div z div D u = div z D u = A D z D u Theorem 5. For all, the PDE ytem 4. can be modeled abtractly a ollow: y t = Ay Bg Gw [DA * ], y H 5.4 where A, B, G are deined a in 4.5 and Moreover, t A i the ininiteimal generator o.c. emigroup e A on H with the domain DA deined in 4.6. When > thi emigroup i analytic. Thi theorem ollow rom the reult preented in the previou ubection. Regarding the analyticity o the emigroup generated by A, when >, thi ollow rom Theorem 5. on the trength o the act that A in 5.38 i jut a topologically equivalent repreentation o A in 4.5. Our preent goal i to veriy that the control problem deined by A,B,Gatiie the aumption impoed by abtract theorem 3.. We thu proceed next with veriication Denoting $v and applying the variational deinition o the operator introduced yield, < $, >=< g, >, $ v, < v, >=< g, > 5.46 We alo note that the econd equation in 5.44 can be written a div $ v = Taking inner product with v give < $,v >= v,v 5.47 Now we are ready to decouple., $ <,v >, $ v, =< g, > < g, > 5.48 and uing 5.47, $ $ v,v

13 Min-max Game Theory or Elatic and Vico-Elatic Fluid The Open Applied Mathematic Journal, 3, Volume 7 3, $ v, =< g, > < g, > 5.49 In order to treat the term with g we ue once more 5.47with v replaced by Dg < $, g >= v, Dg,Dg 5.5 Combining 5.49 and 5.5 give, $, v $ v, v = < g, > v, Dg $,Dg 5.5 and going back to the deinition o $ v = v, hence, $ $ $v,v $ 3 v,v = < g, > v, Dg $,Dg 5.5 Korn inequality and elliptic regularity D LH H $ then give,, C g, v, C g $, 3 v v, 4 v C g $, 5.53 Selecting mall we obtain that or g = g, g H H $ one ha,v, z H H $ H $ H 5.54 Diturbance Operator G The operator G deined in 5.37 involving localized unction i bounded rom L L to H, thu it automatically atiie Aumption H.3 and H.6 or any bounded operator R with DR H. A or the component AN o diturbance operator, we hall have the ame ingular etimate that i dicued below. The explicit orm o the adjoint to the diturbance operator G i given by G * u,v, z = [ u, z,u ] Singular Etimate Property We now invetigate the dynamical property o the pair {A,B} which i critical or applying Theorem 3. to the luid-tructure interaction model. Since the control operator B i not bounded, the tak i not trivial. The main property to be veriied i Output Singular Etimate Property - Hypothee H.5 and H.6 in Theorem 3.. The Singular Etimate provide or quantitative meaure o the unboundedne o control operator. To thi end, we deine the ollowing cale o Hilbert pace parameterized by the parameter : H H $ H $ H, 5.55 We then remark that with the above notation we have: H = H, where H i the energy pace deined in??. The ollowing reult i critical. Theorem I g = and, the emigroup generated in H by the operator A in 4.5 and the control operator B in 5.37atiy the the ollowing Singular Etimate SE e At e At B, g H C T t 4$ g L, < t T 5.56 or every g L, any >, any > and any. t e A. I g, then with > the emigroup generated in H by the operator A in 5.38 and the control operator B in 5.4atiy the ollowing Singular Etimate SE e At Bg,g H C T t g g 34 H $, < t T 5.57 H $ or any > Remark 5. We note that the ingularity o the control operator B in both Neumann and Dirichlet cae i conitent with ingularity etablihed or parabolic dynamic ubject to Neumann rep. Dirichlet boundary control [-3, 6, 37, 38]. The only dierence i that in the Dirichlet cae the preent coupled model require more regular control operator g H rather than L. Thi i due to unbounded coupling. Proo. Step : Structure o the adjoint B *. We conider operator B :U H given by 5.4. For any y = u,v,z DA we have Bg, g, y H = AN g,u A D Dg, z =< g, N * Au > < g, D * A D z > =< g,u > < $ g, $ z > =<< g, g,u, $ z >> U where i irt order tangential operator and we have ued the identiication N * Au = u, D * A D z = z$ The above implie: B * y = [B * y,b * y] = [u, $ z] 5.58 By appealing to 5.58 inequality in 5.56 in equivalent to how B * e A*t Y L C Y H 5.59 Where we denote B g B,g. We provide PDE repreentation o thi tatement.

14 4 The Open Applied Mathematic Journal, 3, Volume 7 Laiecka et al. Conider the ytem * u t = Au L u ANv v t $ v tt = divv v t v t = u on u = u, v = v, v t = v With initial condition 5.58 Y = u, v, v H 5.6. Recalling B * e A*t Y = ut 5.6 where u, v, vt atiie 5.6. Thu, the adjoint operator to B i identiied with the trace o olution. The irt part o Theorem 5.3 When = ollow rom regularity decompoition reult in [36], Theorem 5.. The econd part o thi theorem > ue the analyticity o the emigroup along with a characterization o the ractional power. The detail are given below. The trace unction correponding to 5.6 can be written a ut = N * Ae AL t Y N * t A AL e t AN[$v v t ]d Step : Proo o Theorem 5.3 Part i: 5.6 Cae =. In thi cae we need only to etimate B * aociated with ut Since g =. The ollowing regularity reult ha been proved in [36]. Lemma 5.4 Let =. Then or every > the ollowing decompoition hold v = v v where v $ C Y H,T,L H, v $ C Y C[,T ],H H 5.63 Taking = in 5.6 and uing A N LH $ L 5.64 along with the analyticity o A N * Ae At A u L C u L $ one obtain t N * t A AL e t A A N[v $]d t C t 4$ A N[v ] L d 5.66 C v C Y C[,T ],H $ H 5.67 The contribution o v i evaluated in a imilar manner by uing the act that or all > there exit p > H,T L p,t 5.68 and or p > the convolution t o L p L,T 5.69 Thi complete the proo when =. Cae >. In thi cae we have analyticity o the generator correponding to the operator A. Thi mean, in particular, A e At LH C, [,] 5.7 t The next tep i to ue the characterization o the domain o ractional power o A. DA = {u,v, z H,u H, v H, z H, z = u on } Since 5.7 B * e A*t Y = ut ut 5.7 We can write on the dynamic B * e A*t Y = v t t v t t 5.73 Uing trace theorem B * e A*t Y L = v t t L v t t H $ A 4 Yt H C t 4 Yt H 5.74 Here Yt = e At Y, the olution to the equation ytem 5. with g =. Step 3: Proo o Theorem 5.3. Part ii Cae >, g. Here we hall ue the calculation rom given or the proo o the reolvent etimate 5.4. Indeed, we already know that R,AB LU H $ H H $ H $ The latter pace i compatible topologically with DA modulo Dirichlet boundary condition which are embedded in the deinition o. On the other hand, boundary condition are not recognized below H cale. We alo have due to the analyticity o the emigroup, [, 39, 4]. DA = [DA,H], [, ] For < For < 4 Dirichlet boundary condition z = u, on are no longer recognized due to denity o H in H we then have DA = [DA,H] = H H $ H $ H Thu, R, AB LU DA $, or $ < 4 Analyticity o the emigroup then implie

15 Min-max Game Theory or Elatic and Vico-Elatic Fluid The Open Applied Mathematic Journal, 3, Volume 7 5 e At B AR$,Ae At B L H H L H H = A A R,Ae At B A e At A R,AB L H $H L H $H A e At LH Ct, < 4 Which i a deired ingular etimate etimate. Remark 5. We note a potential lo in the ingular etimate < 4, rather than which reult rom accounting on compatibility condition characterizing the domain o the generator. Thi prevent u ull exploitation o H regularity or the domain o A. From Theorem 5.3, we are ready to obtain the oughtater Output Singular Etimate property or the pair {A,B}. Corollary 5.5 Conider the PDE ytem 4. and correponding abtract model 5.4atiying, in particular, Theorem 5.3, with > abitrarily and henceorth ixed. Let. [Part I] Let g = Neumann control only. Aume that the obervation operator R atiie R LH ; Z 5.75 where Z i the obervation or output pace poibly H = Z. Then 5.56 yield the deired Output Singular Etimate Re At Bg Z C T t g, < t T L [Part II] In the vicoelatic cae, till with g = when > one can take =. In addition, R can be unbounded A R LH, < 3, in which cae one ha 4 Re At Bg Z C T t g, < t T L $ [Part III] Let > and g Dirichlet control. Then Re At Bg Z C T t g, < t T H or any R atiying 5.75 with =. Next, we illutrate example o the obervation operator R atiying the moothing property 5.75, 5.76 in Corollary 4.5. Example 5.: Let v H $, v H $, > mall. Deine the operator R and R on v and v repectively by R v = v, ; R v = v, 5.79 where i, i are ixed vector: [H $ ]; H ; H $ ; L 5.8 Thu, R i bounded in act moothing, R LH $ ; H $, but not moothing above H. Similarly, R i bounded thu moothing: H $ L, R LH $ ; L $, but not moothing above L. Finally, or {u,v,v }H H $ H $, deine R LH ;H but not moothing above H by: R u v v $ I $ u $ = R = R v R R v Example 5.: On, let H H H * L * 5.8 N =, D N = { H $ : = } 5.8 So that D N = H, D N = H,. Let{u,v,v }H, o that, by.3b, v H $. Thu, v H $ and N v $H, N v $L 5.83 R Thu, the operator R LH ;H deined by u v v $ I $ = N N u v v $ u $ = N v *H 5.84 N v where I i the identity on H and atiie property 5.75 with repect to the output pace Z = H. In thi example, R i alo moothing above H, unlike the cae o Example 5.. Remark 5.3 In the vicoelatic cae, >, one can take =. In act, not only every bounded obervation R atiie the hypothee. but R can alo be unbounded up up to the order o 34. In particular RY = u,v,v t r r F, F H 3 i an example o the obervation complying with the hypothee. Thi obervation allow to meaure the tree in addition to diplacement. From the above analyi, we have veriied that the operator A, B and G deined in 4.5atiy the Aumption H.-H.3. Thu we conclude that the luidtructure interaction model 4.atiie the Aumption o Theorem 3. with = 4 or the Neumann cae and with = 3 4 or the Dirichlet cae. Applying Theorem 3. lead to all the tatement in the inal reult Theorem 4.. Remark 5.4 The theory preented above, with >, doe not require any moothing property impoed on the obervation operator R. In act, one can alo conider unbounded obervation eg irt order dierential operator in the Neumann cae. Thi i in contrat with other treatment where uch moothing wa required [5, 35, 36]. 6. CONCLUSIONS AND SOME OPEN PROBLEMS We have tudied the min-max game theory problem o a linear luid-tructure interaction model, with control and diturbance acting on the interace between the two media.