SPECIAL SESSION 4 17

Special Session 4: Nonlinear PDEs and Control Theory with Applications

Barbara Kaltenbacher, University of Alpen-Adria, Klagenfurt, AustriaIrena Lasiecka, University of Virginia, USA

Petronela Radu, University of Nebraska-Lincoln, USALorena Bociu, NC State University, United States

In the large context of nonlinear evolution equations we will focus on systems of PDEs which exhibit ahyperbolic or parabolic-hyperbolic structure. The topics of this special session will revolve around qualitativeand quantitative properties of solutions to these equations, such as existence and uniqueness, regularity ofsolutions, and long time asymptotic behavior. Associated control theoretic questions such as stabilization,controllability and optimal control will be addressed as well. Both bounded and unbounded domain problemswill be under considerations. Of special interest in our discussions are interactions involving nonlinearity andgeometry . Several methods available for investigation of such problems will be presented. We anticipatealso to discuss specific problems that arise in applications such as nonlinear acoustics, traveling waves inelasticity and viscoelasticity, plasma dynamics, and semiconductors.

Rational decay of structural acoustic dynam-ics

George AvalosUniversity of Nebraska-Lincoln, USAgavalos@math.unl.edu

A rate of rational decay is obtained for solutions ofa PDE model which has been used in the literatureto describe structural acoustic flows. This struc-tural acoustics PDE consists partly of a wave equa-tion which is invoked to model the interior acousticflow within a given cavity. Moreover, a structurallydamped elastic equation is invoked to describe time-evolving displacements along the flexible portion ofthe cavity walls. The coupling between these twodistinct dynamics will occur across a boundary inter-face. We obtain the uniform decay rate of this struc-tural acoustic PDE without incorporating any ad-ditional boundary dissipative feedback mechanisms.By way of deriving this stability result, necessarya priori inequalities for a certain static structuralacoustics PDE model are generated, thereby allow-ing for an application of a recently derived resolventcriterion for rational decay.

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Weak and regular solutions for nonlinearwaves with super-critical sources and non-linear dissipations

Lorena BociuNC State University, USAlvbociu@ncsu.edu

We consider nonlinear wave equations characterizedby energy building, super-critical sources and non-linear dissipation terms. We provide sharp resultson the range of parameters for damping terms andsources that determine the exact regions for existenceand uniqueness for both weak and regular solutions,and finite time blow-up.

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Algebraic Riccati equations with unboundedcoe�cients lacking analyticity of the free dy-namics semigroup

Francesca BucciUniversita degli Studi di Firenze, Italyfrancesca.bucci@unifi.it

The talk will report on the recent advances in thestudy of the Linear-Quadratic problem on an infinitetime horizon for composite systems of evolution-ary Partial Di↵erential Equations which compriseboth parabolic and hyperbolic components, suchas models for thermoelastic, acoustic-structure andfluid-solid interactions. Emphasis will be placed onthe far-reaching role of functional analytic methodsat a theoretical level, as well as of parabolic regu-larity and microlocal analysis in order to establishthe appropriate regularity of boundary traces whichultimately allows to ensure well-posedness of the cor-responding algebraic Riccati equations.Most resultspresented in the talk are obtained jointly with IrenaLasiecka (Virginia) and Paolo Acquistapace (Pisa).

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Uniform decay rates for the wave equationwith locally distributed nonlinear damping inunbounded domains with finite measure

Marcelo CavalcantiState University of Maringa, Brazilmmcavalcanti@uem.brValeria Domingos Cavalcanti, Flavio R. DiasSilva

This talk is concerned with the study of the uniformdecay rates of the energy associated with the waveequation with locally distributed nonlinear damping

utt ��u+ a(x) g(ut) = 0 in ⌦⇥ (0,1)

subject to Dirichlet boundary conditions, where ⌦ ⇢Rn n � 2, is an unbounded open set with finite mea-sure and unbounded smooth boundary @⌦ = �. The

18 9th AIMS CONFERENCE – ABSTRACTS

function a(x), responsible for the localized e↵ect ofthe dissipative mechanism, is assumed to be nonneg-ative, essentially bounded and, in addition,

a(x) � a0 > 0 a.e. in !,

where ! = !0 [ {x 2 ⌦; ||x|| > R} (R > 0) and !0

is a neighbourhood in ⌦ of the closure of @⌦ \BR,where BR = {x 2 ⌦; ||x|| < R}.

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Dissipative acoustic solitons

Ivan ChristovPrinceton University, USAchristov@princeton.edu

Lagrangian-averaged models for compressible flowhave recently been proposed [Bhat & Fetecau,DCDS-B 6 (2006) 979–1000]. Like their counter-parts in turbulence, these models introduce a min-imal (“cut-o↵”) length scale beyond which energydissipation cannot occur. When applied to weakly-nonlinear acoustic phenomena in inviscid, losslesssingle-phase fluids, the Lagrangian-averaged modelrepresents a higher-order dispersive regularization ofthe governing equation, which exhibits nonlinear dis-sipation as well [Kei↵er et al., Wave Motion 48 (2011)782–790]. Kink-type solitary wave solutions are de-rived analytically, and an implicit finite-di↵erencescheme with internal iterations is constructed in or-der to study their collisions. It is shown that twokinks can interact and retain their identity after acollision, meaning that these waves represent dis-sipative acoustic solitons. For a di↵erent choice ofparameters, finite-time blow-up can be observed nu-merically. Finally, while the classical equations ofnonlinear acoustics can be reduced to Burgers’ equa-tion, we show that the present model reduces to theKorteweg–de Vries equation.

This work is, in part, a collaboration with R.S.Kei↵er, R. McNorton and P.M. Jordan from theNaval Research Laboratory, Stennis Space Center.

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A unified theory for damped evolutionaryequations

Valeria Domingos CavalcantiState University of Maringa, Brazilvndcavalcanti@uem.brMarcelo Cavalcanti, Vilmos Komornik, JoseHenrique Rodrigues

The main goal of this talk is to establish an abstracttheory which allows us to determine the exponentialdecay in a certain Hilbert space H for locally dampedevolution equations. Our approach can be used fora wide assortment of equations as, for instance, the

heat equation, Airy-Burgers equation, Schrodingerequation, among others.

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Partial continuity for parabolic systems

Mikil FossUniversity of Nebraska-Lincoln, USAmfoss@math.unl.edu

Consider the parabolic system

ut � div[a(x, t, u,Du)] = 0 in ⌦T := ⌦⇥ (�T, 0),

where ⌦ ⇢ Rn is a bounded domain and T > 0.The vector field a : ⌦T ⇥ RN ⇥ RN⇥n ! RN satis-fies natural p-growth and ellipticity assumptions withp > 2n/(n + 2). I will present partial continuity re-sults for weak solutions to the above problem. Moreprecisely, the results establish that there is an openset of full measure in ⌦T in which the solution isHolder continuous. The key assumption for the prob-lem being considered is that the vector field a is con-tinuous with respect to the arguments x, t and u.This distinguishes this result from others which as-sume Holder continuity of a with respect to x, t andu and provide partial continuity for the spatial gra-dient of the solution. The talk will focus mostly onthe subquadratic setting, where 2n/(n+ 2).

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Some elliptic and parabolic problems withboundary conditions of di↵usive type

Ciprian GalFlorida International University, USAcgal@fiu.edu

We will mainly focus on parabolic and elliptic equa-tions with boundary conditions of di↵usive type,which are sometimes dubbed as dynamic(al) orWentzell boundary-type conditions. We wish to dis-cuss issues involving global well-posedness and regu-larity of solutions, the asymptotic behavior as timegoes to infinity in terms of global/exponential at-tractors. Finally, some explicit dimension estimatesare provided to show a di↵erent degree of structuralcomplexity of these systems when compared to thesame equations subject to the usual Dirichlet andNeumann-Robin type of boundary conditions.

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SPECIAL SESSION 4 19

The polarization in a ferroelectric thin film:local and nonlocal limit problems

Antonio GaudielloUniversita’ degli Studi di Cassino e del Lazio merid-ionale, Italygaudiell@unina.itKamel Hamdache

In this joint work with K. Hamdache (Ecole Poly-technique, Palaiseau, France), starting from classi-cal non-convex and nonlocal 3D-variational modelof the electric polarization in a ferroelectric mate-rial, via an asymptotic process we obtain a rigorous2D-variational model for the polarization in a ferro-electric thin film. Depending on the initial boundaryconditions, the limit problem can be either nonlo-cal or local. Thin films of ferroelectric material areused for the realization of “Ram” for computers and“RFID cards ”.

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The wave equation with abstract nonlinearacoustic boundary conditions

Jameson GraberUniversity of Virginia, USApjg9g@virginia.edu

In this talk we introduce a wave equation with“abstract” nonlinear acoustic boundary conditions,which consists of a wave equation coupled with an ab-stract second-order PDE where the coupling occursat the boundary interface. This system resemblesthe original model proposed by Morse and Ingardbut is su�ciently general to cover a wide range ofproblems studied in the literature. We show thatunder quite general assumptions, the system gen-erates a nonlinear semigroup, and, in the presenceof appropriately selected nonlinear boundary damp-ing, the semigroup is uniformly stable. These resultsserve to unify a number of the available studies inthe literature on PDE models of acoustic/structureinteractions. In addition, we study the e↵ects of a”nonlinear coupling” which arises in the case of aporous structure.

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Asymptotic behavior of solutions to nematicliquid crystal models

Maurizio GrasselliPolitecnico di Milano, Italymaurizio.grasselli@polimi.itWu Hao

We present some recent results on the longtime be-havior of solutions to some simplified versions of theEricksen-Leslie model for the nematic liquid crys-tal flow. More precisely, we consider first the one

proposed by F.-H. Lin et al. In this case the evolu-tion system consists of the Navier-Stokes equationscoupled with a convective Ginzburg-Landau typeequation for the (vector-valued) averaged molecularorientations. We discuss the convergence of giventrajectories to single equilibria when the Dirichletboundary conditions for the order parameter and theexternal force acting on the flow are time dependent.Then we introduce a more refined model proposedby C. Liu et al. and we show that the correspondingdynamical system in two spatial dimensions has afinite dimensional global attractor.

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Global attractors for von Karman plate modelwith a localized damping

Pelin Guven GeredeliHacettepe University, Turkeypguven@hacettepe.edu.trIrena Lasiecka, Justin T. Webster

In this study dynamic von Karman equations withlocalized interior damping supported in a boundarycollar are considered. Hadamard well-posedness forvon Karman plates with various types of nonlineardamping are well-known, and the long-time behaviorof nonlinear plates has been a topic of recentinterest.Since the von Karman plate system is of ”hyper-bolictype” with critical nonlinearity (noncompactwith respect to the phase space), this latter topic isparticularly challenging in the case of geometricallyconstrained and nonlinear damping. In this paperwe first show the existence of a compact global at-tractor for finite-energy solutions, and we then provethat the attractoris both smooth and finite dimen-sional. Thus, the hyperbolic-like flow is stabilizedasymptotically to a smooth and finite dimensionalset.

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Free liquid fibers and films

Thomas HagenUniversity of Memphis, USAthagen@memphis.edu

The mathematical description of free liquid fibersand films poses serious analytical challenges. In thesimplest case of a highly viscous material, this de-scription, based on a long-wave approximation of theNavier-Stokes equations with free surface, is essen-tially due to Matovich, Pearson and Yeow. It takesthe form of a nonlinear transport equation coupledto an elliptic momentum equation or system of suchequations.In this presentation I will address recentresults on stability and global existence in the ab-sence of surface tension for equations of this type.Some reduced models will also be addressed.

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20 9th AIMS CONFERENCE – ABSTRACTS

Non-linear plate coupled with Darcy flows forslightly compressible fluid

Akif IbragimovTexas Tech University, USAakif.ibraguimov@ttu.eduE.Aulisa, Y.Kaya-Cekin

In this work, we consider the dynamical response ofa non-linear plate with viscous damping, perturbedin both vertical and axial directions interacting witha Darcy flow. We first study the problem for thenon-linear elastic body with damping coe�cient. Weprove existence and uniqueness of the solution forthe steady state plate problem. We investigate thestability of the dynamical non-linear plate problemunder some condition on the applied loads. Thenwe explore the fluid structure interaction problemwith a Darcy flow in porous media. In an appro-priate Sobolev norm, we build an energy functionalfor the displacement field of the plate and the gra-dient pressure of the fluid flow. We show that for aclass of boundary conditions the energy functionalis bounded by the flux of mass through the inletboundary.

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Nonlinear poroacoustic flow in rigid porousmedia

Pedro JordanU.S. Naval Research Laboratory, USApedro.jordan@nrlssc.navy.milJ. K. Fulford

An acoustic acceleration wave is defined as a propa-gating singular surface (i.e., wavefront) across whichthe first derivatives of the velocity, pressure, or den-sity exhibit jumps. In this talk, the temporal evolu-tion of the amplitude and speed of such waves are in-vestigated in the context of nonlinear, fluid-acousticpropagation in rigid porous media, where the fluid-solid interaction is described by the resistance lawof Darcy. It is shown that there exists a criticalvalue, the constant ↵⇤(> 0), of the initial jump am-plitude. It is then established that the accelerationwave magnitude either goes to zero, as t ! 1, orblows-up, in finite time, depending on whether theinitial jump amplitude is less than or greater than ↵⇤.

Additionally, numerical solutions of an idealizedinitial-boundary value problem involving sinusoidalsignaling in a fluid-saturated porous slab are used toillustrate the finite-time transition from accelerationto shock wave, which occurs when the initial jumpamplitude is greater than ↵⇤, and comparisons withthe linearized case (i.e., the damped wave equation)are presented whenever possible. Finally, the relatedphenomenon of weakly-nonlinear poroacoustic trav-eling waves, where an exact solution is possible interms of the Lambert W -function, is briefly consid-ered and connections to second-sound (i.e., thermal

wave) phenomena are noted. (Work supported byONR funding.)

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Remarks on the Fourier series method

Vilmos KomornikUniversity of Strasbourg, Francevilmos.komornik@math.unistra.fr

We review some recent applications of harmonicand nonharmonic Fourier series in control theory,including various results obtained in collaborationwith C. Baiocchi, P. Loreti, M. Mehrenberger andA. Barhoumi. We discuss several possible general-izations of Parseval type inequalities due to Inghamand Beurling .

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On the exterior problem for nonlinear waveequations in 2D

Hideo KuboTohoku University, Japankubo@math.is.tohoku.ac.jp

In this talk I wish to present a result on the ex-terior problem for the nonlinear wave equations intwo space dimensions. Because the dispersive prop-erty for the solution to the wave equation is ratherweak in 2D, it is not straightforward to get counter-part for the Cauchy problem. Nevertheless, u Underthe geometric assumption on the obstacle, we areable to get an almost global solution for small ini-tial data. Moreover, if we pose the null conditionon the nonlinearity, then the solution actually existsglobally.

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Recovering sound spend and initial data forthe wave equation by a single measurement

Shitao LiuUniversity of Helsinki, Finlandshitao.liu@helsinki.fi

We consider a problem of recovering the sound speedand an initial condition for wave equations which ismotivated from the photo/thermo-acoustic imagingmodel. We will also show the connection betweensuch a problem and the classical inverse hyperbolicproblem with a single measurement.

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SPECIAL SESSION 4 21

Time optimal boundary controls for the heatequation

Sorin MicuUniversity of Craiova, Romaniasd micu@yahoo.comIonel Roventa, Marius Tucsnak

The fact that the time optimal controls for parabolicequations have the bang-bang property has been re-cently proved for controls distributed inside the con-sidered domain (interior control). The main resultin this article asserts that the boundary controls forthe heat equation have the same property, at leastin rectangular domains. This result is proved bycombining methods from traditionally distinct fields:the Lebeau-Robbiano strategy for null controllabilityand estimates of the controllability cost in small timefor parabolic systems, on one side, and a Remez-typeinequality for Muntz spaces and a generalization ofTuran’s inequality, on the other side.

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Inverse problem for the heat equation and theSchrodinger equation on a tree

Ademir PazotoFederal University of Rio de Janeiro, Brazilademir@im.ufrj.brLiviu Ignat, Lionel Rosier

In this paper we establish global Carleman estimatesfor the heat and Schrodinger equations on a network.The heat equation is considered on a general tree andthe Schrodinger equation on a star-shaped tree. TheCarleman inequalities are used to prove the Lips-chitz stability for an inverse problem consisting inretrieving a stationary potential in the heat (resp.Schrodinger) equation from boundary measurements.

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Instability for nonlinear evolution equations

Petronela RaduUniversity of Nebraska-Lincoln, USApradu@math.unl.eduStephen Pankavich

We consider steady solutions of semilinear parabolicand hyperbolic equations; the hyperbolic equationsexhibit a sign changing damping term. We provethat linear instability with a positive eigenfunctiongives rise to nonlinear instability by either exponen-tial growth or finite-time blow-up. We then discussa few examples to which our main theorems areimmediately applicable, including equations with su-percritical and exponential nonlinearities.

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Local and global well-posedness to a nonlin-ear model in viscoelasticity with m-Laplaciandamping

Mohammad RammahaUniversity of Nebraska-Lincoln, USAmrammaha1@math.unl.eduDaniel Toundykov

We consider a general viscoelasticity model in abounded domain ⌦ ⇢ R3 with a smooth boundary �:

utt ��pu��mut = f(u) in ⌦⇥ (0, T ),

subject to Dirichlet boundary condition: u = 0 on�⇥ (0, T ), or subject to p-generalized Robin bound-ary condition. We study the local and global solv-ability in the interesting case when p is between 2and 3. In this case, if the interior source f(u) is oforder r (i.e., |f(u)| c|u|r for all |u| � 1), then f(u)is locally Lipschitz from W 1,p(⌦) into L2(⌦), onlyfor the values 1 r 3p

2(3�p). Thus, our goal is to

prove the local and global solvability of the problemwhen the interior source is of supercritical order, i.e.,

3p2(3�p)

< r < 3p3�p

.

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Asymptotic behavior of the wave equationwith dynamic boundary condition

Belkacem Said-HouariKAUST university, Saudi Arabiasaidhouarib@yahoo.frStephane Gerbi

In this talk, we consider a semi-linear wave equa-tion with a dynamic boundary conditions.

We prove the local existence and uniqueness ofthe solution of our problem. We use the Faedo-Galerkin approximation coupled with the fixed pointtheorem. Concerning the asymptotic behavior of thesolution of this problems we prove the following :

When the initial data are large enough and thedamping is nonlinear, then the energy solution is un-bounded. In fact, we show that the Lp-norm of thesolutions grows as an exponential function.

We prove that in the case of linear boundarydamping, the solution blows up in finite time.

We showed also that if the initial data are in thestable set, the solution continues to live there forever.

In addition, we show that the presence of thestrong damping forces the solution to go to zero uni-formly and with an exponential decay rate. To obtainour results, we combine the potential well methodwith the energy method.

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22 9th AIMS CONFERENCE – ABSTRACTS

Solutions to the Euler equation on a domainwith a moving boundary

Amjad Tu↵ahaThe Petroleum Institute, United Arab Emirates

Igor Kukavica

We present a new proof of local-in-time existenceand regularity of solutions to the free boundaryEuler Equation without surface tension under theTaylor-Rayleigh condition.

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Simultaneous controllability and stabilizationof some uncoupled wave and plate equations

Louis TebouFlorida International University, USAteboul@fiu.edu

First, we discuss the simultaneous controllabilityof uncoupled wave equations with di↵erent speed ofpropagation in a bounded domain. The control islocally distributed, and the control region satisfiesthe geometric control condition of Bardos-Lebeau-Rauch. Thanks to the Hilbert uniqueness methodof Lions, the controllability problem is reduced toan observability one. To solve the observabilityproblem, we employ a combination of the observ-ability result of Bardos-Lebeau-Rauch for a singlewave equation, and a new uniqueness result for theuncoupled system. Then, we address the relatedstabilization problem with the help of a theorem ofHaraux and the new uniqueness result. Afterwards,we investigate similar problems for uncoupled plateequations with the help of the Holmgren uniquenesstheorem. Finally, we discuss some generalizations,and some open problems.

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Carleman estimates and stabilization of hy-perbolic systems in absence of geometric ob-servability conditions

Daniel ToundykovUniversity of Nebraska-Lincoln, USAdtoundykov2@unl.eduMatthias Eller

The sharp su�cient conditions for observability incontrol systems are formulated via geometric opticsand are linked to the structure of closed geodesicsin the underlying physical domain. However, thenecessary unique continuation property for PDEs isan intrinsically weaker requirement and does not im-pose such restrictions. It is, therefore, often possibleto stabilize an evolution system by placing feedbackcontrols on subsets of the domain that fail to satisfy

the geometric conditions. The price to pay is thenecessity to work with smoother solutions and thestabilization rates obtained thereby are no longerexponential.

In this work we present specialized Carleman es-timates and a generalization of a pioneering strategydue to G. Lebeau and L. Robbiano to prove uniformstability (for strong solutions) of 1st-order hyperbolicsystems without reliance on the geometric observabil-ity assumptions.

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An inverse problem for the ultrasound equa-tion: uniqueness and stability

Roberto TriggianiUniversity of Virginia, USArt7u@virginia.edu

We condsider a third order (in time) ultrasoundequation arising in acoustics and we recover a crit-ical coe�cient by means of boundary measurement.More sopecifically, we obtain uniqueness and stabilityof the recovery

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Min-max game problem for coupled PDE sys-tem and an application to the fluid-structureinteraction model

Jing ZhangUniversity of Virginia, USAjz4f@virginia.eduIrena Lasiecka, Roberto Triggiani

We consider an established hyperbolic-parabolicfluid-structure interaction model with control anddisturbance acting at the interface between the twomedia. Here, the structure (modeled by the systemof dynamic elasticity) is immersed in a fluid (mod-eled by the linearized Navier-Stokes equations). Agame theory problem between control and distur-bance is studied, when both act at the interface.To this end, the main mathematical di�culty thatone encounters is the fact that such model fails tosatisfy the ”singular estimate” from, say, control tostate. This is a critical obstacle, as this is preciselythe foundational property for a full theory to includesolvability of the Di↵erential Riccati equation in thestudy of the associated min-max game theory prob-lem. Failure of the ”singular estimate” property isdue to a mismatch between the parabolic and hy-perbolic component of the overall coupled dynamics.By introducing suitable observation or output opera-tors, with incremental smoothness on the trajectory,it is shown that the resulting system satisfies a mod-ified singular estimate, this time from the controlto the observation space. This then allows one toadapt the complete min-max theory to the present

SPECIAL SESSION 4 23

fluid-structure interaction model. More precisely,the approach followed is based on an abstract set-ting, of which the fluid-structure interaction modelis a canonical illustration, which enjoys a desirableproperty not assumed for the abstract model.

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