Tilman Utz
email: tilman.utz@uni-ulm.de

Knut Graichen

Andreas Kugi

Combined Feedforward/Model Predictive Tracking Control Design for Nonlinear Diffusion-Convection-Reaction-Systems

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

A large class of chemical reactors with an interaction of diffusive, convective, and reactive effects leads to infinite-dimensional mathematical models in the form of nonlinear boundary-controlled parabolic partial differential equations (PDEs) [START_REF] Jensen | The bifurcation behavior of tubular reactors[END_REF]. The control design for setpoint transitions of chemical reactors, e. g., for ignition, extinction, or grade-transitions constitutes a challenging problem. In this contribution, the well-known two-degrees-of-freedom (2DOF) control scheme is applied in order to tackle this control task. The basic idea consists in first designing a feedforward control to steer the system along prescribed trajectories. The trajectory planning and feedforward control are complemented with a state feedback tracking control stabilizing the system about the desired trajectories.

In the literature, there exists a variety of concepts for the design of both feedforward and feedback tracking controllers. For the feedforward control design, approaches using the flatness concept [START_REF] Fliess | Flatness and defect of non-linear systems: Introductory theory and examples[END_REF] have found widespread attention. The flatness property allows for a parametrization of the state and input in terms of a so-called flat output and its time derivatives and therefore provides a systematic approach for feedforward control design. Originally proposed for finite-dimensional systems, generalizations of the flatness concept have been successfully carried over to certain classes of PDEs, see, e. g., [START_REF] Laroche | Motion planning for the heat equation[END_REF][START_REF] Lynch | Flatness-based boundary control of a class of quasilinear parabolic distributed parameter systems[END_REF][START_REF] Meurer | Feedforward and feedback tracking control of diffusion-convectionreaction systems using summability methods[END_REF]. In these so-called late lumping approaches the parametrization is directly solved for the underlying PDE. In contrast, the early lumping approach to control design is based on a finite-dimensional approximation of the system. Using suitable finite difference schemes to approximate the spatial derivatives results in a semidiscretization which is differentially flat, and the equivalence of the respective feedforward controls has been shown under certain conditions for some types of PDEs, see, e. g., [START_REF] Ollivier | A generalization of flatness to nonlinear systems of partial differential equations. Application to the command of a flexible rod[END_REF][START_REF] Utz | Trajectory planning for quasilinear parabolic distributed parameter systems based on finite-difference semi-discretizations[END_REF].

In practice, the feedforward part has to be amended by an additional feedback tracking control in order to compensate for model uncertainties or disturbances. For the design of such stabilizing feedback controllers for infinite-dimensional systems, receding horizon optimal control, e. g., [START_REF] Lions | Optimal Control of Systems Governed by Partial Differential Equations[END_REF][START_REF] Tröltzsch | On convergence of a receding horizon method for parabolic boundary control[END_REF], constitutes a promising tool. This method is particularly attractive since it provides, in contrast to most state-of-the-art tracking control design methods for DCRSs, see, e. g., [START_REF] Meurer | Feedforward and feedback tracking control of diffusion-convectionreaction systems using summability methods[END_REF][START_REF] Meurer | Tracking control for boundary controlled parabolic pdes with varying parameters: Combining backstepping and differential flatness[END_REF], the possibility to systematically include constraints as they are frequently encountered in technical systems.

The paper is structured as follows: In Section 1, the diffusion-convectionreaction system (DCRS) and the control task is introduced. Section 2 is devoted to the control design based on a semi-discretization of the considered infinitedimensional system. The paper provides simulation results in Section 3 and conclusions are drawn in Section 4.

Problem formulation

The quasilinear, scalar DCRS, described by the (suitably scaled) parabolic PDE

p(θ(z, t))∂ t θ(z, t) = ∂ z (q(θ(z, t))∂ z θ(z, t)) -ν∂ z θ(z, t) + r(θ(z, t))θ(z, t) , (1)
z ∈ (0, 1), t > 0 is considered. The storage coefficient p(θ(z, t)) = p 0 + p 1 θ(z, t), the diffusion coefficient q(θ(z, t)) = q 0 + q 1 θ(z, t), and the reaction coefficient r(θ(z, t)) = r 0 + r 1 θ(z, t) depend on the state θ(z, t) in an affine way, whereas the convection parameter ν ≥ 0 is constant. The boundary conditions

∂ z θ(z, t)| z=0 = d(t) , (2a) q(θ(1, t)) ∂ z θ(z, t)| z=1 = q(u(t) -θ(1, t)) , (2b)
t > 0, q > 0, and the initial condition

θ(z, 0) = θ init (z) , (3)
z ∈ [0, 1], complete the infinite-dimensional system. In order for (1) to be parabolic it has to be assured that q(θ(z, t)) is positive in the considered range of the state variable. The exogenous input variable d(t) in (2a) represents an additional sink or source term that will be considered as an unknown disturbance and is assumed to be zero for the feedforward controller design. Additionally, the control input u(t) entering via the boundary condition (2b) is supposed to be subject to so-called box constraints, i. e., Denoting the state evaluated at z = 0 as the output of the system, i. e. y(t) = θ(0, t), the control task consists in stably and robustly carrying out transitions between setpoint values and the associated steady-state profiles of ( 1)-(3), within the transition time T = 1. The 2DOF control scheme used to accomplish these transitions is schematically depicted in Figure 1.

u(t) ∈ [u -, u + ] . (4)

Tracking control based on finite difference semi-discretization

The design of both the flatness-based feedforward control and the receding horizon tracking control depends on a semi-discretization of the infinite-dimensional system (1)-( 3), see also [START_REF] Utz | Trajectory planning and receding horizon tracking control of a quasilinear diffusion-convection-reaction system[END_REF]. The methodology to obtain the semi-discretized system pursued in this contribution is to discretize the spatial coordinate z using finite differences on an equidistant grid with N grid elements and the nodes z 0 = 0, z 1 = ∆z, . . . , z k = k∆z, . . . , z N = 1 where ∆z = 1/N . Applying the central finite difference schemes

∂ z θ k = 1 2∆z (θ k+1 -θ k-1 ) + O(∆z 2 ) , (5a)
∂ 2 z θ k = 1 ∆z 2 (θ k+1 -2θ k + θ k-1 ) + O(∆z 2 ) , (5b)
(∂ z θ k ) 2 = 1 ∆z 2 θ k+1 -θ k θ k -θ k-1 + O(∆z 2 ) (5c)
to the PDE (1) and the boundary conditions (2), leads to the following system of N + 1 ODEs for the discretized states3 θ k (t) = θ(z k , t):

(p 0 + p 1 θ 0 ) θ0 = (q 0 + q 1 θ 0 ) 2(θ 1 -θ 0 -∆zd) ∆z 2 + q 1 d 2 -νd + (r 0 + r 1 θ 0 )θ 0 , (6a)
(p 0 + p 1 θ k ) θk = (q 0 + q 1 θ k ) θ k+1 -2θ k + θ k-1 ∆z 2 + q 1 (θ k+1 -θ k )(θ k -θ k-1 ) ∆z 2 -ν θ k+1 -θ k-1 2∆z + (r 0 + r 1 θ k )θ k , k = 1, . . . , N -1 , (6b)
(p 0 + p 1 θ N ) θN = 2(q 0 + q 1 θ N ) ∆z 2 ∆z q q 0 + q 1 θ N (u -θ N ) -θ N + θ N -1 + q 1 ∆z 2 2∆z q q 0 + q 1 θ N (u -θ N ) + θ N -1 -θ N (θ N -θ N -1 ) - ν q q 0 + q 1 θ N (u -θ N ) + (r 0 + r 1 θ N )θ N . (6c)
Note that the scheme (5c) for the squared first derivative with respect to z is only applied to nodes with k > 0, which is advantageous for the parametrization considered in the subsequent section. The initial conditions obtained by evaluating (3) at the nodes

θ k (0) = θ init (z k ) , (7)
k = 0, . . . , N complete the finite-dimensional semi-discretized approximation of the infinite-dimensional system (1)-(3).

Flatness-based state and input parametrization

The semi-discretization ( 6) has the property of being flat, which is shown in the following. The k-th equation of (6a) and (6b) is affine in θ k+1 (t), k = 0, . . . , N -1 and (6c) is affine in the control input u(t). Thus, solving these equations for θ k+1 (t) and u(t), respectively, yields

θ 1 =: Ψ 0 (θ 0 , θ0 ) , (8a)
θ k+1 =: Ψ k (θ k , θk , θ k-1 ) , k = 1, . . . , N -1, (8b)
u =: Ψ N (θ N , θN , θ N -1 ) . ( 8c
)
Considering y(t) = θ 0 (t) it follows from (8a) that θ 1 (t) can be parametrized in terms of y(t) and ẏ(t). Differentiating (8a) with respect to time

θ1 = ∂Ψ 0 ∂θ 0 θ0 + ∂Ψ 0 ∂ θ0 θ0 (9)
and inserting this result into (8b) for k = 1 directly yields a parametrization of θ 2 (t) by y(t), ẏ(t) and ÿ(t). Obviously, this procedure can be analogously continued for k = 2, . . . , N -1 with

θk+1 = ∂Ψ k ∂θ k θk + ∂Ψ k ∂ θk θk + ∂Ψ k ∂θ k-1 θk-1 , (10)
such that every state θ k (t), k = 1, . . . , N as well as the control input u(t) are recursively parametrized by y(t) and its first N + 1 time derivatives. Hence y(t) constitutes a flat output.

As a consequence, nominal state trajectories θ * k (t) and a control input u * (t) can be calculated by recursively evaluating (8) using a sufficiently smooth trajectory for the flat output y * (t), such that the output of ( 6)-( 7) for an exact model and in the absence of disturbances exactly tracks y * (t). Under certain conditions on the growth of the time-derivatives of the prescribed flat output y * (t) (defined by its so-called Gevrey-class) and given a suitable set of system parameters p 0 , p 1 , q 0 , q 1 , ν, r 0 , and r 1 , it can be shown that the control input u * (t) converges for N → ∞ to a suitable control input for the DCRS (1)-( 3), see, e. g., [START_REF] Lynch | Flatness-based boundary control of a class of quasilinear parabolic distributed parameter systems[END_REF][START_REF] Utz | Trajectory planning for quasilinear parabolic distributed parameter systems based on finite-difference semi-discretizations[END_REF].

Receding horizon tracking control

In the case of model uncertainties or in order to account for disturbances or control constraints, the feedforward control Σ ffwd in Figure 1 has to be extended by a feedback controller Σ fb . The receding horizon control strategy thereby is formulated for the same semi-discretization that is used for the flatness-based parametrization.

In view of the 2DOF control structure in Figure 1, the feedback controller Σ fb is designed to stabilize the system Σ ∞ along the reference trajectories θ * k (t), k = 0, . . . , N provided by the flatness-based trajectory planning Σ ffwd . This means that the tracking errors

∆θ k (t) = θ(z k , t) -θ * k (t) ,
k = 0, . . . , N have to be suppressed by the control action ∆u fb (t), which corrects the feedforward control u * (t), see Figure 1. Using θ(t) = [θ 0 (t), . . . , θ N (t)] T ∈ R N +1 to summarize the differential equations ( 6) in the form θ(t) = f (θ(t), u(t)) , the tracking error ∆θ(t) = [∆θ 0 (t), . . . , ∆θ N (t)] T ∈ R N +1 satisfies the error dynamics

∆ θ(t) = f θ * (t)+∆θ(t), u * (t)+∆u fb (t) - θ * (t) =: F(∆θ(t), ∆u fb (t), t) . ( 11
)
The receding horizon controller design accounts for the nonlinear and timevarying error dynamics [START_REF] Mayne | Constrained model predictive control: stability and optimality[END_REF] by solving the following optimal control problem (OCP) in a discrete-time fashion for each instant of time t i = i∆t with the given sampling time ∆t:

min ∆u(•) J(∆u(•), ∆θ i ) = ||∆θ(t i,f )|| 2 P + t i,f ti ||∆θ(t)|| 2 Q + R∆u(t) 2 dt (12a) s.t. ∆ θ(t) = F(∆θ(t), ∆u(t), t) , ∆θ(t i ) = ∆θ i (12b) ∆u(t) ∈ [∆u -(t), ∆u + (t)] , t ∈ [t i , t i,f ]• (12c)
Starting from the tracking error ∆θ i = θ(t i )θ * (t i ) at time t i , the error dynamics in (12b) are used to predict the error trajectory ∆θ(t) over a finite prediction horizon t ∈ [t i , t i,f ] with the final time t i,f = t i + t f , where t f ≥ ∆t denotes the (constant) horizon length. The cost functional (12a) to be minimized penalizes the tracking error ∆θ(t) as well as the feedback control action ∆u(t) with respect to the positive definite matrices4 P, Q and the positive scalar R.

The time-varying input constraints (12c) follow from the original constraints (4) of the feedforward trajectory u * (t) in the form

∆u ± (t) := u ± -u * (t) , t ∈ [t i , t i,f ] . (13)
In the following, it is assumed that the OCP ( 12) possesses an optimal solution ∆ū(t; ∆θ i ), ∆ θ(t; ∆θ i ), t ∈ [t i , t i,f ]. Note that this assumption is not very restrictive due to the absence of terminal and state constraints. The OCP [START_REF] Meurer | Feedforward and feedback tracking control of diffusion-convectionreaction systems using summability methods[END_REF] is solved in each time step t i of the receding horizon scheme and only the first part of the control trajectory ∆ū(t; ∆θ i ) is used as the feedback control in Figure 1, i. e.

∆u fb (t) = ∆ū(t; ∆θ i ) , t ∈ [t i , t i + ∆t) , i ∈ N + 0 . ( 14
)
In the next time step t i+1 , the OCP ( 12) is solved again with respect to the new tracking error ∆θ i+1 that is used as initial condition in (12b).

If the system exactly follows the reference trajectory at time t i , i. e., if ∆θ i = 0, and in the absence of disturbances and model inaccuracies the optimal solution of the OCP ( 12) is

∆ū(t; ∆θ i ) = 0 , ∆ θ(t; ∆θ i ) = 0 , t ∈ [t i , t i,f ] (15)
with J(∆ū(•), ∆θ i ) = 0. Hence, if the system exactly tracks the nominal trajectories the control action of the feedback controller Σ fb is zero, see Figure 1. Important design parameters of the receding horizon control scheme are the choice of the weighting matrices P ∈ R (N +1)×(N +1) and Q ∈ R (N +1)×(N +1) , of the scalar weight R, and the horizon length t f . Receding horizon formulations in model predictive control often use terminal set or equality constraints to achieve stability. In the case of a free end point formulation as it is the case in [START_REF] Meurer | Feedforward and feedback tracking control of diffusion-convectionreaction systems using summability methods[END_REF], stability can be shown, e. g., if the terminal cost function ||∆θ(t i,f )|| 2 P represents a (local) control Lyapunov function [START_REF] Chen | A quasi-infinite horizon nonlinear model predictive control scheme with guaranteed stability[END_REF][START_REF] Mayne | Constrained model predictive control: stability and optimality[END_REF][START_REF] Limon | On the stability of constrained MPC without terminal constraint[END_REF] or if the horizon length t f is sufficiently large [5]. For the error dynamics (12b), which is time-dependent due to the feedforward trajectories, the rigorous proof of stability [START_REF] Graichen | Stability and incremental improvement of suboptimal MPC without terminal constraints[END_REF] as well as the consistency of this finite-dimensional control with the original infinite-dimensional system is subject of current research. In this contribution, the stability and performance of the receding horizon tracking controller are demonstrated by means of simulation studies in the following section.

Simulation example

With the flat output y(t) = θ 0 (t), the control task under consideration consists in realizing the finite-time transition between two setpoints y(0) = y 0 and y(1) = y 1 , which correspond to the (infinite-dimensional) steady-state profiles θ(z, 0) and θ(z, 1). A suitable reference trajectory y * (t) for the trajectory planning has to comply with the desired steady-state output values at the beginning (t = 0) and at the end (t = 1) of the setpoint transition, i. e.

y * (0) = y 0 , y * (1) = y 1 , d l dt l y(t) t=0 = d l dt l y(t) t=1 = 0 (16)
l = 1, . . . , N + 1. The temporal path of the transition can be provided either by a polynomial of suitable order or, especially with regard to the continuous limit of the parametrization, by any smooth function of an appropriate Gevrey-class, see, e. g., [START_REF] Lynch | Flatness-based boundary control of a class of quasilinear parabolic distributed parameter systems[END_REF]. In all simulation results shown in this contribution, the control design relies on a semi-discretization with N = 10 grid elements. The following set of system parameters p 0 = 1.2, p 1 = 0.05, q 0 = 1, q 1 = -0.05, q = 1, ν = 0.1, r 0 = 1, and r 1 = 0.2 is used and the desired initial and final output values are y 0 = 1 and y 1 = 2, respectively. In addition, the values u ± = ±4 are used as box constraints (4) and a non-zero disturbance

d(t) = -0.4 for t ∈ [0.4, 0.6] 0 else (17)
is considered in the simulations. The receding horizon control design is based on the time discretization t i = i∆t with the sampling time ∆t = 0.005 and the horizon length t f = 0.3. In each time step t i , the OCP ( 12) is numerically solved with a tailored gradient projection method [START_REF] Graichen | On the stability of receding horizon control with a general terminal cost[END_REF], such that the OCP may be solved in a computationally very efficient way, see also [START_REF] Utz | Computationally efficient receding horizon trajectory tracking control for a tubular reactor example[END_REF]. The weighting matrix Q is set to a diagonal matrix with the diagonal element values interpolated between 200 for the first error state corresponding to the output and the value 2 for the last error state. The terminal weighting matrix P is set to 0.1Q, while R is chosen as 0.3. The overall feedback controller Σ fb is implemented as a Cmex function in Matlab, and for the simulations, the standard Matlab-solver ode15s is used to solve the semi-discretized system ( 6)-( 7) on a grid with N sim nodes, where N sim ≫ N . In Figure 2, the setpoint transition in the nominal case, i. e., without disturbance or input constraints, as well as a transition with disturbance (17) and a transition with input constraints (4) is shown. The time behaviour of the feedforward control u * (t) as well as the evolution of the cost J(∆ū(•), ∆θ i ) show that the contribution of the tracking controller ∆u fb (t) vanishes in the nominal case. For the non-zero disturbance [START_REF] Utz | Trajectory planning and receding horizon tracking control of a quasilinear diffusion-convection-reaction system[END_REF], this is of course no longer the case. However, it can be seen that after restoring the nominal conditions, the tracking error is reduced very fast, the transition is completed as prescribed and the simulation remains stable. At the same time, the cost J(∆ū(•), ∆θ i ) decreases monotonically to zero. In the case of input constraints (4), deviations from the desired behaviour are also inevitable, since they are not considered in the feedforward control design. It can also be observed that due to the prediction horizon the 0 0.2 0.4 0.6 0. feedback controller becomes aware of the impending violation of the constraints before it actually occurs. This results in a rise of the cost and subsequently in the pre-steering action visible in the control input time behaviour, see Figure 2.

The use of a 2DOF control scheme offers some benefits for the realization of setpoint transition tracking control, which are pointed out in the following. In Figure 3 the same setpoint transition as before is considered with non-zero disturbance [START_REF] Utz | Trajectory planning and receding horizon tracking control of a quasilinear diffusion-convection-reaction system[END_REF] and the input constraints (4). However, the flatness-based trajectory generation and feedforward control are replaced by a pure feedforward of the steady-state reference values of the state variables θ k (t), k = 0, . . . , N and of the input u(t) at t = 0.1. Furthermore, a shorter prediction horizon t f = 0.1 is considered for the receding horizon controller. This could be motivated by the need to reduce the computational cost for the solution of the OCP [START_REF] Meurer | Feedforward and feedback tracking control of diffusion-convectionreaction systems using summability methods[END_REF]. It can be observed on the one hand that with the prediction horizon t f = 0.3, the receding horizon control carries out the transition within a transition time comparable to the one observed for the 2DOF control scheme. However, this also results in significant control action especially at the beginning of the transition. On the other hand, the transition time is increased in the case of the short prediction horizon t f = 0.1 and the desired final output value is not reached within the simulation time. The largely different tracking behaviour seems comprehensible since the transition time is an important tuning parameter of the receding horizon control. This is in contrast to the simulation results obtained if the flatness-based trajectory generation and feedforward control are used in the 0 0.2 0.4 0.6 0.8 1.0 1.2 2DOF control scheme. A slight deterioration of the tracking behaviour can also be seen for a reduced prediction horizon. However, as the main control action for the transition is provided by the feedforward control, this deterioration remains comparatively small. This confirms the observation [START_REF] Utz | Comparative evaluation of nonlinear model predictive and flatness-based two-degree-of-freedom control design in view of industrial application[END_REF] that the disturbance rejection may be designed nearly independently from the setpoint transition in the 2DOF control scheme.

1 1.2 1.4 1.6 1.8 2.0 t f = 0.1 t f = 0.3 t f = 0.1, Σ ffwd

Conclusion

In this contribution, a 2DOF control scheme is presented for setpoint transition tracking control of a quasilinear scalar DCRS. A flatness-based feedforward controller and a receding horizon feedback tracking controller are designed in an early lumping approach using a finite-difference semi-discretization of the DCRS. Thereby, input constraints are systematically incorporated into the receding horizon control design. In the simulation studies, the 2DOF control scheme shows a good performance for both trajectory tracking and disturbance rejection. Furthermore, the 2DOF control scheme allows for a nearly independent tuning of the tracking performance and the disturbance rejection. Stability of the feedback tracking control scheme as well as the further decrease of the computational costs are subject to current research activities.

Fig. 2 .

 2 Fig.2. Output y(t), control input u(t) and cost J(∆ū(•), ∆θ i ) for the tracking control of the DCRS in the nominal case, in the case of non-zero disturbance, and under input constraints.

Fig. 3 .

 3 Fig.3. Output y(t), control input u(t) and cost J(∆ū(•), ∆θ i ) for the tracking control of the DCRS with non-zero disturbance and under input constraints both without flatness-based feedforward control and prediction horizon t f ∈ {0.1, 0.3}, and with flatness-based feedforward control (denoted by Σ ffwd in the figure legend) and prediction horizon t f = 0.1.

  and u * based on desired output trajectories y * , and a tracking controller Σ fb , which provides a correction to the nominal control input trajectories ∆u fb based on the state tracking error ∆θ.

		u *
		θ *
	y * , ẏ * , . . .	Σ ffwd	θ *	-∆θ	Σ fb	∆u fb	u	Σ ∞	θ
	Fig. 1. 2DOF control scheme consisting of the infinite-dimensional system Σ∞, a tra-
	jectory planning and feedforward control Σ ffwd , which provides nominal state and con-
	trol input trajectories θ

*

For the sake of readability, time-dependencies as in θ k (t) are omitted whenever they are clear from the context.

Here ||x|| 2 S = x T Sx denotes the weighted Euclidean norm of the vector x ∈ R n with the matrix S ∈ R n×n being positive definite.


Igor Chueshov
email: chueshov@univer.kharkov.ua

Iryna Ryzhkova
email: ryzhkova@univer.kharkov.ua

Well-Posedness and Long Time Behavior for a Class of Fluid-Plate Interaction Models

Keywords: Fluid-structure interaction, nonlinear shell/plate, linearized 3D Navier-Stokes equations, global attractor, finite dimension

We deal with well-posedness and asymptotic dynamics of a class of coupled systems consisting of linearized 3D Navier-Stokes equations in a bounded domain and a classical (nonlinear) elastic plate/shell equation. We consider three models for plate/shell oscillations: (a) the model which accounts for transversal displacement of a flexible flat part of the boundary only, (b) the model for in-plane motions of a flexible flat part of the boundary, (c) the model which accounts for both transversal and longitudinal displacements. For all three cases we present wellposedness results and prove existence of a compact global attractor. In the first two cases the attractor is of finite dimension and possesses additional smoothness. We do not assume any kind of mechanical damping in the plate component in the case of models (a) and (b). Thus our results means that dissipation of the energy in the fluid due to viscosity is sufficient to stabilize the system in the latter cases.

Introduction

We consider a coupled (hybrid) system, which describes interaction of a homogeneous viscous incompressible fluid that occupies a bounded domain O with elastic plate/shell. Boundary ∂O of O consists of the (solid) walls of the container S and a horizontal (flat) part Ω, on which a thin (nonlinear) elastic shell or plate is placed. The motion of the fluid is described by linearized 3D Navier-Stokes equations. To describe deformations of the shell/plate we use several elastic models. In all cases we assume that large deflections of the shell produce small effect on the fluid. This corresponds to the case when the fluid fills the container which is large in comparison with the size of the shell/plate. We note that the mathematical studies of the problem of interaction of viscous fluids and elastic plates/bodies have a long history. We refer to [START_REF] Chambolle | Existence of weak solutions for the unsteady interaction of a viscous fluid with an elastic plate[END_REF][START_REF] Grobbelaar-Van Dalsen | Strong stability for a fluid-structure model[END_REF][START_REF] Kopachevskii | Small oscillations of a viscous fluid in a vessel bounded by an elastic membrane[END_REF] for the case of plates/membranes, to [START_REF] Coutand | Motion of an elastic solid inside an incompressible viscous fluid[END_REF] in the case of moving elastic bodies, and to [START_REF] Avalos | Semigroup well-posedness in the energy space of a parabolic hyperbolic coupled Stokes-Lamé PDE system of fluid-structure interaction[END_REF][START_REF] Barbu | Smoothness of weak solutions to a nonlinear fluid-structure interaction model[END_REF][START_REF] Du | Analysis of a linear fluid-structure interaction problem[END_REF] in the case of elastic bodies with fixed interface; see also the literature cited in these papers.

Mathematical Description of the Models

Fluid. Let O ⊂ R 3 be a bounded domain with sufficiently smooth boundary ∂O. We assume that ∂O = Ω ∪ S, where Ω ⊂ {x = (x 1 ; x 2 ; 0) : x ′ ≡ (x 1 ; x 2 ) ∈ R 2 } with a smooth contour Γ = ∂Ω and S is a surface lying in R 3 -= {x 3 ≤ 0}. The exterior normal on ∂O is denoted by n. We have that n = (0; 0; 1) on Ω. The surface S corresponds to solid walls of the container with a fluid, and Ω models an elastic shell or plate placed over the fluid.

To describe the fluid we consider the following linear Navier-Stokes equations in O for the fluid velocity field v = v(x, t) = (v 1 (x, t); v 2 (x, t); v 3 (x, t)) and the pressure p(x, t):

v t -ν∆v + ∇p = G f in O × (0, +∞) , (1)
divv = 0 in O × (0, +∞) , (2)
where ν > 0 is the dynamical viscosity and G f is a volume force. We denote by T f (v) the surface force exerted by the fluid on the shell which is equal to T n| Ω , where n is an outer unit normal to ∂O at Ω and

T = {T ij } 3 i,j=1
is the stress tensor of the fluid,

T ij ≡ T ij (v) = ν ( v i xj + v j xi ) -pδ ij , i, j = 1, 2, 3 .
Since n = (0; 0; 1) on Ω, we have that

T f (v) = (ν(v 1 x3 + v 3 x1 ); ν(v 2 x3 + v 3 x2 ); 2νv 3 x3 -p) .
A specific form of this force as well as boundary conditions on Ω for the fluid depend on elastic plate/shell model we choose. General Model (GM). We start with the full von Karman shallow shell model which accounts for both transversal and in-plane displacements (see [START_REF] Vorovich | On some direct methods in nonlinear oscillations of shallow shells[END_REF][START_REF] Lagnese | Boundary Stabilization of Thin Plates[END_REF][START_REF] Koch | Hadamard well-posedness of weak solutions in nonlinear dynamic elasticity-full von Karman systems[END_REF][START_REF] Sedenko | On the uniqueness theorem for generalized solutions of initialboundary problems for the Marguerre-Vlasov vibrations of shallow shells with clamped boundary conditions[END_REF] and the references therein). To this end we equip (1) and (2) with the (nonslip) boundary conditions imposed on the velocity field v = v(x, t):

v = 0 on S, v ≡ (v 1 ; v 2 ; v 3 ) = (u 1 t ; u 2 t ; w t ) on Ω , (3)
where u = u(x, t) ≡ (u 1 ; u 2 ; w)(x, t) is the displacement of the shell occupying Ω. Here w stands for the transversal displacement, ū = (u 1 ; u 2 ) -for the lateral (in-plane) displacements.

To describe the shell motion we use the full von Karman model which takes into account rotational inertia of the filaments and possible presence of in-plane acceleration terms (see the literature cited above):

M α (w tt + γw t ) + ∆ 2 w + trace {KN (u)} -div {N (u)∇w} = G 3 -2νv 3 x3 + p , (4)
and

ϱū tt = div {N (u)} + ( G 1 -ν(v 1 x3 + v 3 x1 ); G 2 -ν(v 2 x3 + v 3 x2 ) ) , (5)
where

M α = 1 -α∆, K = diag (k 1 , k 2 ), and
N (u) ≡ ( N 11 N 12 N 12 N 22 ) = C(ϵ 0 (ū) + wK + f (∇w)) with ū = (u 1 ; u 2 ), C(ϵ) = D [µ trace ϵ • I + (1 -µ)ϵ],
and

ϵ 0 (ū) = 1 2 (∇ū + ∇ T ū), f (s) = 1 2 s ⊗ s, s ∈ R 2 .
Here

D = Eh/(1 -µ 2 ), E is Young's modulus, 0 < µ < 1/2 is Poisson's ratio, h
is the thickness of the shell, α > 0 and ϱ ≥ 0 are constants taking into account rotational inertia and in-plane inertia of the shell, γ ≥ 0 is a parameter which describes intensity of the viscous damping of the shell material. We denote by G sh ≡ (G 1 ; G 2 ; G 3 ) a (given) body force applied to the shell. We impose the clamped boundary conditions on the shell:

u 1 | ∂Ω = u 2 | ∂Ω = 0 (6)
and

w| ∂Ω = ∂w ∂n ∂Ω = 0 . (7)
We supply ( 1)- [START_REF] Chueshov | Von Karman Evolution Equations[END_REF] with the initial data for the velocity field v = (v1 ; v 2 ; v 3 ) and the shell displacement vector u = (u 1 ; u 2 ; w) of the form

1 v t=0 = v 0 , u t=0 = u 0 , w t t=0 = w 1 , ϱ [ ūt t=0 -ū1 ] = 0 , (8)
where ū = (u 1 ; u 2 ). Here v 0 = (v 1 0 ; v 2 0 ; v 3 0 ), u 0 = (u 1 0 ; u 2 0 ; w 0 ), w 1 , and ū1 = (u 1 1 ; u 2 1 ) are given vector functions which we will specify later. From (2) and (3) we can also derive the compatibility condition

∫ Ω w(x ′ , t)dx ′ = const for all t ≥ 0 , (9)
which can be interpreted as preservation of the volume of the fluid.

Simplified Model 1 (SM1). This kind of models arises in the study of the problem of blood flows in large arteries (see, e.g., [START_REF] Grobbelaar-Van Dalsen | Strong stability for a fluid-structure model[END_REF] and the references therein). The model assumes additional hypothesis that the transversal displacement w of the plate is negligible relatively to the in-plane displacement (u 1 ; u 2 ). Thus we consider only longitudinal deformations of the plate and take into account only tangential shear forces which fluid exerts on the plate. Formally this means that we omit equation ( 4) and put w ≡ 0 in (3) and ( 5). Thus we arrive at the following boundary conditions imposed on the velocity field v = v(x, t):

v = 0 on S, v ≡ (v 1 ; v 2 ; v 3 ) = (u t ; 0) ≡ (u 1 t ; u 2 t ; 0) on Ω , (10)
where u = u(x, t) ≡ (u 1 (x, t); u 2 (x, t)) is the in-plane displacement vector of the plate placed on Ω satisfying (5) with N 0 (u) = C(ϵ 0 (u)) instead of N . We assume that for this case the external (in-plane) force (G 1 ; G 2 ) in ( 5) is a nonlinear feedback force represented by a potential Φ:

G i = f i (u 1 , u 2 ) ≡ ∂Φ(u 1 , u 2 ) ∂u i , i = 1, 2 .
Since v 3 (x 1 ; x 2 ; 0) = 0 for (x 1 ; x 2 ) ∈ Ω due to the second relation in [START_REF] Coutand | Motion of an elastic solid inside an incompressible viscous fluid[END_REF], we have v 3 xi = 0 on Ω, i = 1, 2. Thus after rescaling of the elastic constants we arrive at the following equations for the in-plane displacement u = (u 1 ; u 2 ):

u i tt -∆u i -λ∂ xi [div u] + νv i x3 | x3=0 + f i (u) = 0, i = 1, 2 , ( 11
)
where λ is a nonnegative parameter. We impose the clamped boundary conditions [START_REF] Chueshov | Long-Time Behavior of Second Order Evolution Equations with Nonlinear Damping[END_REF] for the displacement u = (u 1 ; u 2 ) on Γ = ∂Ω. Thus we obtain Problem (SM1): Find vector functions v = (v 1 ; v 2 ; v 3 ) and u = (u 1 ; u 2 ) satisfying (in some sense) equations ( 1), ( 2), ( 10), ( 11), ( 6) and the initial data

v| t=0 = v 0 , u| t=0 = u 0 , u t | t=0 = u 1 .
This problem with λ = 0 and f i (u) ≡ 0 was considered in [START_REF] Grobbelaar-Van Dalsen | Strong stability for a fluid-structure model[END_REF] (see also the literature cited there) with the additional strong (Kelvin-Voight type) damping force applied to the interior of the plate. In contrast with [START_REF] Grobbelaar-Van Dalsen | Strong stability for a fluid-structure model[END_REF] we do not assume the presence of mechanical damping terms in the plate component of the system and consider a nonlinearly forced model. Simplified Model 2 (SM2). This model is concerned to dynamics of the transversal displacement w. The corresponding model assumes a special structure of the in-plane displacements ū = ū(x, t) ≡ (u 1 (x, t); u 2 (x, t)) in (4) as a function of the transversal displacement w only. Hence we neglect the equation in [START_REF] Chueshov | Introduction to the Theory of Infinite-Dimensional Dissipative Systems[END_REF] (see, e.g., [START_REF] Lagnese | Boundary Stabilization of Thin Plates[END_REF][START_REF] Vorovich | On some direct methods in nonlinear oscillations of shallow shells[END_REF] and the references therein). We also assume that α = 0. This formal procedure leads to the following boundary conditions imposed on the velocity field v = v(x, t):

v = 0 on S; v ≡ (v 1 ; v 2 ; v 3 ) = (0; 0; w t ) on Ω . (12)
This and also [START_REF] Barbu | Smoothness of weak solutions to a nonlinear fluid-structure interaction model[END_REF] imply that v 3 x3 = 0 and therefore the third (transversal) component T f (v) on ∂Ω is exactly the pressure p of the fluid. Thus the transversal displacement w = w(x, t) of the plate satisfies the following equation:

w tt + ∆ 2 w + F(w) = G pl + p| Ω in Ω × (0, ∞) , (13)
where G pl is a given body force on the plate, F(u) is a nonlinear feedback force which will be specified later. As a result we obtain Problem (SM2): Find the fluid velocity field v = (v 1 ; v 2 ; v 3 ), the pressure p, and the transversal displacement of the plate w satisfying (in some sense) equations ( 1), ( 2), ( 12), ( 13) and also compatibility condition [START_REF] Chueshov | Unsteady interaction of a viscous fluid with an elastic shell modeled by full von Karman equations[END_REF], boundary conditions [START_REF] Chueshov | Von Karman Evolution Equations[END_REF], and initial conditions of the form

v(0) = v 0 , w(0) = w 0 , w t (0) = w 1 .
Remark 1. We emphasize that even in the linear case due to the structure of the surface fluid forces T f (v) we cannot split system (1)-( 8) into two sets of equations describing longitudinal and transversal plate movements separately, i.e., we cannot reduce the model in (GM) to the cases considered in the models in (SM1) and (SM2). The point is that in the case (SM1) equation ( 11) for longitudinal plate deformations does not contain the terms v 3 xi and the model does not require any compatibility conditions like (9) because the volume of the fluid obviously preserves. In the case of purely transversal displacements (see (SM2)) the force exerted on the plate by the fluid contains the pressure only. See also [START_REF] Chueshov | Unsteady interaction of a viscous fluid with an elastic shell modeled by full von Karman equations[END_REF] for a further discussion.

Spaces and notations. To describe fluid velocity fields we introduce the following spaces. Let C(O) be the class of C ∞ vector-valued solenoidal (i.e., divergencefree) functions on O which vanish in a neighborhood of S. We denote by X the closure of C(O) with respect to the L 2 -norm and by V the closure of C(O) with respect to the H 1 (O)-norm. One can see that

X = { v = (v 1 ; v 2 ; v 3 ) ∈ [L 2 (O)] 3 : div v = 0, γ n v ≡ (v, n) = 0 on S } , V = { v = (v 1 ; v 2 ; v 3 ) ∈ [H 1 (O)] 3 : div v = 0, v = 0 on S } .
We refer to [START_REF] Temam | Navier-Stokes Equations: Theory and Numerical Analysis[END_REF], for instance, for the details concerning spaces of this type.

To describe shell/plate displacements we use the Sobolev spaces H s (Ω) and H s 0 (Ω). We also denote H s (Ω) = H s (Ω) ∩ L 2 (Ω) for s ≥ 0, where L 2 (Ω) is the subspace in L 2 (Ω) consisting of functions with zero average over Ω.

For D = either O or Ω we denote by ∥ • ∥ D the norm in L 2 (D) and by ∥ • ∥ s,D the norm in H s (D) and keep the corresponding notations for the inner products.

3 Results: Well-Posedness and Long-Time Dynamics General Model. We deal with weak (variational) solutions to (1)-( 9) and consider the cases ϱ > 0 and ϱ = 0 simultaneously. This is possible due to an additional regularity estimate for the shell velocities, which follows from (3) and from the standard trace theorem. Even in the case ϱ = 0 we have that

||w t (t)|| 2 H 1/2 (Ω) + ||u 1 t (t)|| 2 H 1/2 (Ω) + ||u 2 t (t)|| 2 H 1/2 (Ω) ≤ C||∇v(t)|| 2 O ( 14
)
for every weak solution (v(t); u(t)). We use this observation to suggest unified way to prove a well-posedness result not distinguishing the cases ϱ > 0 and ϱ = 0 in contrast with [START_REF] Vorovich | On some direct methods in nonlinear oscillations of shallow shells[END_REF] (see also [START_REF] Sedenko | On the uniqueness theorem for generalized solutions of initialboundary problems for the Marguerre-Vlasov vibrations of shallow shells with clamped boundary conditions[END_REF]). We also note that in the case we neglect the inertia of longitudinal deformations (ϱ = 0) the equations in (5) become elliptic. However, we keep the initial data for the in-plane displacement (u 1 ; u 2 ). The point is that the first order evolution for (u 1 ; u 2 ) goes from the boundary condition for the fluid velocity in (3).

As a phase space we use

H =    {(v 0 ; u 0 ; u 1 ) ∈ X × W × Y : v 0 = u 1 on Ω} , ϱ > 0 , { (v 0 ; u 0 ; w 1 ) ∈ X × W × H 1 0 (Ω) : (v 0 ) 3 = w 1 on Ω } , ϱ = 0 ,
where w 1 is the third component of the initial displacement velocity u 1 and

W = H 1 0 (Ω) × H 1 0 (Ω) × H 2 0 (Ω), Y = L 2 (Ω) × L 2 (Ω) × H 1 0 (Ω) .
Our main result concerning (GM) is the following well-posedness theorem.

Theorem 1. Assume that α > 0, γ ≥ 0, ϱ ≥ 0, and

U 0 ∈ H, G f ∈ V ′ , G sh ∈ [ H -1/2 (Ω) ] 2 × H -1 (Ω)
(in the case ϱ = 0 the data ū1 are not fixed). Then for any interval [0, T ] there exists a unique weak solution (v(t); u(t)) to (1)-( 9) with the initial data U 0 . This solution satisfies an energy balance equality and generates a continuous (both in strong and weak sense) evolution semigroup S t in the space H. The evolution operator S t is defined as follows

-case ϱ > 0: S t (v 0 ; u 0 ; u 1 ) ≡ U (t) = (v(t); u(t); u t (t))
, where the couple (v(t); u(t)) solves (1)-( 9); -case ϱ = 0: S t (v 0 ; u 0 ; w 1 ) ≡ Ū (t) = (v(t); u(t); w t (t)), where v(t) and u(t) = (u 1 (t); u 2 (t); w(t)) solves (1)-( 9) with ϱ = 0.

Proof. We use the compactness method with Galerkin's approximations, which was inspired by the method developed in [START_REF] Chambolle | Existence of weak solutions for the unsteady interaction of a viscous fluid with an elastic plate[END_REF] for the case of a linear plate interacting with nonlinear Navier-Stokes equations. Uniqueness relies on the same idea as in [START_REF] Sedenko | On the uniqueness theorem for generalized solutions of initialboundary problems for the Marguerre-Vlasov vibrations of shallow shells with clamped boundary conditions[END_REF] and involves Brésis-Gallouet type inequality. We follow the scheme of [START_REF] Koch | Hadamard well-posedness of weak solutions in nonlinear dynamic elasticity-full von Karman systems[END_REF] in the proof of continuity properties of the solution and the energy equality. The details of the proof can be found in [START_REF] Chueshov | Unsteady interaction of a viscous fluid with an elastic shell modeled by full von Karman equations[END_REF].

Remark 2. In the case α = 0 we can prove existence of weak solutions which satisfy an energy inequality using the same type of argument. Uniqueness of the solutions is still an open question. Sedenko's method does not work here because the nonlinearity is strongly supercritical when α = 0.

Our next result deals with global attractors. We recall (see, e.g., [START_REF] Chueshov | Introduction to the Theory of Infinite-Dimensional Dissipative Systems[END_REF][START_REF] Temam | Infinite-Dimensional Dynamical Dystems in Mechanics and Physics[END_REF]) that global attractor of the dynamical system (S t , H) is defined as a bounded closed set A ⊂ H which is invariant (S t A = A for all t > 0) and uniformly attracts all other bounded sets: lim t→∞ sup{dist H (S t y, A) : y ∈ B} = 0 for any bounded set B in H.

Theorem 2. Assume that α > 0, γ > 0, and the external forces satisfy

G f ≡ 0, G 1 sh = G 2 sh ≡ 0, and G 3 sh ≡ g ∈ H -1 (Ω) .
Let the set of the stationary points in H of the problem (1)-( 9) is bounded. Then the corresponding evolution semigroup S t possesses a compact global attractor.

To prove this theorem we apply J.Ball's method (in the form presented in [START_REF] Rosa | Attractors for non-compact semigroups via energy equations[END_REF]).

To this end we need the property γ > 0, i.e., assume a presence of mechanical damping in the transversal component of displacement. The question whether the system under consideration demonstrates compact long-time behavior without mechanical damping in the shell component is still open. The main obstacle in this case is that the dissipation of the energy in the fluid leads to the mechanical damping of order 1/2 in the shell components (see ( 14)). This is not enough to stabilize kinetic energy of the plate, which is of the first order, (at least, uniformly). The same effect is valid for the model (SM2) with rotational inertia (see Remark 3 below). However, in the case of rotational inertia neglected (α = 0) the matter differs. One can compare Remark 3 with the results for models (SM1) and (SM2), in which rotational inertia is not accounted for. We do not require any mechanical damping in these models and compact asymptotic dynamics of the corresponding systems is guaranteed by viscous dissipation in the fluid. Moreover, in these cases we can establish finite-dimensionality of the the corresponding attractors and also their smoothness. See [START_REF] Chueshov | A global attractor for a fluid-plate interaction model accounting only for longitudinal deformations of the plate[END_REF][START_REF] Chueshov | A global attractor for a fluid-plate interaction model[END_REF] for details. Simplified Model 1. We assume that the plate force potential

Φ(u) ∈ C 2 (R 2 )
is a nonnegative polynomially bounded function,

∂Φ(u) ∂u i ∂u j ≤ C (1 + |u| p ) , i, j = 1, 2, u = (u 1 ; u 2 ) ∈ R 2 ,
and the following dissipativity condition holds: for any δ > 0 there exist c 1 (δ) > 0 and c 2 (δ) ≥ 0 such that

∑ i=1,2 u i f i (u) -c 1 (δ)Φ(u) + δ|u| 2 ≥ -c 2 (δ) with f i (u) = ∂Φ(u) ∂u i . (15)
We can consider as examples

Φ(u) = ψ 0 (|u 1 | 2 + |u 2 | 2 ) or Φ(u) = ψ 1 (u 1 ) + ψ 2 (u 2 ),
where ψ i (s) are nonnegative polynomials. We use the following phase space for this model:

H = {v ∈ X : (v, n) = 0 on Ω} × [ H 1 0 (Ω) ] 2 × [L 2 (Ω)] 2 .
Theorem 3 (Well-Posedness). Let U 0 ∈ H. Then for any interval [0, T ] there exists a unique weak solution (v(t); u(t)) to problem (SM1) for which an energy balance equality holds. Moreover, this solution defines a continuous evolution operator S t : H → H by the formula

S t (v 0 ; u 0 ; u 1 ) = (v(t); u(t); u t (t)) ,
where the couple (v(t); u(t)) solves problem (SM1), and there exists a constant a R,T > 0 such that for any couple of initial data possessing the property ∥U ∥ H , ∥ Û ∥ H ≤ R we have

∥S t U -S t Û ∥ 2 H + ∫ t 0 ∥∇(v -v)∥ 2 O dτ ≤ a R,T ∥U -Û ∥ 2 H , ∀ t ∈ [0, T ] ,
where S t U = (v(t); u(t); u t (t)) and S t Û = (v(t); û(t); ût (t)). In the linear case Φ(u) ≡ 0 problem (SM1) generates an exponentially stable C 0 -semigroup of contractions e -tA in H: there exist C, α > 0 such that ∥e -tA U ∥ L(H,H) ≤ Ce -αt for all t > 0 .

We refer to [START_REF] Chueshov | A global attractor for a fluid-plate interaction model accounting only for longitudinal deformations of the plate[END_REF] for the details . We note that property [START_REF] Rosa | Attractors for non-compact semigroups via energy equations[END_REF] improves the result in [START_REF] Grobbelaar-Van Dalsen | Strong stability for a fluid-structure model[END_REF] which states the strong stability only.

The following result shows that model (SM1) demonstrates finite dimensional long-time dynamics.

Theorem 4 (Global Attractor). The dynamical system (H, S t ) generated by (SM1) possesses a compact global attractor A of finite fractal dimension 2 . If relation [START_REF] Lagnese | Boundary Stabilization of Thin Plates[END_REF] holds with c 2 (δ) ≡ 0, then the global attractor A consists of a single point, A = {(0; 0; 0)}, which is exponentially attractive, i.e. there exist c R > 0 and α > 0 such that

∥S t U ∥ H ≤ c R e -αt for any U ∈ H such that ∥U ∥ H ≤ R .
It is well-known (see, e.g., [START_REF] Temam | Infinite-Dimensional Dynamical Dystems in Mechanics and Physics[END_REF]), that to prove the existence of a compact global attractor it is sufficient to show that the system is dissipative and asymptotically smooth. To prove dissipativity we use an appropriate Lyapunov function. As for asymptotic smoothness of the system and finite-dimensionality of the attractor, we rely on recently developed approach based on stabilizability estimates (see [START_REF] Chueshov | Long-Time Behavior of Second Order Evolution Equations with Nonlinear Damping[END_REF][START_REF] Chueshov | Von Karman Evolution Equations[END_REF] and the references therein). We refer to [START_REF] Chueshov | A global attractor for a fluid-plate interaction model accounting only for longitudinal deformations of the plate[END_REF] for further details.

Simplified Model 2. We impose the following hypotheses concerning the nonlinear feedback force F(u) in the plate equation [START_REF] Koch | Hadamard well-posedness of weak solutions in nonlinear dynamic elasticity-full von Karman systems[END_REF].

(F1) F(u) is locally Lipschitz from H 2-ϵ 0 (Ω) into H -1/2 (Ω) for some ϵ > 0. (F2) There exists a C 1 -functional Π(u) on H 2 0 (Ω) such that F(u) = Π ′ (u). (F3) The plate force potential Π is bounded on bounded subsets of H 2 0 (Ω), and there exist η < 1/2 and c ≥ 0 such that

η∥∆w∥ 2 Ω + Π(w) ≥ -c, η∥∆w∥ 2 Ω + (w, F(w)) Ω ≥ -c ∀ w ∈ H 2 0 (Ω) .
We can consider Kirchhoff, von Karman, or Berger nonlinearities as examples of nonlinear feedback (elastic) force F(u), see [START_REF] Chueshov | A global attractor for a fluid-plate interaction model[END_REF] for the details.

We use

H = { (v 0 ; w 0 ; w 1 ) ∈ X × H 2 0 (Ω) × L 2 (Ω) : v 3 0 = w 1 on Ω }
as a phase space and deal with weak (variational) solutions.

Theorem 5 (Well-Posedness). Assume that

U 0 = (v 0 ; w 0 ; w 1 ) ∈ H, G f ∈ V ′ ,
∫ t 0 ||∇v|| 2 O dτ = E(v 0 , w 0 , w 1 ) + ∫ t 0 (G f , v) O dτ
for every t > 0, where the energy functional E is defined by the relation

E(v, w, w t ) = 1 2
( ∥v∥ 2 O + ∥w t ∥ 2 Ω + ∥∆w∥ 2 Ω ) + ∫ Ω Π(w(x))dx -(G pl , w) Ω .
Moreover, there exists a constant a R,T > 0 such that for any couple of weak solutions U (t) = (v(t); w(t); w t (t)) and Û (t) = (v(t); ŵ(t); ŵt (t)) with the initial data possessing the property ∥U 0 ∥ H , ∥ Û0 ∥ H ≤ R we have

∥U (t) -Û (t)∥ 2 H + ∫ t 0 ∥∇(v -v)∥ 2 O dτ ≤ a R,T ∥U 0 -Û0 ∥ 2 H , t ∈ [0, T ] . ( 17
)
In the case G f ≡ 0, G pl ≡ 0, F(u) ≡ 0 the problem generates a strongly continuous exponentially stable contraction semigroup T t on H.

Using the same approach as in the proof of Theorem 4 we can establish the following result (see [START_REF] Chueshov | A global attractor for a fluid-plate interaction model[END_REF] for details).

Theorem 6 (Global Attractor). The dynamical system (S t , H) generated by (SM2) possesses a compact global attractor A. Moreover, (1) A is the unstable set emanating from the set of equilibria N , A = M + (N );

(2) the attractor has finite fractal dimension;

(3) any trajectory γ = {(v(t); w(t); w t (t)) : t ∈ R} from the attractor A possesses the property (v t ; w t ; w tt ) ∈ L ∞ (R; X × H 2 0 (Ω) × L 2 (Ω)), and there is R > 0 such that ( ∥v t ∥ 2 O + ∥w t ∥ 2 2,Ω + ∥w tt ∥ 2 Ω ) ≤ R 2 for all t ∈ R and γ ⊂ A.

We recall (see, e.g., [START_REF] Temam | Infinite-Dimensional Dynamical Dystems in Mechanics and Physics[END_REF]) that the unstable set M + (N ) emanating from some set N ⊂ H is a subset of H such that for each z ∈ M + (N ) there exists a full trajectory {y(t) : t ∈ R} satisfying y(0) = z and dist(y(t), N ) → 0 as t → -∞.

Remark 3. We can consider model (SM2) with the rotational inertia accounted for (i. e., with the additional inertial term -α∆w tt in the plate equation). In this case the phase space is {(v 0 ; w 0 ; w 1 ) ∈ H : w 1 ∈ H 1 0 (Ω)} and the corresponding analog of Theorem 5 remains true, except the property of exponential stability of the linear semigroup T t (the case G f ≡ 0, G pl ≡ 0, F(u) ≡ 0). To prove the existence of the attractor for (SM2) with the rotational inertia as in the case of (GM), we need to assume presence of rotational mechanical damping in the plate equation (see Theorem 2 and the comments after its statement). Whether model (SM2) with rotational inertia term and without mechanical dissipation demonstrates a compact long-time dynamics is still an open question.

  and G pl ∈ H -1/2 (Ω). Then for any interval [0, T ] there exists a unique weak solution (v(t); w(t)) to problem (SM2) with the initial data U 0 .

	The solution
	possesses the property
	U (t) ≡ (v(t); w(t); w t (t)) ∈ C(0, T ; H)
	and satisfies the energy balance equality
	E(v(t), w(t), w t (t)) + ν

We put the multiplier ϱ in the fourth relation of (8) to emphasize that this relation is not needed in the case of negligibly small in-plane inertia (ϱ = 0).

For the definition and basic properties of the fractal dimension see, e.g.,[START_REF] Temam | Infinite-Dimensional Dynamical Dystems in Mechanics and Physics[END_REF].


Andrei Fursikov

On the Normal Semilinear Parabolic Equations Corresponding to 3D Navier-Stokes System

Keywords: Equations of normal type, Navier-Stokes system, structure of dynamical flow

The semilinear normal parabolic equations corresponding to 3D Navier-Stokes system have been derived. The explicit formula for solution of normal parabolic equations with periodic boundary conditions has been obtained. It was shown that phase space of corresponding dynamical system consists of the set of stability (where solutions tends to zero as time t → ∞), the set of explosions (where solutions blow up during finite time) and intermediate set. Exact description of these sets has been given.

Introduction

As well known (see e.g. [START_REF] Ladyzhenskaya | The Mathematical Theory of Viscous Incompressible Flow[END_REF], [START_REF] Temam | Navier-Stokes Equations -Theory and Numerical Analysis[END_REF]), existence of weak solution to 3D Navier-Stokes equations is proved with help of energy estimate, which is true because the image B(v) of nonlinear operator from Navier-Stokes equations consists of vectors tangent to sphere in the L 2 -space with the centrum in origin. If these vectors would be tangent to sphere in Soblev H 1 -space, one could prove existence of strong solution to 3D Navier-Stokes system by the methods similar to ones used to prove existence of a weak solution. But this is not the matter: in this case B(v) = B τ (v)+B n (v) where B τ (v) is the component tangent to sphere in H 1 and B n (v) is normal component. In this paper we change nonlinear operator B(v) of input system on its normal part B n (v). Obtained equations, which we call Normal Parabolic Equations do not satisfy to analog in H 1 of energy estimate "'in the most degree"'. We hope that investigation of these equations can help to understand better the problems connected with solvability of 3d Navier-Stokes system in the class of strong solutions.

In this paper we study Normal Parabolic Equations (NPE ) corresponding to 3D Navier-Stokes system. In Section 2 we derive NPE. In Section 3 we study some properties of NPE. The key property obtained there is existence of explicit formula for solution to NPE. In section 4 the structure of dynamical flow corresponding to NPE is investigated.

Note that NPE has been introduced in [START_REF] Fursikov | On one semilinear parabolic equation of normal type[END_REF] where normal parabolic equation corresponding to Burgers equation was studied. Here we generalize results from [START_REF] Fursikov | On one semilinear parabolic equation of normal type[END_REF] on the case of NPE corresponding to Navier-Stokes system and, besides, continuer to develop the theory of NPE.

Semilinear Parabolic Equations of Normal Type

Our aim is to try to understand better how to investigate 3D Navier-Stokes system in phase space of one time differentiable vector fields where energy estimate is not true. To this end we derive some semilinear parabolic equation.

Navier-Stokes System and Helmholtz Equations

Let consider 3D Navier-Stokes equations with periodic boundary conditions:

∂ t v(t, x) -∆v + (v, ∇)v + ∇p(t, x) = 0, div v = 0, (1)
v(t, . . . , x i , . . .) = v(t, . . . , x i + 2π, . . .), i = 1, 2, 3, (2)
v(t, x)| t=0 = v 0 (x) (3)
where

t ∈ R + , x = (x 1 , x 2 , x 3 ) ∈ R 3 , v(t, x) = (v 1 , v 2 , v 3 ) is the velocity vector field of fluid flow, ∇p is the gradient of pressure, ∆ is Laplace operator, (v, ∇)v = 3 j=1 v j ∂ xj v.
Periodic boundary conditions (2) mean in fact that Navier-Stokes equations (1) and initial conditions (3) are defined on torus T 3 = (R/2πZ) 3 . We transform problem (1)-(3) for velocity to the problem for curl of velocity as unknown function:

ω(t, x) = curl v(t, x) = (∂ x2 v 3 -∂ x3 v 2 , ∂ x3 v 1 -∂ x1 v 3 , ∂ x1 v 2 -∂ x2 v 1 ) (4)
Recall the following well-known formulas of vectorial analysis:

(v, ∇)v = ω × v + ∇ |v| 2 2 , (5) curl
(ω × v) = (v, ∇)ω -(ω, ∇)v, if div v = 0, div ω = 0 (6)
where

ω × v = (ω 2 v 3 -ω 3 v 2 , ω 3 v 1 -ω 1 v 3 , ω 1 v 2 -ω 2 v 3 ) is vector product and |v| 2 = v 2 1 + v 2 2 + v 2
3 . Let substitute (5) into the first equality of (1) and apply to both parts of obtained equality operator curl. Then in virtue of (4),(6), and formula curl∇F = 0 we obtain the Helmholtz equations

∂ t ω(t, x) -∆ω + (v, ∇)ω -(ω, ∇)v = 0 (7)
We add these equations with initial conditions

ω(t, x)| t=0 = ω 0 (x) (8)
where ω 0 = curl v 0 .

Normal Parabolic Equations (NPE) and Their Derivation

For each m ∈ Z + = {j ∈ Z : j ≥ 0} we define the space

V m = V m (T 3 ) = {v(x) ∈ (H m (T 3 )) 3 : divv = 0, T 3 v(x)dx = 0} (9)
where H m (T 3 ) is Sobolev space. Multiplying Navier-Stokes system (1) on v scalarly in L 2 (T 3 ) we obtain after integration by parts (on x) and integration on t well-known energy estimate

T 3 |v(t, x)| 2 dx + 2 t 0 T 3 |∇ x v(τ, x)| 2 dxdτ ≤ T 3 |v 0 (x)| 2 dx (10)
that gives opportunity to prove existence of weak solutions to problem (1)-( 3).

Unfortunately, this solution is not smooth enough to establish its uniqueness. If in a hope to get existence of smooth solution to (1) we would try to get analog of energy estimate in phase space V 1 , multiplying (1) on v scalarly in V 1 (T 3 ) we will not get analog of bound (10). Let try to understand situation passing from Navier-Stokes to Helmholtz equations. Using decomposition in Fourier series

v(x) = k∈Z 3 v(k)e ix•k , where v(k) = (2π) -3 T 3 v(x)e -ix•k dx, x • k = 3 j=1 x j k j , k = (k 1 , k 2 , k 3 )
, and well-known formula curl curl v = -∆v if div v = 0, we see that on space V m inverse operator to curl is well-defined and is determined by the formula

curl -1 ω(x) = i k∈Z 3 k × ω(k) |k| 2 e ix•k (11)
That is why operator

curl : V 1 -→ V 0
realized isomorphism of the spaces. Therefore sphere in V 1 for problem (1), ( 3) is equivalent to sphere in V 0 for problem (7), (8). Let denote nonlinear term of Helmholtz equation by B:

B(ω) = (v, ∇)ω -(ω, ∇)v (12)
(we did not indicate dependence B on v because it can be expressed via ω by (11)).

Multiplying equality (12) on ω = (ω 1 , ω 2 , ω 3 ) scalarly in V 0 and integrating by parts we get

(B(ω), ω) V 0 = - T 3 3 j,k=1 ω j ∂ j v k ω k dx (13)
that, generally saying, is not equal to zero. Just because of this 3D Helmholtz equations do not possess energy estimate. In other words, operator B admits the decomposition

B(ω) = B n (ω) + B τ (ω) (14)
where vector B n (ω) is orthogonal to the sphere Let construct now normal parabolic equations with respect to sphere in V 0 , corresponding to problem (7), (8).

Σ ω = {u ∈ V 0 : u V 0 = ω V 0 }
Since summand (v, ∇)ω from ( 7) is tangential operator:

T 3 (v, ∇)ω • ωdx = 0,
normal part of nonlinear operator from ( 7) is defined by nonlinear term (ω, ∇)v.

We look it for in the form Φ(ω)ω where Φ is unknown functional that is found by equation

T 3 Φ(ω)ω(x) • ω(x)dx = T 3 (ω(x), ∇)v(x) • ω(x)dx (15)
Relation (15) implies desired formula for Φ:

Φ(ω) = T 3 (ω(x), ∇)curl -1 ω(x) • ω(x)dx T 3 |ω(x)| 2 dx, ω = 0, 0, ω ≡ 0 ( 16
)
where curl -1 ω(x) is defined in (11). So, normal parabolic equations corresponding to system (7) are defined as follows:

∂ t ω(t, x) -∆ω -Φ(ω)ω = 0, div ω = 0 (17)
where functional Φ is defined in (16). These equations supplied with initial conditions (8) and periodic boundary conditions are the main object of our investigation in this paper.

3 Properties of Normal Parabolic Equations

Explicit Formula for Solution of NPE

In this subsection we derive explicit formula for NPE solution. This is the key result because it gives the possibility to establish many important properties on NPE. Some of them will be obtained below in next sections. The following assertion is true:

Lemma 1. Let S(t, x, y 0 ) be resolving operator of the following Stokes system with periodic boundary conditions:

∂ t y(t, x) -∆y(t, x) = 0, div y = 0, y(t, x)| t=0 = y 0 (x), (18)
i.e. S(t, x, y 0 ) = y(t, x) (we assume, of course, that div y 0 = 0). The solution of problem ( 17),( 8) has the form

ω(t, x; ω 0 ) = S(t, x; ω 0 ) 1 - t 0 Φ(S(τ, x; ω 0 ))dτ (19)
The proof of this lemma is reduced to substitution (19) into (17) and direct checking of obtained equality.

Properties of Functional Φ

Let s ∈ R. Recall that Sobolev space H s (T 3 ) is the space of periodic real-valued distributions z(x) possessing with the finite norm

z 2 H s (T 3 ) ≡ z 2 s = k∈Z 3 \{0} |k| 2s | z(k)| 2 < ∞ (20)
where z(k) are Fourier coefficients of z.

We will use the following generalization of spaces (9) of solenoidal vector fields:

V s ≡ V s (T 3 ) = {v(x) ∈ (H s (T 3 )) 3 : divv(x) = 0, T 3 v(x)dx = 0}, s ∈ R (21)
Lemma 2. Let Φ(u) be functional (16). There exists a constant c > 0 such that for each u ∈

V 3/2 |Φ(u)| ≤ c u 3/2 ( 22
)
This lemma is proved similarly to analogous bound from [START_REF] Fursikov | On one semilinear parabolic equation of normal type[END_REF].

Lemma 3. Let Φ be functional (16). For each β < 1/2 there exists a constant c 1 > 0 such that for each

y 0 ∈ V -β (T 3 ), t > 0 t 0 Φ(S(τ, •, y 0 ))dτ ≤ c 1 y 0 -β (23)
where S(t, •, y 0 ) is resolving operator of problem (18).

Proof. Using (22) we get

t 0 Φ(S(τ, •, y 0 ))dτ ≤ c t 0 e -τ /2   k =0 (| y 0 (k)| 2 |k| -2β )|k| 3+2β e -(k 2 -1)τ   1/2 dτ ( 24
)
where y 0 (k) are Fourier coefficients of y 0 . Solution ρ = ρ(t) of extremal problem

f (t, ρ) = ρ 3+2β e -(ρ 2 -1)t → max, ρ ≥ 1
is defined with expression ρ(t) = 3+2β 2t , and

f (t, ρ(t)) =    3+2β 2t 3+2β 2 e -(3+2β-2t)/2 , t ≤ 3+2β 2 , 1, t ≥ 3+2β 2 (25)
Substitution ( 25) into ( 24) implies (23).

Remark 1. Lemma 3 implies that the functional from left side of bound ( 23) is well defined for y 0 ∈ V -β (T 3 ) with β < 1/2. In particular, in virtue of this Lemma and ( 19) solution of problem ( 17),( 8) is well defined for each initial condition ω 0 ∈ V 0 , and therefore our choice V 0 as phase space for corresponding dynamical system is correct. Note also that simple modification of Lemma 3 proof gives continuity of the functional from left side in (23) with respect to

y 0 ∈ V -β , β < 1/2.

The Structure of NPE Dynamics

The aim of this section is to find out the main feature of dynamical flow corresponding to NPE. We decompose the phase space of the dynamical system on three sets with different behavior of dynamical flow inside each of them.

Distinctive Sets of Phase Space

Let give definitions of three subsets of phase space for NPE. Recall that we take V 0 (T 3 ) ≡ V 0 as the phase space for problem (17),(8).

Definition 1. The set M -≡ M -(α) ⊂ V 0 of initial conditions ω 0 such that the solution ω(t, x; ω 0 ) of problem ( 17),(8) exists and satisfies inequality

ω(t, •; ω 0 ) 0 ≤ α ω 0 0 e -t ∀t > 0 (26)
is called the set of stability. Here α > 1 is a certain fixed number.

The following simple sufficient condition for belonging to

M -(α) is true in virtue of (19): If ω 0 ∈ V 0 satisfies the bound sup t∈R+ t 0 Φ(S(τ, •; ω 0 ))dτ ≤ α -1 α (27) then ω 0 ∈ M -(α).
Definition 2. The set M + ⊂ V 0 of initial conditions ω 0 from ( 17),( 8) such that corresponding solution ω(t, x; ω 0 ) exists only on a finite interval t ∈ (0, t 0 ) with t 0 > 0 depending on ω 0 , and blows up at t = t 0 is called the set of explosions.

In virtue of formula (19) for solution ω(t, x; ω 0 )

M + = {ω 0 ∈ V 0 : ∃t 0 > 0 t0 0 Φ(S(τ, •; ω 0 ))dτ = 1} (28)
The minimal magnitude from the set {t 0 } for which equality in (28) holds is called the time of explosion.

Definition 3. The collection

M (α) = V 0 \ {M -(α) ∪ M + } ( 29
)
is called intermediate set.

Remark 2. Definitions of stability and intermediate sets include parameter α > 1 and from this point of view they are not absolute. Nevertheless they are convenient for using.

We study below the properties of these sets and, in particular, we show that all these sets are nonempty. We begin from the set of stability. This set is the most important for us.

Subsets Belonging to the Set of Stability

Let ρ > 0, β < 1/2. Introduce the set

El β ρ = {v ∈ V 0 (T 3 ) : v 2 -β = k∈Z 3 \{0} | v(k)| 2 |k| 2β ≤ ρ 2 } = {v ∈ V 0 (T 3 ) : ∞ k∈Z 3 \{0} | v(k)| 2 ρ 2 |k| 2β ≤ 1}, (30)
which we can interpret as ellipsoid in V 0 (T 3 ) with length of axes directed along functions e ik•x , e -ik•x equal to ρ|k| β . Since

ρ|k| β → ∞ as |k| → ∞, this ellipsoid is unbounded in V 0 . Lemma 4. Let c 1 ρ < 1 and ρ ≤ (α -1)/(αc 1 ) where c 1 is the constant from (23). Then El β ρ ⊂ M -(α) ( 31
)
where the set El β ρ is defined in (30).

and

A -= ∩ t≥0 A -(t), A + = ∩ t≥0 A + (t), A 0 = ∩ t≥0 A 0 (t) (37)
All these sets are closed and nonempty. For instance, A 0 = ∅ in virtue of Lemma 5. Sets A ± (t), A ± possess nonempty interior in topology of Σ, i.e. in the topology induced on Σ by topology of the space V 0 . This assertion follows from continuity of the functional V -β ∋ v → t 0 Φ(S(τ, •, v))dτ with β < 1/2 and in particular for β = 0 (see Remark 1). Evidently, A 0 = A -∩ A + .

Linearity on v of operator S(t, x, v) and oddness of Φ(v) with respect to v imply Lemma 6.

v ∈ A -if and only ifv ∈ A + Introduce also the sets The proof of this theorem will be given in some other place. Theorem 1 implies that M + = ∅ and M I (α) = ∅.

B + = Σ \ A -≡ {v ∈ Σ : ∃t 0 > 0 t0 0 Φ(S(τ, •, v))dτ > 0}, B -= Σ \ A + (
B + ∋ v → Γ ρ (v) = ρ b(v) v ∈ V 0 (

0 Φ

 0 38)It is easy to see that the set B + is open in topology of Σ. Moreover, the boundary ∂B + of set B + is defined by the relation∂B + = {v ∈ Σ : ∀t > 0 t 0 Φ(S(τ, •, v))dτ ≤ 0, ∃t 0 > 0 : t0 0 Φ(S(τ, •, v))dτ = 0} It is clear that A 0 ⊂ ∂B + and ∂B + \ A 0 = ∅.4.4 On Structure of Phase Space V 0Let us introduce the following function defined on the set B + of sphere Σ:B + ∋ v → b(v) = max t≥0 t (S(τ, v))dτ(39)Evidently, b(v) > 0 and b(v) → 0 as v → ∂B + . Let ρ ∈ (0, 1]. We define the following map Γ ρ (v) that plays the key role in description of structure of phase flow generated by boundary value problem (17),(8):

Theorem 1 .

 1 40) where b(v) is function(39). Note that Γ ρ (v) 0 → ∞ as v → ∂B + . Let α > 1 be a parameter from definition of the stability set M -(α) and ρ = (α -1)/α. Then the image Γ ρ (v), v ∈ B + of the map Γ ρ divides the space V 0 on two separate parts. The part containing origin coincides with the set of stability M -(α). The part of V 0 between Γ ρ (v), v ∈ B + and Γ 1 (v), v ∈ B + coincides with intermediate space M I (α), and the rest part of V 0 coincides with the set of explosions M + .

  at the point ω, and vector B τ (ω) is tangential to Σ ω at ω. Generally saying, both operators B n , B τ in (14) are not equal to zero. Note that just the component B n = 0 prevents to derivation of energy bound and therefore it is quite possible that the main difficulties obstructing to investigation of Navier-Stokes equations are connected just with this operator. That is why there is reason to omit in Helmholtz equations the component B τ and to study on the first stage analog of equations (7) in which nonlinear operator B(ω) is changed on its normal component B n (ω). Such equations we call Normal Parabolic Equations (NPE).

Proof. Note that solution S(t, x, ω 0 ) of problem (18) with y 0 = ω 0 satisfies inequality:

) and inequalities (32), ( 23) imply (26) if α ≥ 1/(1c 1 ρ). But the last inequality is equivalent to ρ ≤ (α -1)/(αc 1 ). This Lemma is analog of local existence theorem for 3D Navier-Stokes equations obtained in [START_REF] Fursikov | Local Existence Theorems with Unbounded Set of Input Data and Unboundedness of Stable Invariant Manifolds for 3D Navier-Stokes Equations[END_REF] with help of Fujita-Kato approach [START_REF] Fujita | On the Navier-Stokes initial value problem[END_REF] and of local existence theorem for NPE connected with Burgers equation (see [START_REF] Fursikov | On one semilinear parabolic equation of normal type[END_REF]). The proof of Lemma 4 is essentially easier then proofs of aforementioned results because here we use explicit formula (19) for solutions.

We show in this subsection that, actually, M -(α) is essentially wider than El β ρ . For this goal we consider one infinite-dimensional subspace and show that it belongs to M -(α)

Let introduce the following subset U L of Z 3 \ {0}:

An example of the subset belonging to U L is the following set:

The subspace

The proof of this Lemma is similar to analogous Lemma from [START_REF] Fursikov | On one semilinear parabolic equation of normal type[END_REF]. Lemmas 4, 5 imply that M -(α) = ∅.

Certain Sets of Unit Sphere of V 0

Let denote the unit sphere of the phase space V 0 as follows:


Michael Hinze
email: michael.hinze@uni-hamburg.de

Christian Kahle
email: christian.kahle@uni-hamburg.de

A Nonlinear Model Predictive Concept for Control of Two-Phase Flows Governed by the Cahn-Hilliard Navier-Stokes System

Keywords: Flow control, Navier-Stokes, Cahn-Hilliard, Model Predictive Control, Instantaneous Control, Adaptive Finite Elements

. In the present work we provide numerical investigations which indicate that distributed instantaneous control also is well suited to stabilize the Cahn-Hilliard Navier-Stokes system.

Introduction

The aim of this work is the development of numerical methods for closed-loop control of two-phase flows governed by the Cahn-Hilliard Navier-Stokes system. The approach is based on an inexact variant of model predictive control called instantaneous control. Instantaneous control in the context of flow control is proposed in e.g. [START_REF] Choi | Suboptimal control of turbulent flow using control theory[END_REF][START_REF] Choi | Instantaneous control of backward-facing step flows[END_REF][START_REF] Hinze | Three control methods for time -dependent Fluid Flow[END_REF], and for distributed control of the Navier-Stokes system is analyzed in [START_REF] Hinze | Instantaneous closed loop control of the Navier-Stokes system[END_REF], where among other things it is shown that the method is able to exponentially stabilize given solution states supposing certain smallness assumptions on the initial conditions. For an overview in the field of nonlinear model predicitve control we refer to [START_REF] Nevistic | Finite Receding Horizon Control: A General Framework for Stability and Performance Analysis[END_REF][START_REF] Qin | A survey of industrial model predictive control technology[END_REF] and also to the monograph [START_REF] Grüne | Nonlinear Model Predictive Control[END_REF], where also further references can be found.

The outline of this paper is as follows. In section 2 we describe the concept of nonlinear model predictive control as it is used in the present work, and also introduce instantaneous control. In section 3 we present a brief introduction to the Cahn-Hilliard Navier-Stokes system, including its numerical treatment. In section 4 we discribe the instantaneous control strategy for the Cahn-Hilliard Navier-Stokes system and demonstrate its performance at morphing a circle into a square in section 5. We end with some conclusions formulated in section 6

Nonlinear Model Predictive Control

The aim of model predictive control consist in steering or keeping the state of a dynamical system to or at a given desired trajectory. To fix the concept, we are going to apply it to the Cahn-Hilliard Navier-Stokes system. Let us first consider an abstract dynamical system with initial condition x 0 , state x(t), observation y(t)a n dc o n t r o lu(t):

ẋ(t)+Ax(t)=b(x, t)+Bu(t), (t>0) state, y(t)=Cx(t)o b s e r v a t i o n , x(0) = x 0 initial condition. (1)
Our aim consists in constructing a nonlinear feedback control law K with Bu(t)=K(x(t)) which steers the dynamical system to the desired trajectory x(t), i.e. x(t) ! → x(t)( t →∞ ). To simplify notations we from here onwards use B = id and C = id, i.e. we do not distinguish between state and observation and we allow fully distributed controls.

To prepare for model predicitve control, system (1) is discretized in time using the semi-implicit Euler method on an equidistant time grid 0 = t 0 ≤ t 1 ≤ ... with t k+1 -t k = τ for k =0, 1,....H e r ex k denotes the state at time t k and b k denotes the nonlinearity b(x k ,t k ). We obtain the time discrete model

(I + τA)x k+1 = x k + τb k + u k+1 ,k =0, 1,.... (2)
For L ∈ N and x j given, we consider the optimal control problem min J(x j+1 ,...,x j+L ,u j+1 ,...,u j+L ) s.t. (2) for j = k,...,k+ L -1, (P k ) where J(x j+1 ,...,x j+L ,u j+1 ,...,u j+L ):

= L i=1 1 2 x k+i -xk+i 2 + α 2 u k+i 2
Let us note that for L =1problem(P k ) admits a unique solution. However, in the case L>1 the solution might not be unique due to the nonlinear character of the transition constraints [START_REF] Blowey | The Cahn-Hilliard gradient theory for phase separation with non-smooth free energy. Part I: Mathematical analysis[END_REF]. In this case we assume that (P k )a d m i t sa solution.

Now we define the abstract model predictive control strategy using a computing oracle called RECIP E.

Model Predicitve Control:

1. Initialization: Specify time grid (t j ) and discrete state x,s e tk =0a n d specify L 0 > 0. 2. Given u k ,x k ,s e t u k+1 = RECIP E(u k ,x k ,L k ).

3. Solve (2) with u k+1 , i.e. compute x k+1 according to (I + τA)

x k+1 = x k + τb(x k ,t k )+u k+1 . 4. Set k = k + 1, goto 2.
In the classical model predictive control context [START_REF] Nevistic | Finite Receding Horizon Control: A General Framework for Stability and Performance Analysis[END_REF][START_REF] Qin | A survey of industrial model predictive control technology[END_REF] the oracle RECIP E(u, x, L), for given u, x, L, solves the optimal control problem (P k ) with x k ≡ x, u k ≡ u and L the length of the control horizon. From the solution (x k+1 ,...,x k+L ,u k+1 ,...,u k+L ) only x k+1 and u k+1 are actually used to steer the discrete system to the next time instance t k+1 .

In practical applications often quick response to system changes is necessary. In such cases it may be too time consuming to solve problem (P k )e x a c t l y .I nstead, an approximate solution could be used. This leads to the so called concept of instantaneous control [START_REF] Choi | Instantaneous control of backward-facing step flows[END_REF][START_REF] Hinze | Instantaneous closed loop control of the Navier-Stokes system[END_REF], whose oracle is described next.

Instantaneous Control: Given L, x, v,thenu = RECIP E(v, x, L)iffu = u k+1
,where(x k+1 ,...,x k+L ,u k+1 ,...,u k+L ) is the result of a steepest descent step applied to the solution of (P k ) with x k ≡ x and u k ≡ v.

In the case L = 1, instantaneous control realizes the steps:

-Solve (I + τA)z = x + τb(x, t k )+v, -solve (I + τA * )λ = -(x -z), -set d = αv -λ, -determine ρ>0 (step size for steepest descent), -set RECIP E = v -ρd,
where we have used the adjoint calculus to expose the derivations of the functional J, see e.g. [START_REF] Hinze | Optimization with PDE constraints[END_REF].

Instantaneous control with L = 1 is analytically investigated in [START_REF] Hinze | Instantaneous control for the Burgers equation: Convergence analysis and numerical implementation[END_REF] for the control of Burger's equation and in [START_REF] Hinze | Instantaneous closed loop control of the Navier-Stokes system[END_REF] for control of the two-dimensional Navier-Stokes system. Among other things it is shown in [START_REF] Hinze | Instantaneous closed loop control of the Navier-Stokes system[END_REF]Thm. 4.4,4.5] that

x k -xk H 1 (Ω) ≤ cκ k for some κ ∈ (0, 1),
and also that instantaneous control may be regarded as the discrete realization of a nonlinear feedback operator K which is able to steer x(t) exponentially fast to the desired trajectory x(t), i.e. with u(t)=K(x(t)) in (1) there holds

x(t) -x(t) H 1 (Ω) ≤ c exp - ρ τ t ,
where ρ denotes the step size in the steepest descent algorithm and τ is the time step in the discretization, see [START_REF] Hinze | Instantaneous closed loop control of the Navier-Stokes system[END_REF]Thm. 4.1,4.2] for the details.

The Cahn-Hilliard Navier-Stokes System

The Cahn-Hilliard Navier-Stokes equations are a diffuse interface model for describing two-phase flows. In comparison to sharp interface models (see e.g. [START_REF] Gross | Numerical ethods for two-phase incompressible flows[END_REF])

which model the interface between the two components as a sharp line, diffuse interface models allow a partial mixing of the components yielding a small diffuse interface. A derivation of the model used here can be found e.g. in [START_REF] Abels | Diffuse interface models for two-phase flows of viscous incompressible fluids[END_REF]. It is related to the model 'H' for two-phase flows in the classification of Hohenberg and Halperin [START_REF] Hohenberg | Theory of dynamic critical phenomena[END_REF]. We note that there are also models for flows with three components, see e.g [START_REF] Boyer | Cahn-Hilliard/Navier-Stokes model for the simulation of three-phase flows[END_REF].

For Ω ⊂ R n , (n =2 , 3) and T>0 we here consider the following weak form of the Cahn-Hilliard Navier-Stokes system with double-obstacle free energy according to [START_REF] Blowey | The Cahn-Hilliard gradient theory for phase separation with non-smooth free energy. Part I: Mathematical analysis[END_REF].

Find (c(t),w(t),y(t),p(t)) in K×H 1 (Ω) × H 1 0 (Ω) × L 2 (0) (Ω) such that (∂ t y, v)+ 1 Re (∇y : ∇v) +((y •∇)y, v) -(p, div v)+(c∇w, v)=0 ∀v ∈ H 1 0 (Ω),a . e .t∈ (0,T], (3) (-div y, v)=0 ∀v ∈ L 2 (0) (Ω),a . e .t∈ (0,T], (4) c(t) ∈K a.e. t ∈ (0,T], (5)
(∂ t c, v)+ 1 Pe (∇w, ∇v)-(cy, ∇v)=0 ∀v ∈ H 1 (Ω),a . e .t∈ (0,T], (6)
γ 2 (∇c, ∇(v -c))-(w + c, v -c) ≥ 0 ∀v ∈K,a . e .t∈ (0,T], (7)
c(x, 0) = c 0 (x) ∀x ∈ Ω, (8)
∂ ν c =0,∂ ν w =0 on∂Ω × (0,T], (9) y
(x, 0) = y 0 (x) ∀x ∈ Ω, (10)
y =0 on∂Ω × (0,T]. (11)
Here, for v, w ∈ H 1 (Ω)

∈ L 2 (0) (Ω)={v ∈ L 2 (Ω) | (v, 1) = 0}
denotes the mean free pressure of the fluid. The flow profile at t = 0 is given by y 0 .T h eP éclet number Pe and the Reynolds number Re are given physical constants. The parameter γ is related to the width of the diffuse interface region which is of size O(γ 2 ) [START_REF] Cahn | Free energy of a nonuniform system-I: Interfacial free energy[END_REF] .

For an analytical treatment of the above system we refer to [1, Chapter 6.5]. Especially in two space dimensions there exists a unique solution (c, w, y, p)t o this system and we have

y ∈ C 0 ([0,T], H 1 0 (Ω)),c ∈ BC ω ([0,T],H 1 (Ω)), ∇w ∈ L 2 (0,T; L 2 (Ω)).
Here, BC ω ([0,T],H 1 (Ω)) is the space of bounded and weakly continuous functions from [0,T] with values in H 1 (Ω).

3.1 Time discretization and treatment of the variational inequality in [START_REF] Eyre | Unconditionally gradient stable time marching the Cahn-Hilliard equation[END_REF] For the discretization of ( 3)-( 11) we use the semi-implicit approach proposed in e.g. [START_REF] Eyre | Unconditionally gradient stable time marching the Cahn-Hilliard equation[END_REF][START_REF] Kay | Finite element approximation of a Cahn-Hilliard-Navier-Stokes system[END_REF] with constant step size τ>0. The variational inequality in [START_REF] Eyre | Unconditionally gradient stable time marching the Cahn-Hilliard equation[END_REF] according to [START_REF] Hintermüller | An adaptive finite element Moreau-Yosida-based solver for a coupled Cahn-Hilliard/Navier-Stokes system[END_REF] is relaxed using Moreau-Yosida regularization. At time instance t this results in finding (y, p, c, w)

∈ H 1 (Ω) n × L 2 (0) (Ω) × H 1 (Ω) × H 1 (Ω)s u c h that (y -y old ,v)+ τ Re (∇y : ∇v)+τ ((y old •∇)y old ,v) +τ (c old ∇w old ,v) -τ (p, div v)=0 ∀v ∈ H 1 0 (Ω), (12)
(-div y, v)=0 ∀v ∈ L 2 (0) (Ω), ( 13
) (c, v)+ τ Pe (∇w, ∇v) -τ (c old y, ∇v) -(c old ,v)=0 ∀v ∈ H 1 (Ω), (14)
γ 2 (∇c, ∇v) -(w, v)+(λ s (c),v) -(c old ,v)=0 ∀v ∈ H 1 (Ω). (15)
is satisfied. Here the subscript old refers to the value of the respective function at time t old = t -τ and λ s (c)=λ + s (c)+λ - s (c)=s(max(0,c-1) + min(0,c+1)) stems from Moreau-Yosida regularization of [START_REF] Eyre | Unconditionally gradient stable time marching the Cahn-Hilliard equation[END_REF], see e.g. [START_REF] Hintermüller | An adaptive finite element Moreau-Yosida-based solver for a coupled Cahn-Hilliard/Navier-Stokes system[END_REF][START_REF] Hintermüller | An adaptive finite element Moreau-Yosida-based solver for a non-smooth Cahn-Hilliard problem[END_REF] for details. Let us emphasize that ( 12)-( 15) is decoupled in the sense that using y old ,c old and w old , the flow y and the pressure p at time t can be computed from ( 12)-( 13), and then using this flow vector y, the concentration c and the chemical potential w are obtained from ( 14)- [START_REF] Hinze | Optimization with PDE constraints[END_REF]. Furthermore, normalizing p by (p, 1) = 0, it can be shown that ( 12)-( 15) admits a unique solution (y, p, c, w), compare [11, Thm 4.1]. The system ( 14)-( 15) is nonlinear and can be treated by semi-smooth Newton methods, see [START_REF] Hintermüller | An adaptive finite element Moreau-Yosida-based solver for a non-smooth Cahn-Hilliard problem[END_REF].

Spatial Discretization

The spatial discretization is performed by linear finite elements for both the concentration and the chemical potential to obtain finite element approximations c h ,w h . For the flowfield and the pressure we use the LBB-stable Taylor-Hood P 2 -P 1 finite elements, see e.g. [START_REF] Hood | Navier-Stokes equations using mixed interpolation[END_REF][START_REF] Verfürth | A posteriori error analysis of space-time finite element discretizations of the time-dependent Stokes equations[END_REF], to obtain finite element approximations y h ,p h . For the spatial treatment of the Cahn-Hilliard part ( 14)-( 15) we use the adaptive approach presented in [START_REF] Hintermüller | An adaptive finite element Moreau-Yosida-based solver for a coupled Cahn-Hilliard/Navier-Stokes system[END_REF][START_REF] Hintermüller | An adaptive finite element Moreau-Yosida-based solver for a non-smooth Cahn-Hilliard problem[END_REF]. We emphasize that we use different spatial meshes for the Cahn-Hilliard and the Navier-Stokes part.

Instantaneous Control of the Cahn-Hilliard Navier-Stokes System

The aim of this section is to develop a simple distributed closed-loop control strategy for the Cahn-Hilliard Navier-Stokes system. It uses instantaneous control with L = 1 on a control horizon which coincides with the time interval of one time step. We consider the idealized situation that the flow can be controlled by a vector field which is distributed over the whole spatial domain and that the concentration c can be measured in the whole spatial domain. The control goal consists in steering the concentration c towards a prescribed concentration trajectory c d by applying volume forces to the flow field. Now let t 0 =0a n dk ∈ N. At time instance t = t k ,f o rα>0, we consider the minimization problem

min J k (u):= 1 2 c -c d 2 L 2 (Ω) + α 2 u 2 L 2 (Ω) (P k ) s.t. (y -y old ,v)+ τ Re (∇y : ∇v)+τ ((y old •∇)y old ,v) +τ (c old ∇w old ,v) -τ (p, div v)=u ∀v ∈ H 1 0 (Ω), (16)
(-div y, v)=0 ∀v ∈ L 2 (0) (Ω), ( 17
) (c, v)+ τ Pe (∇w, ∇v) -τ (c old y, ∇v) -(c old ,v)=0 ∀v ∈ H 1 (Ω), (18)
γ 2 (∇c, ∇v) -(w, v)+(λ s (c old ),v) -(c old ,v)=0 ∀v ∈ H 1 (Ω). (19)
where now old refers to the time instance t k-1 .

It is not hard to show that problem (P k ) admits a unique solution u ∈ L 2 (Ω) n , which together with (y, p, c, w) satisfies the adjoint system

(p 1 ,v)+γ 2 (∇p 2 , ∇v)=(c -c d ,v) ∀v ∈ H 1 (Ω), (20)
(p 2 ,v)= τ Pe (∇p 1 , ∇v) ∀v ∈ H 1 (Ω), (21)
(p 3 ,v)+ τ Re (∇p 3 , ∇v) -τ (p 4 , div v)=τ (p 1 ∇c old ,v) ∀v ∈ (H 1 0 (Ω)) n , (22) (div p 3 ,v)=0 ∀v ∈ L 2 (0) (Ω), (23)
αu + p 3 =0. (24)
i.e. there exists a uniquely determined adjoint (p 20)-( 24). The gradient of J k is given by ∇J k (v)= αv + p 3 . Its evaluation for a given v ∈ L 2 (Ω) n amounts to first solving ( 16)-( 19) for (y, p, c, w) and then solving (20)-( 23) for (p 1 ,p 2 ,p 3 ,p 4 ).

1 ,p 2 ,p 3 ,p 4 ) ∈ H 1 (Ω)×H 1 (Ω)× H 1 (Ω) n ×L 2 (0) (Ω) which solves (
Let us note that ( 16)-( 19) differ from ( 12)-( 15) in the explicit treatment of the nonlinearity λ s , which in ( 19) is frozen at t old . We emphasize that ( 16)- [START_REF] Kay | Finite element approximation of a Cahn-Hilliard-Navier-Stokes system[END_REF] is not used to simulate the controlled Cahn-Hilliard Navier-Stokes system, but to construct a feedback operator K such that u = K(y, p, c, w). With this feedback operator available, system (3)- [START_REF] Hintermüller | An adaptive finite element Moreau-Yosida-based solver for a non-smooth Cahn-Hilliard problem[END_REF] then is controlled through inserting u = K(y, p, c, w)) as right hand in (3). The resulting system then is treated on the time discrete level as in ( 12)- [START_REF] Hinze | Optimization with PDE constraints[END_REF], where K is evaluated at t old , i.e. u = K(y old ,p old ,c old ,w old ).

Next we describe the construction of K from ( 16)-(24). For this pupose let us denote by B the solution operator associated to the quasi-Stokes problem ( 12)- [START_REF] Hinze | A variational discretization concept in control constrained optimization: the linear quadratic case[END_REF] and by C the linear, fourth order solution operator of the Cahn-Hilliard system ( 18)- [START_REF] Kay | Finite element approximation of a Cahn-Hilliard-Navier-Stokes system[END_REF]. Then the system to obtain and is inserted in [START_REF] Hinze | Instantaneous closed loop control of the Navier-Stokes system[END_REF]. Note that this system does not depend on α since u k 0 =0. The spatially discrete treatment of ( 12)-( 15) with this control is similar as in the uncontrolled case. We note that (24) motivates to use Taylor-Hood finite elements for the discretization of the control, see the concept of variational discretization proposed in [START_REF] Hinze | A variational discretization concept in control constrained optimization: the linear quadratic case[END_REF].

Numerical Results -Circle2Square

To demonstrate the effectiveness of our control method we now morph a circle into a square with the following setup. As domain we use Ω := (0, 1) 2 ,t h e initial concentration c 0 is chosen as 1 in

B 1 4 ( 1 2 , 1 2 )a n da s-1i nΩ \ B 1 4 ( 1 2 , 1
2 ), see Fig. 1 (left). Control is applied to the flow field with control gain of steering c to the desired state c d with values 1 in the square centered at ( 7 20 , 7 20 ), see Fig. 1 (right), with edge length such that (c 0 , 1) = (c d , 1). This requirement is meaningful, since our time discretization scheme is mass-conserving. We choose Re = 10, Pe = 100, γ =1 /(40π), α =1 e -4, τ =0 .01 and use ρ =1a ss t e p size in the steepest descent method.

Fig. 2 presents snapshots of the concentration after 40 time steps (left) and after 500 time steps (right, where also the controlled flow is depicted). Clearly, the corners formed in c d cannot be reached by the controlled concentration, since the phasefield approach used here always delivers a smooth diffuse interface in the present situation. Fig. 3 presents the evaluation of d(t):= c(t) -c d (t) L 2 (Ω) for various Reynolds numbers ranging in the interval [START_REF] Hintermüller | An adaptive finite element Moreau-Yosida-based solver for a coupled Cahn-Hilliard/Navier-Stokes system[END_REF]200]. We observe a clear decrease in d(t), till a certain value around 1e -1 is reached. The method is not able to further reduce d(t) due to the unreachability of c d . The oscillations of d for larger Reynolds numbers can be explained by the indirect control method (flow is controlled, concentration should be steerd to c d ), which becomes more sensible with increasing Reynolds number.

6C o n c l u s i o n

We have presented a general inexact model predictive control concept called instantaneous control and sketched its interpretation as closed loop controller.

For the Cahn-Hilliard Navier-Stokes system we have derived a nonlinear feedback u = K(y, p, c, w) which realizes instantaneous control on the continuous level. With morphing the circle into a square we have numerically demonstrated the scope and effectiveness of our approach in control of two-phase flows.

  ij (∇w) ij dx, and c 0 ∈K:= {v ∈ H 1 (Ω) ||v|≤1 a.e. in Ω}),y 0 ∈ H 1 0 (Ω). The function c is called order parameter and satisfies c = c(t, x) ∈ [-1, 1], with c ≡ 1 on the pure A-phase and c ≡-1 on the pure B-phase region, respectively, where A and B are the two components of the fluid. Initially, i.e. for t = 0, we assume that the concentration equals c 0 .Thequan tit yw represents the chemical potential, y denotes the mean flow velocity field, i.e. y = 1+c 2 y A + 1-c 2 y B ,w h erey A and y B denote the fluid velocities in the fluid phases A and B, respectively, and p

  p 3 c a nb ew r i t t e ni nt h ef o r m y = B (y old -τ (y old ∇y old )) -τc old ∇w old + u) , c = C c old -τy∇c old + τ Pe ∆(λ s (c old ) -c old ) , p 1 = C(c -c d ), and p 3 = -τB(p 1 ∇c old ). Using these abbreviations, the control u obtained by the instantaneous control strategy for u k 0 = 0 is given by ỹ = B(y old + τb(y old ) -τc old ∇w old ), c = C c old -τ ∇c old ỹ + τ Pe ∆(λ s (c old ) -c old ) , u = ρτ B∇c old C (c -c d )=:K(y old ,p old ,w old ,w old ) (25)

Fig. 1 .

 1 Fig. 1. Initial state c0 (left) and desired state c d (right). Black indicates -1, white indicates +1.

Fig. 2 .

 2 Fig. 2. Controlled state at t =40τ and t = 500τ .

Fig. 3 .

 3 Fig. 3. The reduction of cc d for various Reynolds numbers.