BOUNDARY STABILIZATION OF THE NAVIER-STOKES EQUATIONS WITH FEEDBACK CONTROLLER VIA A GALERKIN METHOD

In this work we study the exponential stabilization of the two and three-dimensional Navier-Stokes equations in a bounded domain Ω, around a given steady-state flow, by means of a boundary control. In order to determine a feedback law, we consider an extended system coupling the Navier-Stokes equations with an equation satisfied by the control on the domain boundary. While most traditional approaches apply a feedback controller via an algebraic Riccati equation, the Stokes-Oseen operator or extension operators, a Galerkin method is proposed instead in this study. The Galerkin method permits to construct a stabilizing boundary control and by using energy a priori estimation technics, the exponential decay is obtained. A compactness result then allows us to pass to the limit in the system satisfied by the approximated solutions. The resulting feedback control is proven to be globally exponentially stabilizing the steady states of the two and three-dimensional Navier-Stokes equations.


Introduction.
Let Ω be a bounded and connected domain in R d (d = 2, 3), with a boundary Γ of class C 2 , and composed of two connected components Γ l and Γ b such that Γ = Γ l ∪ Γ b , in order to impose two different boundary conditions specified in (1).In particular, the boundary Γ b is the part of Γ, where a Dirichlet boundary control in feedback form has to be determined.
We denote by • | • and • = • L 2 (Ω) , the scalar product and norm in L 2 (Ω), respectively.Moreover, if u ∈ L 2 (Ω) is such that ∇ • u ∈ L 2 (Ω), then we denote the normal trace of u in H − 1 2 (Γ) by u • n, where n denotes the unit outer normal vector to Γ.
We consider a stationary motion of an incompressible fluid described by the velocity and pressure couple (v s , q s ), which is the solution to the stationary Navier-Stokes equations In this setting, ν > 0 is the viscosity, f s is a function in L 2 (Ω), v b belongs to V 3 2 (Γ) defined as V 3 2 (Γ) = u ∈ H 3/2 (Γ) : Γ u • n dζ = 0 .Recall [17] that a solution (v s , q s ) to ( 1) is known to exist in H 2 (Ω) × (H 1 (Ω) ∩ L 2 0 (Ω)).For T > 0 fixed, let Q = [0, T [×Ω, Σ l = [0, T [×Γ l and Σ b = [0, T [×Γ b and consider a trajectory (u, q) solution of the non stationary Navier-Stokes equations with x = (x, y, z) if d = 3.Consequently, the couple (v = u − v s , p = q − q s ) satisfies the following non stationary system in Ω. ( In order to stabilize the unsteady solution u of (2), for a prescribed rate of decrease σ > 0, we need to find a control u b such that the components v of the solution (v, ∇p) to the boundary value problem (3) satisfies the exponential decay: for a constant C > 0 independent of v 0 (x).It's worth noticing that, in the present paper, we let C = 1.
The control u b (t) is called a feedback if there exists a mapping F : and the corresponding feedback law in (5) is pointwise in time.However, the feedback law may be chosen in a different manner, for example as where F 0 is a mapping belonging to L(X(Ω), U(Γ b )), but in that case, the feedback law is not pointwise in time.The spaces X(Ω) and U(Γ b ) will be defined accordingly.
Pointwise feedback laws are usually needed in engineering applications as they are more robust with respect to perturbations in the models.Different approaches have been pursued in the past, which first determine a linear feedback law by solving a linear control problem for the linearized system of equations (for example the Oseen system) and then use this linear feedback law in order to stabilize the original non linear system (for example the Navier-Stokes system).In such a framework, several significant questions have to be addressed.First, do we obtain a pointwise feedback law able to stabilize the linearized system?Secondly, by assuming that F is a pointwise (in time) feedback law able to stabilize the linear system in X(Ω), does F also stabilize the nonlinear system for v 0 (x) in a subspace of {u ∈ L 2 (Ω) : ∇ • u = 0}, with v 0 (x) small enough ?Finally, assuming that the existence of a feedback law stabilizing the linear system is proved, is it possible to obtain a well posed equation characterizing F , for example a Riccati equation, which can be numerically solved by classical methods?
These questions of stabilizing the Navier-Stokes equations with a boundary control have been first addressed by A.V. Fursikov in [14,15], where stability results for the two and three-dimensional Navier-Stokes equations are proved by employing an extension operator.With an adequate extension procedure for the initial velocity condition v 0 (x) in (3), which requires the knowledge of the eigenfunctions and the eigenvalues of the Oseen operator, the author obtains a boundary control of the form with k ≥ 1.However, if the feedback controls are well characterized, the corresponding laws are not pointwise in time.
In [24], as far as the two-dimensional case is concerned, J.-P.Raymond has obtained boundary feedback control laws, pointwise in time, where the feedback controller is determined by solving an algebraic Riccati equation obtained via the solution of an optimal control problem with where 0 < ε < 1/4 and m ∈ C 2 (Γ).Unfortunately, the three-dimensional case is more demanding in terms of velocity regularity, as explained in [23], and it cannot be treated in the same manner as the two-dimensional case.Indeed, in the three-dimensional case the feedback controller needs to satisfy F (v) belonging to H 1/4+ε/2 (0, ∞; L 2 (Γ)) with 1/2 ≤ ε, and in the particular case 1/2 < ε, the space , implying that the velocity v has to satisfy the initial compatibility condition v 0 | Γ = F (v 0 ).This is the reason why the feedback law used in [24] cannot be employed in the three-dimensional case, and why this difficulty has been overcome in [23] by introducing a time dependent feedback law in an initial transitory time interval.In order to obtain a stabilization result via the Riccati approach, particular spaces of initial conditions have to be employed that are given in [3].
The study, performed in [23], also improves in some way the results obtained in [8,9], where a tangential boundary stabilization of two and three-dimensional Navier-Stokes equations is employed with both Riccati-based and spectral-based (tangential) feedback controllers.In [9], for the three-dimensional case which is highly demanding in terms of velocity regularity, the existence of boundary feedback laws, pointwise in time, is established by solving an optimal control problem with a cost functional involving the L 2 (0, ∞; H 3/2+ε (Ω)) norm of the velocity field, for some 0 < ε small enough.However, such a feedback law cannot be characterized by a well posed Riccati equation, as shown in [9], and the numerical calculation of the feedback control thus becomes problematic.In [23], for the three-dimensional Navier-Stokes system, J.-P.Raymond chooses a functional involving a very weak norm of the state variable which leads to a well posed Riccati equation.
Recall in [23], a time dependent feedback law in an initial transitory time interval was introduced.As mentioned in [2], the problem of finding a time independent feedback controller satisfying v 0 | Γ = F (v 0 ), for a sufficiently large class of initial conditions v 0 , is not obvious.This problem has been examined in [2] for the two and three-dimensional case, and it has led to search for solutions u b satisfying an extended system composed of the evolution system coupled with the original Navier-Stokes equations, where the feedback controller F now acts on the pair (v, u b ) and ∆ B is the vector-valued Laplace Beltrami operator.The space X(Ω) is now defined as , the oprerator F is found from a well-posed Riccati equation and the controller u b , localized on an arbitrary small part of Γ, can be obtained.
In the purpose of stabilizing the Navier-Stokes equations around a stationary state, the feedback control laws are determined by solving a Riccati equation in most of the studies cited above [2,3,7,8,9,23,24], except in the Fursikov's papers [14,15].The Riccati equation is obtained via the solution of an optimal control problem and it is stated in a space of infinite dimension.Although our study is only concerned with the construction of boundary controllers, the Riccati approach described above, stated in a space of infinite dimension, applies as well to the case of internal control [5,11].
In the case the feedback controller lies in an infinite-dimensional space, an optimal control problem has to be solved, involving the minimization of an objective functional.In practice, the control is calculated through approximation via the solution of an algebraic Riccati equation, which is computationally expensive.Consequently, the use of finite-dimensional controllers may be more appropriate to stabilize the Navier-Stokes equations.Such an approach is performed in [10], in the case of an internal control, and in [1,7,8,9,22], in the case of a boundary control.Recall the Riccati equation is stated in a space of infinite dimension in [7,8,9].In [1,10,22], the authors search for a boundary control u b of finite dimension of the form where (ϕ j ) j=1,2,3,...,N is a finite-dimensional basis obtained from the eigenfunctions of some operator and ū = (u 1 , u 2 , u 3 , . . ., u N ) is a control function expressed with a feedback formulation.In [22], where d = 2, the feedback control is obtained from the solution of a finite-dimensional Riccati equation stated in R n c ×n c , where n c is the dimension of the unstable space of the Oseen operator.The same approach is then extended in [1] for the three-dimensional case.However, in [10,22] the minimal value of N is a priori unknown while in [1], N is greater or equal to the maximum of the geometric multiplicities of the unstable modes of the Oseen operator.Finally, finitedimensional stabilizing feedback laws of the form of ( 7) are obtained in [6] and [4], in the case of internal and boundary control, respectively.Instead of employing the Riccati approach, a stochastic-based stabilization technique is employed in [6] which avoids the difficult computation problems related to infinite-dimensional Riccati equations.The procedure employed in [4] ressembles the form of stabilizing noise controllers designed in [6].In all the above-mentioned studies, a linear feedback law is first determined by solving a linear control problem for the linearized system of equations and then this linear feedback is used in order to stabilize the original non linear system.However, such a procedure imposes to choose the initial velocity small enough.Further, the employed methods (e.g. the Riccati approach) require to search for the control u b and the initial condition in sufficiently regular spaces, depending on whether d = 2 or d = 3.For example, in [4, Theorem 2.3], we have in the case d = 2 and, for v 0 ∈ X(Ω), with v 0 X(Ω) < ρ and ρ sufficiently small, the function v satisfies the following stability estimate v X(Ω) ≤ Ce −σt v 0 X(Ω) , for all t ≥ 0 and for some σ > 0, but the value of C is not precisely given.Note that, in the case d = 3, no control is proposed in [4] to stabilize the non linear Navier-Stokes equations.Further, in [1, Theorem 2], we have , where P is the Leray projector, and , and stability estimates are also obtained.
In this paper, a new approach is proposed.Instead of obtaining the feedback law by first solving a linear control problem for the linearized system of equations, eventually via the resolution of a Riccati equation, an extended system is considered.Indeed, in (3) the boundary control u b is rewritten on the form The quantity α(t) is a priori unknown.In order to stabilize (3), with u b = α(t)g(x) on Σ b , by employing energy a priori estimation technics, the quantity α(t) is found to satisfy the relation where f is a polynomial in α(t) of degree 2. Note that α(t) depends nonlinearly on v and hence α(t), which reads α(v(t)), satisfies a nonlinear feedback law.Such a feedback, pointwise in time, ressembles to (5) but the mapping F is nonlinear here.
The system (3) is then extended by adding (10), and the extended system, namely (3) and (10) with u b = α(t)g(x) on Σ b , is then solved in order to determined α(t), leading to the determination of the boundary control u b .Such a boundary representation of u b is also employed in [21] in the two-dimensional case, where a linear feedback control dα(t)/dt is obtained via the solution of a Riccati equation stated in a space of infinite dimension.In the present paper, however, the quantities α(t), and hence u b , are computed at the discrete level.Further, contrary to (7) and [21], where u j (t), j = 1, 2, 3, . . ., N , and dα(t)/dt, respectively, are linear feedbacks, α(t) is nonlinear here and it is thus calculated through a Galerkin procedure instead of being the solution of a finite-dimensional Riccati equation, for example.
Note that the Galerkin procedure first consists of building a sequence of approximated solutions via an adequate Galerkin basis.Because the energy bounds are not sufficient to pass to the limit in the weak formulation, additional bounds are obtained.A compactness result then permits to pass to the limit in the system satisfied by the approximated solution, leading to the existence of at least one weak solution.Such a procedure relies on technics previously introduced in [19], but it is worth to note that the work performed in [19] is not related to a stabilization problem.
The approach proposed in this paper has several advantages.First, the stabilization result in (4), i.e. v(t, x) ≤ C e −σt v 0 (x) , for t ∈ (0, ∞), is obtained with C = 1 and for an arbitrary initial data v 0 belonging to implying less regularity on v 0 than in the case of the previous studies cited above, for example see (9).Further, the regularity results are independent of d and they are thus obtained in the two and three-dimensional case as well.
The paper is organized as follows.In section 2, the notations and mathematical preliminaries are introduced.The stabilization problem is formulated in Section 3, and the existence of the solution of the nonlinear Navier-Stokes system is established and the existence analysis is carried out by applying the Galerkin method.Finally, some concluding remarks complete the study in Section 4.

Notation and Preliminaries.
2.1.Function Spaces.Several spaces of free divergence functions are now introduced: ) be the space of trace functions that, if extended by zero over Γ, belongs to , the solution of (3) coupled with (10) is searched in The following lemma [19], will be used in the sequel.
Lemma 2.2.There exists a constant C b > 0 such that, for all (v, α) ∈ W (Q), we have We now define an Hilbertian basis for the space W (Q).
2.2.An Hilbertian basis for the space W (Q). Let {z j , λ j , j = 1, 2, 3, • • • } be the eigenfunctions and eigenvalues of the following spectral problem for the Stokes operator: As shown in [25], 0 , and {z j } forms an orthonormal basis in V 0 (Ω) verifying: The space W (Q), defined in (14), is then rewritten as where w satisfies the following system Since g satisfy Γ b g • n dζ = 0, system (19) hence admits a unique solution (w, q) ∈ V(Ω) × L 2 0 (Ω), where L 2 0 (Ω) is the pressure space with zero mean value: Note that the existence and uniqueness of (w, q) in ( 19) can be deduced from [25].
2.3.Linear Forms.In order to define a weak form of the Navier-Stokes equations, we introduce the continuous bilinear forms and the trilinear form: By integration by parts, the following properties hold true Thanks to Hölder inequality, we obtain 3. Stability Result.
3.1.The stabilization Problem.In order to stabilize the non stationary Navier-Stokes System (3), we choose to search the solution v in the form v = z+αw, where z ∈ V 0 (Ω), and α and w satisfy ( 10) and ( 19), respectively.We then have v = αg on Γ b as z = 0 on Γ.Consequently, the state (v, p) satisfies the following extended coupled system: where with σ 0 > 0 is a constant, λ 1 is the smallest positive eigenvalue of ( 16) and Recall that α is a priori unknown and thanks to (23-f), it satisfies a nonlinear feedback law leading to search for α(v(t)).Because (23-f) is independent of x, α(v(t)) is a function of t only.For the sake of simplicity, α(v(t)) is written α in the sequel.
3.2.The variational formulation.We first state to consider the variational formulation of the extended Navier-Stokes system.
Note that the rate of decrease σ(t) depends on the control α and σ 0 may be regarded as an accelerator.
Proof.Let us begin with the proof of the stability estimates followed by the existence result.

A priori estimates. Taking
Let us estimate the terms in the left-hand side of (30).According to ( 20)-( 22), we obtain Using ( 24) and ( 31)-( 33) in (30), leads to 1 2 Due to (19), we have ∇w, ∇z = 0 and from (34) we deduce 1 2 Since and using v = z + αw, we obtain from (35) 1 2 For all σ 1 such that 0 < σ 1 ≤ σ = νλ 1 − ∇v s ∞ , we have 1 2 and omitting the second term in the left hand side of (37) leads to Multiplying ( 38) by e 2σ(t) , where σ(t) = σ 1 t + σ 0 t 0 α 2 (s)ds ≥ 0, we obtain d dt e 2σ(t) v 2 ≤ 0 and consequently, By omitting the third term in the left hand side of (37) we deduce 1 2 and integrating from 0 to t yields Since v = z + αw, we substitute w 2 α 2 + 2α w, z = v 2 − z 2 in the two last terms in the right hand side of (34), and this leads to 1 2 Integrating (41) from 0 to t yields and employing (39) and (40) we obtain Because σ(t) = σ 1 t + σ 0 t 0 α 2 (s)ds, we have σ(t) ≥ σ 1 t, and hence Therefore, we obtain the a priori estimate 3.4.Existence.The proof of the existence follows a standard procedure.In a first step a sequence of approximate solutions using a Galerkin method is built.A compactness result from [20] allows us to pass to the limit in the system satisfied by the approximated solutions.
3.4.1.The Galerkin Method.For all m ∈ N, we define the space W m as: where w 0 = w and φ im w i and we define the following finite-dimensional problem where δ ij defined the Kronecker symbol and Moreover this solution satisfies : where C is a positive constant independent of m.
Proof.We rewrite (44) in terms of the unknown φ im , i = 0 • • • m, and we obtain Because the matrix with elements w i , w j (0 ≤ i, j ≤ m) is nonsingular, (47) reduces to a nonlinear system with constant coefficients where X ij , Y ijk , Z ij , ∈ R.Then, there exists T m (0 < T m ≤ T ) such that the nonlinear differential system (48) has a maximal solution defined on some interval [0, T m ].
In order to show that T m is independent of m, it is sufficient to verify the boundedness of φ im , and hence the boundedness of the L 2 -norm of v m independently of m.Following the same procedure as for the derivation of the a priori estimates (39) and (43), yields Consequently, according to (49-a), we obtain T m = T .
Moreover, a consequence of the a priori estimates (49) is that (v m ) m is bounded in L 2 (0, T ; V(Ω)) and L ∞ (0, T ; H(Ω)).Therefore, for a subsequence of v m (still denoted by v m ), the estimates in (49) yield the following weak convergences as m tends to ∞ : Nevertheless, the convergences in (50) are not sufficient to pass to the limit in the weak formulation (44), because of the presence of the convection term.Consequently, we need to obtain additional bounds in order to utilize the compactness theory on the sequence of approximated solution (v m ) m .

Additional bounds.
As in [20], let us assume that B 0 , B and B 1 are three Hilbert spaces such that Let us recall the following identity about the Fourier transform of differential operators: , for a given γ > 0, and let us define the space .
We also define H γ (0, T ; B 0 , B 1 ), as the space of functions obtained by restriction to [0, T ] of functions of H γ (R; B 0 , B 1 ).Further, we recall the following result [20]: Lemma 3.4.Let B 0 , B and B 1 be three Hilbert spaces such that B 0 ⊂ B ⊂ B 1 and B 0 is compactly embedded in B. Then for all γ > 0, the injection H γ (0, T ; B 0 , B 1 ) → L 2 (0, T ; B) is compact.
For small enough ε, this lemma is used later with The main result of the present section, based on utilizing Lemma 3.4, is furnished by the following lemma: We denote by vm the extension of v m by zero 0 for t < 0 and t > T , and v m the Fourier transform with respect to time of vm .It is classical that since vm has two discontinuities at 0 and T , in the distributional sense, the derivative of vm is given by where δ 0 , δ T are Dirac distributions at 0 and T , and

BOUNDARY FEEDBACK STABILIZATION OF THE NAVIER-STOKES 13
After a Fourier transformation, (51) gives where v m and u m denote the Fourier transforms of vm and ūm respectively.Since we already know that v m is uniformly bounded in L 2 (0, T, V(Ω)), it remains to prove that We have that vm satisfies where ).We now apply the Fourier transform to the equation (53) and take ( v m , φ 0m ) as a test function, it yields 2iπτ where G m , G 0 m , G 1 m and H m are respectively the Fourier transform with respect to time of G m , G 0 m , G 1 m and H m .Note that where F m is the Fourier transform with respect to time of φ 0m v m 2 .Thanks to lemma 2.2, we have By using (55) in (54) and taking the imaginary part of (54) leads to Note that in the sequel, C stands for different positive constants.We now prove that the right hand side of (56) is bounded.First, we have and thanks to the energy estimate (49) satisfied by v m , G m and G s m remain bounded in L 1 (R; V (Ω)) and the functions G m , G s m are bounded in L ∞ (R; V (Ω)).Consequently, we have and the second line of ( 56) is hence bounded.We now show that the first four terms in the right hand side of (56) are bounded.
where C stands for different positive constants.For 0 < γ < 1  4 , we now estimate the norm Now, applying Lemmas 3.4 and 3.5, there is a subsequence of (v m ) m∈N which converges strongly in L 2 (0, T, H(Ω)).
3.4.3.Passage to the limit.The compactness result obtained in the previous section implies the following strong convergence (at least for a subsequence of v m still denoted v m ) v m → v strongly in L 2 (0, T ; L 2 (Ω)).This convergence result together with (50) enable us to pass to the limit in the following weak formulation, obtained from (44) by multiplication by ϕ ∈ D(]0, T [) and integration by parts with respect to time Using the weak estimates (50) leads to for the linear terms.Further, since v m converges to v in L 2 (0, T ; V(Ω)) weakly, and in L 2 (0, T ; L 2 (Ω)) strongly, we can pass to the limit in the nonlinear term to obtain Using Lemma 2.2 and according to (49-a), φ 0m ∈ L ∞ (0, T ).Then for a subsequence of φ 0m (still denoted by φ 0m ): As far as the right hand side of (61) is concerned.Let us notice that the convergence of v m in L 2 ([0, T ] × Ω) implies its convergence in L 1 (0, T ; L 2 (Ω)).Hence Due to lemma 2.2, we have and φ 0m is then a Cauchy sequence in L 1 (0, T ) and Further, according to (63) we have φ 0 = α ∈ L ∞ (0, T ) from [12,Proposition II.1.26].
Since v m and φ 0m are bounded in L ∞ (0, T ), using (64) and ( 65) we obtain from [12, Corollaire II.1.24],for all p ∈]1, +∞[.Now we can pass to the limit in the following terms: Passing to the limit in (61) then gives for all v = v j , ∀j = 0, 1, 2, • • • , m.By linearity, equation (69) holds true for all v combination of finite v j and by density, for any element of W (Q).
Finally, it remains to retrieve the stabilized problem (23), which requires to prove the existence of pressure.
4. Concluding remarks.In this work the exponential stabilization of the two and three-dimensional Navier-Stokes equations in a bounded domain is studied around a given steady-state flow, using a boundary feedback control.In order to determine a feedback law, an extended system coupling the Navier-Stokes equations with an equation satisfied by the control on the domain boundary is considered.We first assume that on Σ b (a part of the domain boundary), the trace of the fluid velocity is proportional to a given velocity profile g.The proportionality coefficient α measures the velocity flux at the interface, it is an unknown of the problem and is written in feedback form.By using the Galerkin method, α is determined such that the Dirichlet boundary control u b = αg is satisfied on Σ b , and the stabilizing boundary control is built.The resulting nonlinear feedback control is proven to be globally exponentially stabilizing the steady states of the two and three-dimensional Navier-Stokes equations.This feedback control was shown to guarantee global stability in the L 2 -norm.Finally, in order to take into account (23-f) in the variational formulation, the test functions, for example v, need to be written on the form v = αg.This requires to construct a finite-element basis which allows such a requirement and hence at least one element of the basis, for example w, such that w = g on Γ b .A number
Thanks to lemma 2.2 and estimate (49), φ 2 0m and F m = φ 0m v m 2 are bounded in L 1 (R), and hence φ 2 0m and F m are bounded in L ∞ (R) with: sup Thanks to the energy estimate (49-a) satisfied by v m , we have v m (T ) ≤ C and v m (0) ≤ C. Inequation (56) thus finally reduces to