EXPONENTIAL STABILIZATION OF THE STOCHASTIC BURGERS EQUATION BY BOUNDARY PROPORTIONAL FEEDBACK

. In the present paper it is designed a simple, ﬁnite-dimensional, linear deterministic stabilizing boundary feedback law for the stochastic Burg- ers equation with unbounded time-dependent coeﬃcients. The stability of the system is guaranteed no matter how large the level of the noise is.

x)dW (t), t > 0, x ∈ (0, L), Y x (t, 0) = v(t), Y x (t, L) = 0, t > 0, Y (0, x) = y o (x), x ∈ (0, L). (1) The unknown is the function Y ; dW denotes a Gaussian time noise, that is usually understood as the distribution derivative of the Brownian sheet W (t) on a probability space (Ω, F, P) with normal filtration (F t ) t≥0 ; Y x stands for the partial derivative with respect to the space variable, i.e. ∂Y ∂x ; the initial data y o is F 0 −measurable; ν and θ are positive constants known as the viscosity coefficient and the level of the noise, respectively.
The function b is such that there exist C b > 0 and 0 ≤ m 1 ≤ m 2 ≤ ... ≤ m S , for some S ∈ N, for which sup x∈(0,L) Moreover, we assume that m S and θ are such that: θ can be split as where θ 1 > 0. Burgers equation is often refereed to as a one-dimensional "cartoon" of the Navier-Stokes equation because it does not exhibit turbulence. By contrary, it turns out that its stochastic version, (1), models turbulence, for details one can see [7,9]. A principal tool in order to attenuate or even eliminate the turbulence is to control the equations. In this paper we propose a Neumann boundary-type control v, that will be described with details below. The aim of the present paper is to find such a feedback v such that, once inserted into the equation (1), the corresponding solution of the closed-loop equation (1) satisfies e γt L 0 Y 2 (t, x)dx < const., ∀t ≥ 0, P − a.s., for some positive constant γ, provided that the initial data y o is small enough in the L 2 −norm (that is the main result stated in Theorem 2.2 below). (Note that this is an almost sure pathwise local boundary stabilization type result.) In the literature there are plenty of results concerning the stabilization of the deterministic Burgers equation, for example we refer to [15] and [13]. The last one provides a global stabilization result, with some consequences on the stabilizability of the stochastic version. The control problem, associated to the stochastic version, has been addressed as-well in many works such as [7,8,9]. However, to our knowledge, the present work represents a first result on boundary stabilization for the stochastic Burgers equation. For more details about stochastic Burgers equation, we refer to [10,12].
In (1), let us consider the transformation where Γ(t) : L 2 (0, L) → L 2 (0, L) is the linear continuous operator defined by the equations that can be equivalently expressed as Frequently below we shall use the obvious inequality By the law of the iterated logarithm, arguing as in Lemma 3.4 in [3], it follows that there exists a constant C Γ > 0 such that where we have used that 1 2 θ 2 = m S + 1 4 + θ 1 . Then, by (2), we have that Applying Itö's formula in (1) (the justification for this is as in [1]), we obtain for y the random differential equation (8) Taking into account that Γ(t) commutes with the space derivatives, it follows by (8) that y obeys the equation The idea is to design a boundary stabilizer feedback u(t) for the equation (9), then v(t) = Γ(t)u(t) will achieve the path-wise stability desired. The method to design such an u relies on the ideas in [17], where a proportional type stabilizer was proposed to stabilize, in mean, the stochastic heat equation. We emphasize that, unlike to the equation in [17], now we deal with a nonlinear one of order two. In order to overcome this complexity, we further develop the ideas in [17]. Roughly speaking, we design a similar feedback as in [17], whose simple form allows us to write the corresponding solution of the closed-loop system in a mild formulation, via a kernel. Then, in order to show the stability achieved, we use the estimates of the magnitude of the controller (similarly as in [17]) and, in addition, a fixed point argument. The idea to use fixed point arguments in order to show the stability of deterministic or stochastic equations has been previously used in papers like [6,16]. Proportional type feedback, similar to the one we shall design here, has its origins in the works [2,19], while in the papers [20,21,22,23,24,18], it has been used to stabilize other important parabolic-type equations, such as the Navier-Stokes equations (also with delays), the Magnetohydrodynaimc equations, the phase filed equations, and even for stabilization to trajectories.

The boundary feedback stabilizer and the main result of the work.
In what follows L 2 (0, L) stands for the space of all functions that are square Lebesgue integrable on (0, L), and | · | 2 , ·, · the classical norm and scalar product, respectively. Also, we denote by ·, · N , the standard scalar product in R N , N ∈ N. We shall denote by C different constants that may change from line to line, though we keep denote them by the same letter C, for the sake of the simplicity of the writing.
Let us denote by Ay = −νy xx , ∀y ∈ D(A), : y x (0) = y x (L) = 0 , the Neumann-Laplace operator on (0, L). It is well known that it has a countable set of eigenvalues, namely with the corresponding eigenfunctions that form an orthonormal basis in L 2 (0, L). Setting the Gaussian heat kernel, it is well-known the relation where p H is the heat kernel In particular, it follows that we have that Let N ∈ N a large constant. Next, likewise in [17, Eq. (3)], we define the Neumann operators as: D γ k , k = 1, 2, ..., N , solution to the equation Here, with 7 4 < α < 2. We go on following the ideas in [17]. We denote by B the next square matrix and multiply it on both sides by and finally, introduce the matrix which, as claimed in [17], is well-defined. At this stage we are able to introduce the feedback law we propose here for stabilization. To this end, for each k = 1, ..., N , we set the following feedback forms then, introduce u as u(y) Finally, arguing similarly as in [17,Eqs. (17)- (19)], we equivalently rewrite (9) as an internal-type control problem: By [17, Lemma 3.1] (together with the details from its proof), we have the following result related to the linear operator that governs equation (18): can be written in a mild formulation as Here and

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The quantities q ji (t) and w j i (t) involved in the definition of p, obey the estimates: for some C q > 0, depending on N , for all i, j = 1, 2, ..., N , for some positive constant c < ν π 2 L 2 (because of later purpose); and for some C w > 0, depending on N , for all i = 1, 2, ..., N and j = N + 1, N + 2, .... Moreover, for all z o ∈ L 2 (0, L), we have that Proof. We shall show only (23), since all the others items are exactly the results obtained already in [17,Lemma 3.1]. Setting It yields that ∀t ≥ 0. That is exactly what we have claimed.
Relying on the above lemma, we may now proceed to state and prove the main existence & stabilization results of the present work. Theorem 2.2. Let η > 0, depending on ω and sufficiently small. Then, for each y o ∈ L 2 (0, L) with |y 0 | 2 < η, there exists a unique solution y to the random deterministic equation (18) belonging to the space Y, In particular, the stochastic Burgers equation has a unique solution Y = Γ −1 y, that path-wise almost sure is exponentially decaying in the L 2 −norm.
Proof. The space It is clear that, for all y ∈ Y, we have  In what follows we aim to show that G is a contraction on B r (0), that maps the ball B r (0) into itself, for r > 0 properly chosen. Then, via the contraction mappings theorem, we shall deduce that G has a unique fixed point y ∈ B r (0). Then, easily one arrives to the wanted conclusion claimed by the theorem.
We need to estimate the | · | Y -norm, of Gy. So, in particular, we need to estimate the | · | Y -norm of Fy, for y ∈ Y. We begin with the L 2 -norm of Fy. We aim to use 2180 IONUŢ MUNTEANU the Parseval identity, so, in order to do this, based on the kernel's form (20), we conveniently rewrite the term Fy as where (28) It follows via the Parseval identity, that (using that the eigenfunctions are uniformly bounded and relation (7) We go on with (by Schwarz's inequality) (by the inequality between the heat kernel and the Gaussian kernel in (10)) Finally, we deal with (by (7), (22) and the uniform boundedness of the eigenfunctions) We conclude that, (29)-(31) imply that where B(x, y) is the classical beta function. By the exponential semi-group property, we have as-well that

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We go on with the estimates in the H 1 -norm. Using the above notations, we have (arguing as in (29)) Next,  (arguing as in (30)) Finally, (with similar arguments as before) Therefore, (34)-(36) imply that Heading towards the end of the proof, we note that since the presence of the µ j , in the infinite summation, is controlled as in (35) by the presence of t 1 2 . Consequently, via the semi-group property, we deduce that Now, gathering together the relations (31), (33), (37) and (38), we arrive to the fact that, there exists a constant C 1 > 0 such that for all y ∈ Y.
Easily seen, similar arguments as above lead as-well to for some constant C 2 > 0.
Recall that y o < η. It then follows that, if η is small enough such that taking r = 2C 1 η, we get from (40) that G is a contraction and by (39) that G maps the ball B r (0) into itself, as claimed.
Note that, C 1 , C 2 depend on ω, since in the above, ω−estimations for Γ were used. Thus, η should depend on ω too. This means that, in fact, y o must depend on ω.
Since the solution of the system, y, is a fixed point of the map G, we deduce from (39) that |y| Y ≤ C 1 ( y o + |y| 2 Y ). Seeing this as a second order inequality in |y| Y , we immediately obtain that Now, recalling the norm in Y, we readily see that To conclude with the proof, in virtue of the relation between the solution Y of the stochastic Burgers equation (25) and the solution y of the random equation (18), namely Y = Γ −1 y, the above relation yields Using again the law of the iterated logarithm, we can be sure with probability 1 that e − N 2 t Γ −1 (t) (for large enough N ) is bounded. Thereby, completing the proof.
3. Conclusions. In this work, based on the ideas of constructing proportional type stabilizing feedbacks in [17] associated to the stochastic heat equation, together with a fixed point argument, we managed to obtain a first result of boundary stabilization for the stochastic Burgers equation. We emphasize that, the proposed feedback is linear, of finite-dimensional structure, involving only the eigenfunctions of the Laplace operator, being easily to manipulate from the computational point of view.
Regarding the stabilization of deterministic Navier-Stokes equations, with finitely many controllers, we mention the works [4,5], and emphasize that the ideas used in the stochastic context bear a lot of similarities. Due to the presence of the space-time function b, the system (9) is not stabilizable by collocated boundary feedback like in [13]. Indeed, let us consider, as in [13], equation (9) It is clear that the presence of the integral term 1 3 Γ(t) L 0 b x (t, x)y(t, x) 3 dx destroys the argument in [13]. Consequently the collocated feedback laws u 0 , u 1 , proposed in [13, Eqs. (3.2)-(3.3)], do not imply the stability of the system. Indeed, the feedback law, we propose here, requires full state knowledge. However, taking into account the numerical results in [15], where a similar feedback is proposed for the stabilization of the Fischer equation, we may expect that only the knowledge on a part of the domain is enough. In [15] it is shown that measurements are needed only on (a, 1) instead of the whole interval (0, 1), where a ≤ 0.25.