Global Stabilization of Two Dimensional Viscous Burgers' Equation by Nonlinear Neumann Boundary Feedback Control and its Finite Element Analysis

In this article, global stabilization results for the two dimensional viscous Burgers' equation that is, convergence of unsteady solution to its constant steady state solution with any initial data, are established using a nonlinear Neumann boundary feedback control law. Then, using $C^0$-conforming finite element method, global stabilization results for the semidiscrete solution are shown. Moreover, optimal error estimates in $L^\infty(L^2)$ and in $L^\infty(H^1)$-norms for the state variable are obtained. Further, convergence result is derived for the boundary feedback control law. All the results preserve exponential stabilization property. Finally, some numerical experiments are conducted to confirm our theoretical findings.


Introduction
Consider the following Neumann boundary control problem for the two-dimensional viscous Burgers' equation : seek u = u(x, t), t > 0 which satisfies on (x, t) ∈ ∂Ω × (0, ∞), (1.2) u(x, 0) = u 0 (x) in x ∈ Ω, (1.3) where ν > 0 is a constant, v 2 is scalar control input, 1 = (1, 1), u(∇u · 1) = u 2 i=1 u xi is the nonlinear term and u 0 is a given function and Ω ∈ R 2 is a bounded domain with smooth boundary. Related to local stabilization result of one dimensional version of (1.1), see [5], [6] for distributed and Dirichlet boundary control, and [7] for Neumann boundary control with sufficiently small initial data. For more references regarding local stabilization result including results on existence and uniqueness, refer to [10], [11] and [16]. In [18], authors have shown local stabilization results for Navier-Stokes equation around a nonconstant steady state solution by constructing a linear feedback control law for the corresponding linearized equation. This, in turn, locally stabilizes the original nonlinear system. Thus, local stabilizability results can be proved in a similar fashion for the two dimensional Burgers' equation using linear feedback control law. In [21], authors have shown local stabilization results for the two dimensional Burgers' equation directly through a nonlinear feedback control law. For instance, convert the two dimensional Burgers' equation into the abstract form where A, with domain D(A), is the infinitesimal generator of an analytic semigroup on a Hilbert space W , F (w) is the nonlinear term, B m is the operator from a control space V into W . If the corresponding cost functional is of the form then corresponding linear feedback control law is v = −B * m P w, where P is the solution to the algebraic Riccati equation for the LQR problem. Through solving Hamilton-Jacobi-Bellman equation using Taylor series expansion, one can obtain the nonlinear feedback control law as where (A−B m B * m P ) − * is the inverse of (A−B m B * m P ) * , see [21] for more details. Buchot et al. [4] have discussed local stabilization result in the case of partial information for the two dimensional Burgers' type equation. Regarding global stabilization result for one dimensional Burgers' equation, refer to [12] and [3] for both Dirichlet and Neumann boundary control law. For adaptive control (when the coefficient of viscosity is unknown) for one dimensional Burgers' equation, see [15], [19], and [20]. For one dimensional version of (1.1), we in [13], have shown optimal error estimates in the context of finite element method for the state variable and superconvergence result for the feedback control laws. To the best of our knowledge, there is hardly any result on global stabilization for the two dimensional Burgers' equation. Further, when applying finite element method, it is interesting to know the rate of convergence analysis. Hence, in this paper, an attempt has been made to fill this gap. The major contribution of this article are summarized as follows: • Global stabilization results for the problem (1.1)-(1.3) in L ∞ (H i ) (i = 0, 1, 2) norms are established using Lyapunov function to derive feedback control laws.
• Based on C 0 -conforming finite element method, global stabilization results are also obtained for the semidiscrete solution keeping the time variable continuous. Moreover, optimal error estimates in different norms for the state variable and feedback control law are derived.
For the rest of the article, denote H m (Ω) = W m,2 (Ω) to be the standard Sobolev space with norm · m , and seminorm | · | m . For m = 0, it corresponds to the usual L 2 norm and is denoted by · . The space L p ((0, T ); X) 1 ≤ p ≤ ∞, consists of all strongly measurable functions v : and v L ∞ ((0,T );X) := ess sup 0≤t≤T v(t) X < ∞.
The following trace embedding result holds for 2D. Boundary Trace Imbedding Theorem (page 164, [1]): Below, we recall for the following inequalities for our subsequent use Friedrichs's inequality: For y ∈ H 1 (Ω), there holds where φ(x) = 1 4 |x| 2 so that ∆φ = 1. Now integrate by parts to obtain Hence Hence the Friedrichs's inequality constant can be taken as C F = max{sup x∈∂Ω |x| 2 , sup x∈∂Ω |x|}. Gagliardo-Nireberg inequality (see [17]): For w ∈ H 1 (Ω) Agmon's inequality (see [2]): For z ∈ H 2 (Ω), there holds Now the corresponding equilibrium or steady state problem becomes: find u ∞ as a solution of Note that any constant w d satisfies (1.6)-(1.7). Without loss of generality, we assume that w d ≥ 0. To achieve lim it is enough to consider lim t→∞ w = 0, where w = u − w d and w satisfies The motivation behind choosing Neumann boundary control comes from the physical situation. In thermal problem, one cannot actuate the temperature w on the boundary, but the heat flux ∂w ∂n which makes the stabilization problem nontrivial because w d is not asymptotically stable with zero Neumann boundary. Also for our analysis, compatibility conditions for w 0 on the boundary namely ∂w0 ∂n = v 2 (x, 0) and are needed. For motivating to choose the control law, we construct Lyapunov functional of the following form Using the Young's inequality, it is valid that where c 0 is a positive constant. Therefore Now, choose the Neumann boundary feedback control law as where C Lyp = 2 C F min ν, c0 2 + w d > 0. Setting B v; w, φ as B v; w, φ = v ∇w · 1 , φ , w satisfies a weak form of (1.8)-(1.10) as For our subsequent analysis, we assume that there exists a unique weak solution w of (1.13) satisfying the following regularity results (1.14) The rest of the paper is organized as follows. In Section 2, we focus on global stabilization results using nonlinear feedback control law. Section 3 deals with finite element approximation and global stabilization results for the semidiscrete system. Further, optimal error estimates are obtained for the state variable and convergence result is derived for the feedback control law. Finally, Section 4 concludes with some numerical experiments.

Stabilization results
In this section, we establish global stabilization results for the state variable w(t) of the continuous problem (1.13). Throughout the paper, all the results hold with the same decay rate α: Lemma 2.1. Let w 0 ∈ L 2 (Ω). Then, there holds where β = min{2(ν −αC F ), (c 0 +2w d −2αC F )} > 0, and C F > 0 is the constant in the Friedrichs's inequality (1.5).
For the first term on the right hand side of (2.2), we use integration by parts and then bound it as Similarly, using the Young's inequality, the second term on the right hand side of (2.2) is bounded by Now, using the Friedrichs's inequality (1.5), it follows that Hence, from (2.2), we arrive using (2.3), (2.4) and (2.5) at From (2.1), the coefficients on the left hand side are non-negative. Integrate (2.6) with respect to time from 0 to t, and then, multiply the resulting inequality by e −2αt to obtain This completes the proof.
Remark 2.1. The above Lemma also holds for α = 0, that is, Moreover, using the Friedrichs's inequality, it follows that Remark 2.2. Now instead of taking the control on the whole boundary, if we take the above mentioned Neumann control on some part of the boundary (Γ N ) with remaining part zero Dirichlet boundary condition, still the stabilization result holds. For instance, consider where Γ D and Γ N are sufficiently smooth. With this setting, from (2.2), we arrive at d dt e αt w 2 −2α e αt w 2 + 2ν e αt ∇w 2 + 2e 2αt Proceed as before to complete the rest of the proof for L 2 -stabilization result. In higher order norm, stabilization result also holds similarly when control works on some part of the boundary.
Proof. Form an L 2 -inner product between (1.8) and −e 2αt ∆w to obtain The fourth term on the left hand side of (2.9) can be rewritten as The terms on the right hand side of (2.9) are bounded by and using Lemma 2.1 Finally, from (2.9), we arrive at Integrate the above inequality from 0 to t, and then use the Grönwall's inequality with Lemma 2.1 to obtain Use Remark 2.1 for the integral term under the exponential sign, and then multiply the resulting inequality by e −2αt to complete the rest of the proof.
Then, there exists a positive constant C = C w 0 1 such that the following estimate holds.
The terms on the right hand side of (2.11) are bounded by and using Lemma 2.1 Hence, rewriting the boundary integral term in (2.11) as in previous Lemma 2.2, we arrive from (2.11) at Apply Lemmas 2.1 and 2.2, and the Grönwall's inequality to the above inequality to complete the rest of the proof.
Then there exists a positive constant C = C w 0 2 such that Proof. Differentiate (1.8) with respect to t and then take the inner product with e 2αt w t to obtain The right hand side terms in (2.12) Hence, from (2.12), we arrive at To calculate w t (0) , take the inner product between (1.8) and w t to obtain Integrate the inequality (2.13) from 0 to t and then use Lemmas 2.1-2.3 to complete the rest of the proof.
Lemma 2.5. Let w 0 ∈ H 3 (Ω). Then there exists a positive constant C = C w 0 3 such that Proof. Differentiate (1.8) with respect to t and then take inner product with −e 2αt ∆w t to obtain 14) The first three terms on the right hand side of (2.14) are bounded by and using Lemma 2.1 The boundary terms on the right hand side of (2.14) are bounded by Therefore, from (2.14), we arrive at Integrate the above inequality from 0 to t and then apply the Grönwall's inequality along with Lemmas 2.1-2.4 to obtain Again, use of Lemmas 2.1, 2.2 and 2.4 for the above inequality (2.15) completes the proof.

Finite element method
In this section, we discuss semidiscrete Galerkin approximation keeping time variable continuous and prove optimal error estimates for both state variable and feedback controller. Given a regular triangulation T h of Ω, let h K = diam(K) for all K ∈ T h and h = max The semidiscrete approximation corresponding to the problem (1.13) is to seek w h (t) ∈ V h such that Since V h is finite dimensional, (3.1) leads to a system of nonlinear ODEs. Hence, an application of Picard's theorem ensures the existence of a unique solution locally, that is, there exists an interval (0, t n ) such that w h exists for t ∈ (0, t n ). Then, using the boundedness of the discrete solutions from Lemmas 3.1-3.2 below, the continuation arguments yields existence of a unique solution for all t > 0. In a similar fashion as in continuous case, the following stabilization results hold for the semidiscrete solution.
Proof. Forming an L 2 -inner product between (3.4) and −e 2αt ∆ h w h , we obtain Bounding the integral term on the left hand side and the terms on the right hand side of (3.7) as in continuous case, we arrive at Integrate from 0 to t, and then use the Grönwall's inequality with Lemma 3.1 to obtain Apply Lemma 3.1 for α = 0 to the integral term under the exponential sign, and then multiply the resulting inequality by e −2αt to complete the rest of the proof.
Proof. Proof follows as in continuous case, namely; Lemma 2.3.
Lemma 3.4. Let w 0 ∈ H 2 (Ω). Then there exists C = C w 0 2 , a positive constant such that Proof. Differentiating (3.4) with respect to t and then taking inner product with e 2αt w ht , we obtain d dt e αt w ht 2 −2α e αt w ht 2 + 2ν e αt ∇w ht Hence bounding the right hand side terms of (3.8), it follows that To obtain w ht (0) , take the inner product between (3.4) and w t to arrive at w ht (0) 2 ≤ C ∇w 0h 2 + ∆ h w 0h 2 + w 0h 2 ∇w 0h 4 .
Integrate the inequality (3.9) from 0 to t and then use Lemmas 3.1-3.3 to complete the proof.
Proof. Differentiate (3.4) with respect to t and then take inner product with −e 2αt ∆ h w ht to obtain d dt e αt ∇w ht 2 − 2α e αt ∇w ht Therefore, bounding the right hand side term of (3.10) as in continuous case, we arrive at d dt e αt ∇w ht 2 + (c 0 + 2w d ) e αt w ht Integrate the above inequality from 0 to t and then apply the Grönwall's inequality with Lemmas 3.1-3.4 to obtain As in continuous case, we can find the value ∇w ht (0) . The other two terms namely w ht (0) L 2 (∂Ω) and w h (0)w ht (0) 2 L 2 (∂Ω) are bounded respectively by C w ht (0)

Error estimates
Define an auxiliary projectionw h ∈ V h of w through the following form where λ ≥ 1 is some fixed positive number. For a given w, the existence of a uniquew h follows by the Lax-Milgram Lemma. Let η := w −w h be the error involved in the auxiliary projection. Then, the following error estimates hold: For a proof, we refer to Thomée [22]. Following Lemma 3.6 is needed to establish error estimates. Lemma 3.6. Let F ∈ H 3/2+ (Ω), for some > 0, and G ∈ H 1/2 (∂Ω). Then F G ∈ H 1/2 (∂Ω) and Proof. For a proof see [9].
In addition, for proving error estimates for state variable and feedback controllers, we need the following estimate of η and η t at boundary.
Consider an auxiliary function φ satisfying the following problem . For a proof of this regularity result see [14].
Since estimates of η are known from (3.12) and Lemma 3.7, it is sufficient to estimate θ. Subtracting the weak formulation (1.13) from (3.1), a use of (3.11) yields In the following theorem, we estimate θ(t) .
Proof. Set χ = θ in (3.16) to obtain The first term I 1 (θ) on the right hand side of (3.17) is bounded by where > 0 is a positive number which we choose later. For the second term I 2 (θ) on the right hand side of (3.17) a use of the Cauchy-Schwarz inequality with Young's inequality and θ H 1/2 (∂Ω) ≤ C θ 1 yields The third term I 3 (θ) on the right hand side is bounded by For the fourth term I 4 (θ), first we bound the following sub-terms as The other sub-term in I 4 (θ) can be bounded by For I 5 (θ), we note that Finally, using Lemmas 2.1-2.4, 3.1, 3.2 and 3.7, we arrive from (3.17) at Integrate the above inequality from 0 to t and choose = β1 2C F . Then use the Grönwall's inequality to obtain

ds .
A use of Lemmas 2.1-2.5, 3.1 and 3.2 to the above inequality with a multiplication of e −2αt completes the proof.
Proof. Set χ = θ t in (3.16) to obtain The first term I 1 (θ t ) on the right hand side of (3.16) is bounded by The second term I 2 (θ t ) on the right hand side of (3.16) can be rewritten as and hence, The third term I 3 (θ t ) on the right hand side of (3.16) is bounded by For the fourth term I 4 (θ t ) on the right hand side of (3.16), first we rewrite the sub terms as and using integration by parts Similarly, The other two sub-terms in the fourth term are bounded by For the last term I 5 (θ t ) on the right hand side of (3.16), the first sub-term is bounded by Similarly, the other sub-terms are bounded by , Hence, from (3.16), we arrive at Multiply the above inequality by e 2αt and use Lemmas 2.1, 2.4, 3.1-3.3 and 3.6 to obtain Integrate the above inequality from 0 to t. Then multiply the resulting inequality by e −2αt and use Lemmas 2.2, 2.4, 2.5, 3.2-3.6, and Theorem 3.1 to arrive at Hence, we obtain after using kickback arguments This completes the rest of the proof.
Proof. First part of the proof follows from estimates of η in (3.12) and Theorems 3.1 and 3.2 with a use of triangle inequality. For the second part, we note that A use of Lemmas 2.2, 3.2, 3.7 and Theorem 3.2 completes the proof.

Numerical experiments
In this section, we conduct some numerical experiments to show the convergence of the unsteady solution to constant steady state solution using nonlinear Neumann feedback control law for different values of feedback parameters c 0 . Moreover, we obtain the order of convergence for both state variable and feedback control law. For complete discrete scheme, the time variable is discretized by replacing the time derivative by difference quotient. Let 0 < k < 1 denote the time step size and t n = nk, where n is nonnegative integer. For smooth function φ defined on [0, ∞), k . Based on backward Euler method, we seek W n , an approximation of w(t) at t = t n as with W 0 = w 0h . Using Brouwer's fixed point theorem, there exists a solution of the discrete problem (4.1). Here for implicit method, CFL condition is not needed as it is an implicit scheme, but at each time level, we now solve a nonlinear algebraic system using the Newton's method taking initial guess as W n−1 .   From Figure 1, we can easily see that with the control (1.12), the solution for the problem (1.13) in L 2 norm goes to zero exponentially. In Table 2, it is noted that the order of convergence of nonlinear Neumann feedback control law (1.12) is 2, which theoreticallly 3/2 in Theorem 3.3. Also numerically it can be shown that for other values of c 0 , the system (1.13) is stabilizable. Since the exact solution is unknown in this case, we have taken very refined mesh solution as exact solution and derived the order of convergence. From Table 1, it follows that L 2 and H 1 orders of convergence for state variable w(t) are 2 and 1, respectively, which confirms our theoretical results in Theorem 3.3.
From Figure 3, it is observed that steady state solution w d = 0 is unstable in the first case denoted as "Uncontrolled solution ". But in other cases, using the control law (1.12), it is shown that w goes to zero. Figure 4 indicates how nonlinear control law (1.12) behave with time. Below we discuss another example, where steady state solution is not a constant for the forced Burgers' equation. It will be shown that, even with the linear control law, system can be stabilizable computationally.  We now consider a case when the steady state solution is not constant: where, f ∞ and g ∞ , independent of t are functions of x 1 and x 2 only. Corresponding equilibrium or steady state solution u ∞ of the unsteady state problem satisfies Let w = u − u ∞ . Then w satisfies w t − ν∆w + u ∞ (∇w · 1) + w(∇u ∞ · 1) + w(∇w · 1) = 0 in (x, t) ∈ Ω × (0, ∞), (4.4) ∂w ∂n (., t) = v 2 (x, t), on ∂Ω × (0, ∞), In this case, backward Euler method applied to (4.4) for time discretization yields (∂ t W n , ϕ h ) + ν(∇W n , ∇ϕ h ) + u ∞ ∇W n · 1, ϕ h + W n ∇u ∞ · 1, ϕ h + W n (∇W n · 1), ϕ h )

5)
where v 2 = 1 ν c 0 W n . For the numerical experiment, we take ν = 0.1, u ∞ = −0.2x 1 , f ∞ = 0.04x 1 and g ∞ = −0.2n 1 with w 0 = sin(πx 1 )sin(πx 2 ) + 0.  From the first draw line in Figure 5, we observe that nonconstant steady state solution is uncontrollable with zero Neumann boundary. But, using the linear control law 1 ν c 0 W n with c 0 = 1, u ∞ = −0.2x 1 is stable which is documented in Figure 5. Figure 6 shows that the linear control law 1 ν c 0 W n decays to zero as time increases. However, we do not have a theoretical result to substantiate this observation. We believe that the system is locally stabilizable with this linear control law.

Concluding Remarks.
In this paper, global stabilization results for the two dimensional viscous Burgers' equation are established in L ∞ (H i ), i = 0, 1, 2 norms, when the steady state solution is constant. Global Stabilization results are also discussed for the semidiscrete solution in the context of C 0 -conforming finite element method. Optimal error estimates in L ∞ (L 2 ) and in L ∞ (H 1 ) for the state variable are established. Further, error estimate for the feedback controller is also shown. Now under addition of forcing function in the two dimensional Burgers' equation, the steady state solution is no more constant and as such the present analysis does not hold for nonconstant steady state case. Hence the analysis for two dimensional generalized forced viscous Burgers' equation will be our future project.