Global Stabilization of BBM-Burgers' Type Equations by Nonlinear Boundary Feedback Control Laws: Theory and Finite Element Error Analysis

In this article, global stabilization results for the Benjamin-Bona-Mahony-Burgers' (BBM-B) type equations are obtained using nonlinear Neumann boundary feedback control laws. Based on the $C^0$-conforming finite element method, global stabilization results for the semidiscrete solution are also discussed. Optimal error estimates in $L^\infty(L^2)$, $L^\infty(H^1)$ and $L^\infty(L^\infty)$-norms for the state variable are derived, which preserve exponential stabilization property. Moreover, for the first time in the literature, superconvergence results for the boundary feedback control laws are established. Finally, several numerical experiments are conducted to confirm our theoretical findings.

Based on distributed and Dirichlet boundary control in feedback form through Riccati operator, local stabilization results for the Burgers' equation with sufficiently small initial data are established in [4], [5]. Moreover, for local stabilization results using Neumann boundary control, we refer to [7], [11] and [12]. It is to be noted that for viscous Burgers' equation, global existence and uniqueness results with Dirichlet and Neumann boundary conditions are derived for any initial data in L 2 in [18]. Subsequently, based on nonlinear Neumann and Dirichlet boundary control laws, global stabilization results for the Burgers' equation are proved using a suitable application Lyapunov type functional in Krstic [14], Balogh and Krstic [1]. Later on, adaptive (when ν is unknown) and nonadaptive (when ν is known) stabilization results for generalized Burgers' equations are established in [17], [23] and [24] with different types of boundary conditions. For existence of solution to the problem (1.1)-(1.4), when µ = 0, we refer to [1] and [17].
For stabilization of the BBM-B equation, the authors in [10] have shown global stabilization results corresponding to µ = 1 with zero Dirichlet boundary condition at one end and Neumann boundary control on the other end. Using a reduced order model, distributed feedback control for the BBM-B equation is discussed in [21]. Also, quadratic B-spline finite element method followed by linear quadratic regulator theory to design feedback control, is used to stabilize in [22] without any convergence analysis. In [15], we have shown that, under the uniqueness assumption of the steady state solution, the steady state solution of the problem (1.1) with zero Dirichlet boundary condition is exponentially stable.
In this paper, we discuss global stabilization results using nonlinear Neumann feedback control law. Our second objective is to apply C 0 -finite element method to the stabilization problem (1.1)-(1.4) using nonlinear Neumann boundary control laws and discuss convergence analysis. Since to the best of our knowledge, there is hardly any discussion in the literature on the rate of convergence, hence, in this paper, an effort has been made to prove optimal order of convergence of the state variable along with superconvergence result for the feedback control laws. The main contributions of this article are summarized as: • Global stabilization for problem (1.1)- (1.4), that is, convergence of the unsteady solution to the problem (1.1) to its constant steady state solution under nonlinear Neumann boundary control laws (1.2)-(1.3) is proved.
• Based on the C 0 -conforming finite element method, global stabilization results for the semidiscrete solution are discussed and optimal error estimates are established in L ∞ (L 2 ), L ∞ (H 1 ), and L ∞ (L ∞ ) norms for the state variable. Moreover, superconvergence results are derived for the nonlinear Neumann feedback control laws.
• Finally, some numerical experiments are conducted to confirm our theoretical results.
For related issues of finite element analysis of the viscous Burgers' equation using nonlinear Neumann boundary feedback control law, we refer to our recent article [16]. Compared to [16], special care has been taken to establish global stabilization results in L ∞ (H i )(i = 0, 1, 2) norms as µ → 0. It is further observed that the decay rate for the BBM-B type equation is less than the decay rate for the viscous Burgers' equation and as the dispersion coefficient µ approaches zero, the decay rate also converges to the decay rate for the Burgers' equation. Finite element error analysis holds for fixed µ.
For the rest of this article, we denote by H m (I = [0, 1]) the standard Sobolev spaces with norm · m and for m = 0, · denotes the corresponding L 2 norm. The space L p ((0, T ); X) 1 ≤ p ≤ ∞, consists of all strongly measurable functions v : When there is no confusion, L p ((0, T ); X) is simply denoted by L p (X). The equilibrium or steady state solution u ∞ of (1. Note that any constant w d is a solution of the steady state problem (1.5)-(1.6). Without loss of generality, we assume that w d ≥ 0.
w(x, 0) = w 0 (x), x ∈ (0, 1), (1.10) where, w is the state variable and v 0 and v 1 are feedback control variables. Since for the problem with zero Neumann boundary condition, the steady state constant solution w d is not asymptotically stable, we plan to achieve stabilization result through boundary feedback law. The present analysis can be easily extended to the problem with one side control law say for example: when w(0, t) = 0, w x (1, t) = v 1 (t), see [10]. The weak formulation of the problem (1.7)-(1.10) is to seek w(t) ∈ H 1 (0, 1), w t ∈ L 2 (L 2 ) and µw t ∈ L 2 (H 1 ) such that for almost all t > 0 with w(x, 0) = w 0 (x). For motivation to choose the control laws v 0t and v 1t using construction of Lyapunov functional, see [14]. Based on the nonlinear Neumann control law propose in our earlier article in Burgers' equation, see [16], which is a modification of control law in [14], we now choose the feedback control law as where K 0 and K 1 represent feedback control laws, and c 0 and c 1 are positive constants. Using (1.12)-(1.13), we obtain a typical nonlinear problem (1.7)-(1.10) with boundary conditions (1.12)-(1.13). Its weak formulation (1.11) becomes Throughout the paper, we use the following norm which is equivalent to the usual H 1 -norm: and C is used as a generic positive constant. We now recall some results to be use in our subsequent sections.
In addition, the following regularity result holds Subsequently for our error estimates in L ∞ (L ∞ ) norm, we further assumed that w(t) ∈ W 2,∞ with its norm denoted by · 2,∞ .
The rest of the article is organized as follows. Section 2 deals with global stabilization results and the existence and uniqueness of strong solution. Section 3 is devoted to optimal error estimates for the semidiscrete solution with superconvergence results for feedback controllers. Finally in section 4, some numerical examples are considered to confirm our theoretical results.

Stabilization and continuous dependence result
In this subsection, we discuss a priori bounds for the problem (1.14) and derive stabilization results. In addition, these estimates are needed to prove optimal error estimates for the state variable and feedback controllers. All estimates throughout the paper are valid for the same α with The other terms in (2.16) are bounded by Therefore, from (2.16), we arrive at Now multiply the above inequality by e 2αt to obtain By the Gronwall's inequality, it follows from above with a use of Lemmas 2.1 and 2.3 that Also after putting χ = w t in the weak formulation (1.14), we arrive at Therefore, we can find the value of w t (t) Hence, we arrive at Multiply the above inequality by e −2αt to complete the proof.
Proof. Form the L 2 -inner product between (1.7) and −w xxt to obtain where we use the bound of w 2 (i, t) and w 4 (i, t) for i = 0, 1 from Lemma 2.2. Multiply (2.17) by e 2αt to obtain Integrate from 0 to t and then multiply the resulting inequality by e −2αt with a use of Lemmas 2.2 and 2.4 to arrive at This completes the proof.

Continuous dependence property
Below, we show a continuous dependence property from which uniqueness follows.
Lemma 2.6. For two different initial conditions w 10 and w 20 ∈ H 1 (0, 1), the following continuous dependence property holds where z = w 1 − w 2 , and E 4 (t) is same as in (2.23).
Proof. Let w 1 and w 2 be two solutions of (1.7) with boundary conditions (1.12), (1.13) and initial conditions w 10 and w 20 , and set z = w 1 − w 2 . Then, z satisfies In its weak formulation, seek z ∈ H 1 such that (2.22), and bound the fourth and fifth terms on the left hand side, respectively, as Now to bound the other terms on the left hand side of (2.22), we rewrite the following terms as for i = 0, 1 Therefore, from (2.22), we arrive at Setting Applying Gronwall's inequality to the above inequality yields

ds .
A use of Lemmas 2.2-2.4 gives the desired result.
As a consequence, when w 10 = w 20 , it follows that w 1 (t) = w 2 (t) for all t > 0. Hence, the solution is unique.

Finite element approximation
In this section, we discuss semidiscrete Galerkin approximation keeping the time variable continuous. Moreover, optimal error estimates for the state variable and superconvergence results for feedback controllers are established.
For any positive integer N, We define a finite dimensional subspace V h of H 1 as follows where P 1 (I j ) is the set of linear polynomials in I j . Now, the corresponding semidiscrete formulation for the problem , an approximation of w 0 . For our analysis, we assume that w 0h is the H 1 projection of w 0 onto V h . Now since V h is finite dimensional, the semidiscrete problem (3.1) leads to a system of nonlinear ODEs. Then an appeal to the Picard's theorem yields the existence of a unique solution w h (t) in t ∈ (0, t * ) for some t > 0. Since from Lemma 3.1, w h (t) is bounded for all t > 0, using a continuation argument, the global existence of w h (t) is established. Below, we state four Lemmas for the semidiscrete problem (1.7)-(1.10), which imply global stabilization result for the semidiscrete solution.
and β is the same as in (2.2).
Proof. For the proof we can proceed as in continuous case.

Error estimates
To bound the error, we first introduce an auxiliary projectionw h ∈ V h of w through the following form where λ is some fixed positive number. For a given w ∈ H 1 , the existence of a uniquew h follows from the Lax-Milgram Lemma. Let η := w −w h be the error involved in the auxiliary projection. Then, the following standard error estimates hold η(t) j ≤ Ch min(2,m)−j w(t) m , and η t (t) j ≤ Ch min(2,m)−j w t (t) m , j = 0, 1 and m = 1, 2. (3.9) and For a proof, we refer to Thomée [25]. In addition, for proving optimal error estimates, we need the following estimates of η and η t at the boundary points x = 0, 1 whose proof can be found out in [16] and [19].
Proof. Set χ = θ in (3.11) to obtain where I 4 (θ) and I 5 (θ) are last two summation term respectively. The first term on the right hand side of (3.12) is bounded using the Cauchy-Schwarz inequality and the Young's inequality in where constant > 0 we choose later. For the second term on the right hand side of (3.12), integration by parts, the Cauchy-Schwarz inequality, and Young's inequality yield For the third term, we note that First subterms of the fourth and fifth term on the right hand side of (3.12) are bounded by For second subterm of the fourth term on the right hand side, we note that for i = 0, 1 Using Young's inequality, implies that for i = 0, 1 Again, a use of Young's inequality yields Hence, the contribution of the second subterm of the fourth term on the right hand side of (3.12) after applying Lemmas 2.2, 2.4 and 3.5, can be bounded as Expanding the second subterm of the fifth term, we note that for i = 0, 1 and using Lemmas 2.2 and 2.4, we obtain Also, it holds that Rewrite and use the Young's inequality to obtain Similarly, Hence, from (3.12), we arrive using Lemmas 2.2, 2.4, 3.2, 3.4 and 3.5 at Multiply the above inequality by e 2αt . Use Poincaré-Wirtinger's inequality where φ(t) = |||w(t)||| 2 + w xx (t) 2 + ∆ h w h (t) 2 . Now integrate from 0 to t and choose = β1 Then, an application of Gronwall's inequality to (3.14) shows This completes the proof.
Lemma 3.7. Assume that w 0 ∈ H 2 (0, 1). Then, there exists a positive constant C independent of h such that Proof. Set χ = θ t in (3.11) to obtain The first term on the right hand side of (3.17) is bounded by For the second term I 2 (θ t ), first rewrite it as A use of Young's inequality shows For the third term I 3 (θ t ) on the right hand side of (3.17), we first rewrite it as and then an application of Young's inequality with Lemmas 2.2, 2.4 and 3.2 yields The first subterm of the fourth term I 4 (θ t ) on the right hand side of (3.17) can be rewritten for i = 0, 1 as Hence, we obtain For second subterm of the fourth term on the right hand side, we note that for i = 0, 1 Using Lemma 2.2, it follows that for i = 0, 1 Also, we obtain We note that The first subterm of the fifth term I 5 (θ t ) on the right hand side is bounded by For the second subterm of the fifth term I 5 (θ t ), we note that for i = 0, 1 Using Lemmas 2.2 and 3.5, it follows that nonlinear boundary terms as in Lemma 3.6 to arrive at Integrate from 0 to t and then multiply the resulting inequality by e −2αt to obtain t 0 e 2αs θ x (t) 2 + w 2 ht (0, s)θ 2 (0, s) + w 2 ht (1, s)θ 2 (1, s) + θ 4 (0, s) + θ 4 (1, s) ds Use Young's inequality and Lemma 2.2 to obtain Again using Young's inequality and Lemma 2.2, we arrive at Bounding in a similar fashion as in Lemma 3.6, we obtain a bound for the nonlinear boundary terms as follows Finally, apply Grönwall's inequality to (3.18) to arrive using Lemmas 2.2, 2.4, 3.1-3.3 and 3.6 at This completes the proof.
Remark 3.2. As a consequence of Lemma 3.7, we obtain superconvergence result for |||θ(t)||| which depends on 1 √ µ . However, for proving optimal estimate, only one modification may be made to which does not depend on 1 √ µ . Now using triangle inequality with Lemmas 3.6 and 3.7 and (3.19), we obtain the following result.
Theorem 3.1. Let w 0 ∈ H 2 (0, 1). Then, the following error estimates hold for the state and control variables

20)
where r = 0, 1 and Proof. The proof follows from Lemmas 2.4, 3.6 and 3.7 with a use of triangle inequality and (3.9).
Proof. From Lemma 3.7, we obtain a superconvergence result for |||θ(t)|||. Using the Poincaré-Wirtinger's inequality, it follows that θ(t) L ∞ (I) ≤ C |||θ(t)||| . Now a use of triangle inequality with estimates of η(t) L ∞ and θ(t) L ∞ , we arrive at the estimate (3.21). To find (3.22), we note that the error in the control law is given by Similarly, it follows that This completes the proof.

Numerical experiments
In this section, we discuss the fully discrete finite element formulation of (1.7) using backward Euler method with Neumann boundary control laws. Here, the time variable is discretized by replacing the time derivative by difference quotient. Let W n be the approximation of w(t) in V h at t = t n = nk. 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, ∞), set φ n = φ(t n ) and∂ t φ n = (φ n −φ n−1 ) k . Using backward Euler method, the fully discrete scheme corresponding {W n } n≥1 ∈ V h is a solution of with W 0 = w 0h . At each time level t n , the nonlinear algebraic system (4.1) is solved by Newton's method with initial guess W n−1 . For implicit scheme (4.1) in our case, CFL condition is not needed. We take time step k = 0.0001 and mesh size h = 1/60.  From the line denoted as 'uncontrolled soln' in Figure 1, we can clearly observe that W n does not go to zero, that is, constant steady state solution w d = 3 is not asymptotically stable with zero Neumann boundaries. We now observe that for various combination of c 0 and c 1 , the discrete solution goes to zero exponentially, see Figure 1. Moreover from Figure 1, we can see that the optimal decay rate α (with w d = 3), 0 < α ≤ 1 2 min ν µ+1 , ν 2µ+ν , ν(4+ci) ν+(4+ci)µ (i = 0, 1) happens when c 0 = 1 = c 1 , which verify our theoretical result in Lemma 2.1. When c i (i = 0, 1) < 1, then decay rate for the state is slow compare to the case when c i (i = 0, 1) ≥ 1. and v 1 (t) (|v 0 (t n ) − v 0h (t n )| and |v 1 (t n ) − v 1h (t n )| ) in L ∞ norm at t = 1. Exact solution is obtained through refined mesh solution. Figures 2 and 3 indicate the error plot for the state variable w in L 2 and L ∞ norms respectively, for various values of c 0 and c 1 . We can easily observe from Figure 2 that the convergence rate in the L 2 -norm for error in state variable is of order 2 as predicted by Theorem 3.1. From Figure  3, it is also noticeable that the order of convergence for error in state variable in L ∞ norm is 2 as expected from Theorem 3.2. For error in feedback controllers at x = 0 and x = 1, it is observed from Figures 4 and 5 that for various values of c 0 and c 1 , the order of convergence is 2 which confirms the result in Theorem 3.2.
In Figures 6 and 7, we present the behavior of the feedback controllers at x = 0 and x = 1 with respect to time for various positive values of c 0 and c 1 . Absolute value of the feedback controllers go to zero as time increases. So for c i (i = 0, 1) < 1 in the feedback control law, it will take more time for the control and state to settle down to zero (See Figures 1, 6 and 7). The next example consists of different type feedback control which is stated below. In the following example, we consider the solution of (1.7) with one part zero Dirichlet boundary and another part different Neumann conditions.   Figure 8 shows that solution with zero boundary conditions (w(0, t) = 0 and w x (1, t) = 0) oscillate. But using above mentioned type of control with different values of c 1 , solution goes to zero. With the initial condition of Example 4.2, decay of the state w in L 2 -norm varying µ with fixed ν = 0.1, c 0 = 1 = c 1 is shown in Figure 9. We observe that as µ decreases, L 2 -norm of the state w for BBM-B equation converges to the L 2 -norm of the state w with µ = 0 that is to the L 2 -norm of the state of Burgers' equation.

Conclusion
In this article, under the assumption of the existence of solution, we show stabilization estimate in higher order norms which is crucial to obtain optimal error estimates in the context of C 0conforming finite element analysis. Optimal error estimates for the state variable w in L ∞ (L 2 ), L ∞ (H 1 ) and L ∞ (L ∞ ) norms are established. Furthermore, superconvergence results for error in feedback controllers are also proved. Following points which are itemized below will be addressed in a separate paper.
• When the coefficient of viscosity is unknown (in the case of adaptive control), we believe that the control law as in Smaoui [23] will also work for BBM-B equation. Also when ν = 0, it is interesting to extend the analysis modifying the control law appropriately.
• On the other hand, we have not discussed rigorously the existence of solution of problem (1.7)-(1.10), namely Theorem 1.1.
• In addition, for the fully discrete scheme (4.1), it is interesting to know the large time behavior of the solution and how the corresponding time step size k behaves in error estimates for fully discrete solution in addition to the space step size h.