Boundary control of stochastic Korteweg-de Vries-Burgers equations

The boundary control problem is considered for stochastic Korteweg-de Vries-Burgers equations. First, a boundary controller is proposed, and a criterion is obtained for mean square exponential stability by using the Lyapunov functional method and inequality techniques. Then, when there exist uncertainties in the system parameters, the robust mean square exponential stability is considered, and a sufficient criterion is obtained. Furthermore, if there are also additive noises in the considered system, the H-infinity performance is investigated and a sufficient condition is obtained to ensure the mean square H-infinity performance. Numerical examples illustrate the validity of the theoretical results.

with unidirectional motion, and it is also a typical representative of nonlinear dispersive equations. The Burgers equation can simulate the propagation and reflection of shock waves, and it is also a typical representative of nonlinear dissipative equations. The study of KdV equations or Burgers equations has been an active research topic because of its potential applications [3,4,6,22,28,40].
In many actual physical problems, the case of pure dispersion and pure dissipation is really rare. When adding a diffusion term, the KdV equation becomes the Korteweg-de Vries-Burgers (KdVB) equation. KdVB equations are widely applied in the fields of engineering, fluid mechanics, biological mathematics and so on. For example, in fluid mechanics, the corresponding KdVB equation is where y(x, t) represents the height from the horizontal plane to the free surface of water wave, t represents time, x represents the distance of water wave moving along the direction of propagation, ε is the dissipation coefficient and δ is the dispersion coefficient.
For partial differential equations, the control problem has been a research hotspot [10,21,23,30,37,41]. Generally, the control strategies for partial differential systems can be divided into two categories, the distributed control and the boundary control. Distributed control means that the actuators are placed at each point of the spatial domain, which is not easy to be realized in engineering [18]. Boundary control only needs to apply the actuators on the boundary of the spatial region, which has the characteristics of low cost and easy to realize. In recent years, boundary control has been studied by many scholars [1,5,8,9,12,13]. For the boundary control of deterministic systems, a classic method is the backstepping method. Many researchers have made a lot of excellent results for deterministic KdVB equations which have taken on an upsurge [2,[32][33][34]. Global boundary stabilization of KdVB equations was studied in [25]. Smaoui et al. proposed a boundary control strategy for the generalized KdVB equations on the interval [0, 1] in [31]. For the boundary control of stochastic systems, it is difficult to use the backstepping method. There exists an essential difficulty in dealing with the Itô's formula. Then, it is worth considering to use the Lyapunov functional method for the boundary control of stochastic systems.
In real applications, there are uncertainties of system parameters in modeling, which will naturally affect the behavior analysis of the system. Since parameter uncertainties may destroy the stability, the robust stability for partial differential equations is an interesting topic [15,26]. Sakthivel studied the problem of robust global stabilization by a nonlinear boundary feedback control for KdVB equations on the domain [0, 1] in [29]. Moreover, there may also be external noises, which enter the system in the form of non-Gaussian additive noises, and they will also affect the behavior of the system. H-infinity control is a feasible strategy to reduce the effect of external disturbances on system states (or system output), which has been widely concerned [11,20,24,38,39]. In [19], Kang et al. considered the observer-based H-infinity control problem for a stochastic KdVB equation under point or averaged measurements.
Random noises are ubiquitous. KdVB equations may be disturbed by random environmental factors, so it is important to study stochastic KdVB (SKdVB) equations. Stochastic systems driven by Brownian motions have received extensive attention in population growth, electronic circuits, financial investment and so on. The stability of this kind of systems has been an important subject [7,16,17]. Although many researchers have made excellent results for deterministic KdVB equations, there are few results on the SKdVB equations. In [19], Kang et al. considered the distributed control problem for a SKdVB equation, not the boundary control. The research results of SKdVB equations with boundary control are rare. In this research, we have made a meaningful attempt. The boundary controller is designed, and the sufficient criteria for mean square exponential stability, robust stabilization and H-infinity performance of SKdVB equations are obtained.
There will be many difficulties and challenges when studying the boundary control of SKdVB equations. It is difficult to deal with the nonlinear terms in SKdVB equations. The boundary control strategy only applies a controller on the boundary of the spatial region and does not directly affect the state equations of the system. Using the general systems analysis methods, it is difficult to analyze the behavior of the system. In addition, due to the addition of stochastic items, it brings extra difficulties to the performance analysis of the system. Meanwhile, it is not feasible to use the single-point controller or the integral controller in [25,36] to study the stability of SKdVB equations. The results in [25] are suitable for small noises and small disturbances, which limit the application of the theoretical results. Moreover, the integral controller in [36] can't handle the nonlinear terms in SKdVB equations. Therefore, a new controller design scheme should be proposed to study the stability of SKdVB equations.
Basing on the above discussions, we aim to investigate the boundary control for SKdVB equations. A boundary controller is designed first based on the combination of the single-point form and the integral form. By the Lyapunov functional method and inequality techniques, a sufficient criterion is established to ensure the mean square exponential stability. Then, when there exist uncertainties in the system parameters, the robust boundary stabilization is considered. In addition, when the system is disturbed by external additive noises, a criterion is presented to ensure the mean square H-infinity performance under our designed boundary controller. The major contributions are listed as follows.

A boundary controller is designed based on the
combination of the single-point form and the integral form. Under the designed controller, a sufficient condition is presented to achieve mean square exponential stability of SKdVB equations. 2. A sufficient condition is obtained to achieve robust stabilization of the uncertain SKdVB equations. 3. A sufficient condition is obtained to achieve mean square finite horizon H-infinity performance of SKdVB equations with additive noises.
Notation: Throughout this paper, W l,2 ([0, L]; R n ) stands for a Sobolev space of absolutely continuous functions σ (x) : [0, L] → R n with square integrable derivatives d l σ (x) dx l up to order l. R n is the n-dimensional Euclidean space. L 2 (0, l) stands for the Hilbert space of square integrable scalar functions over (0, l) with the corresponding norm z 2 L 2 (0,l) = l 0 |z(x)| 2 dx. Let (Ω, F, P) be a complete probability space adapted to the filtration {F t } t≥0 satisfying the usual conditions. E denotes the mathematical expectation with respect the probability measure P.

Preliminaries and model formulation
We consider the following SKdVB equation where ε, δ and C are positive constants, y(x, t) is the state, x and t are the space variable and time variable, φ(x) is the initial state of the system, and u(t) is the boundary control input, W (t) is the one-dimensional standard Brownian motion, which satisfies For the further analysis, the following definition and lemmas are needed. (1) is said to be mean square exponentially stable if there exist positive numbers M > 0 and η > 0 such that

Definition 1 ([19]) System
Lemma 2 (Itô's formula [27]) Let X t be an Itô progress with the stochastic differential is also an Itô progress and satisfies Moreover,

Main results
In this section, first, a boundary controller is designed and a sufficient condition is obtained to ensure the mean square exponential stabilization for SKdVB equations. Then, the robust stabilization is studied for the uncertain SKdVB equations. Finally, H-infinity control is investigated for the disturbed SKdVB equations.

Mean square exponential stabilization for SKdVB equations
In this subsection, a boundary controller is designed for system (1). Based on the designed controller, a sufficient condition is obtained to ensure the mean square exponential stabilization. We take the following boundary controller for system (1) where K 1 and K 2 are the boundary control gain. We present the following theorem to guarantee that system (1) achieves the mean square exponential stability under the boundary controller (2). Theorem 1 If constants K 1 and K 2 satisfy the following inequalities then system (1) under the boundary controller (2) achieves mean square exponential stability.

Thus, we obtain
that is, which completes the proof.
Remark 1 For SKdVB equations, a novel controller design scheme (2) is presented to instead of using the single-point controller or the integral controller alone which was adopted in [25,36]. The reasons are as follows. When the single-point controller u(t) = k 1 y 3 (1, t) + k 2 y(1, t) in [25] is used to study the stability of system (1), a stability criterion is obtained.
The following inequality must be held: C 2 − επ 2 2 < 0, and this restriction implies that this criterion is workable only for small random interference C. This limits the application of the theoretical results, and many systems may not meet this criterion, so this design is inappropriate. Meanwhile, the integral controller u(t) = K 1 0 y(x, t)dx in [36] cannot handle the nonlinear items in system (1). Therefore, the pure integral boundary controller is not workable here.
Remark 2 It should be noted that the controller (2) with integral form is designed to suppress the item C 2 . The existence of y 2 (1, t) in controller design is to deal with the third-order term in system (1) on the stability.

Remark 3
In system (1), according to criterion (3), one can see that the larger the dispersion coefficient δ, the more difficult it is to find the satisfied K 1 . In the real applications, the dispersion coefficient is equivalent to the speed of water wave, that is, the faster the amplitude of water wave, the more difficult to make it stable. At the same time, the larger C means the large random disturbances, and the more difficult it is to find the satisfied K 2 , the more difficult it is to stabilize the system.

Robust stabilization for SKdVB equations
In this subsection, a sufficient condition is derived for robust stabilization of SKdVB equations under our designed boundary controller (2).
When the system (1) have uncertainties in the parameters ε and C, and we assume that they satisfy the following conditions where ε 0 > 0, C 0 > 0, η 1 > 0 and η 2 > 0 are given constants, ε 0 and C 0 represent the nominal value (see, for instance, [29]), 1 η 1 and 1 η 2 represent the maximum magnitude of the allowable uncertainty, respectively.
System (1) is rewritten as where λ ∈ [−1, 1] represents the uncertainty. In order to derive a sufficient condition for mean square exponential stability of the uncertain system (14), we introduce new signals z, ω and let z = y x x , ω = λz. The system (14) is written as where ω represents the uncertainty. For the subsequent discussion, the following lemma about parameter uncertainties is given.

Lemma 4
Let α = | y 2η 1 y xx |, then for z = y x x , ω = λz, we have the following inequality Proof The proof is motivated by the one in [29]. From z = y x x , we have From ω = λz, we have Substituting (17) and (18) into (16), which is guaranteed by λ ∈ [−1, 1]. This proof is complete.
Theorem 2 Suppose that constants K 1 and K 2 satisfy the following inequalities then uncertain system (14) with the boundary controller (2) achieves mean square exponential stability.
Proof Consider the following functional Then, along system (14) and using inequality (16), we have Using integration by parts, applying the completion of squares to (22) and the boundary conditions of system (14), we have Noting y(0, t) = 0, by virtue of Lemma 1, we obtain Substituting (24) into (23) gives Then it follows from condition (20) that whereλ = (ε 0 η 1 +1)π 2 4η 1 . Hence, the uncertain terms were eliminated. Following a similar line to that of Theorem 1, and the results can be carried out.

Remark 4
The introduction of the scaling α enables us to eliminate the uncertainty λ in the time-derivative of the Lyapunov function, which is motivated by the one in [29].

H-infinity control for SKdVB equations
When non-Gaussian additive noises enter SKdVB equations, properties of the system may be degraded or even destroyed. Under this situation, the abilities of the system to resist the external disturbances need to be studied, and that is the H-infinity control problem.
When system (1) is subject to exogenous disturbances, it turns into where ν(x, t) is the external disturbance, and we assume that for a positive time constant t f . Now, we present the definition of mean square Hinfinity performance for system (27). The following theorem gives the result of the Hinfinity performance for system (27).
Theorem 3 Assume that constants K 1 and K 2 satisfying then system (27) with the boundary control (2) has mean square H-infinity performance over the finite horizon [0, t f ].
Proof Taking the Lyapunov functional as the one defined in (4), and computing dV along system (27) yields For the given constants γ and t f , using the fact that V (y(·, 0)) = 0 when y(x, 0) = 0, we have It follows from (7) that By using the techniques as in the proof of Theorem 1, condition (29) is equivalent which completes our proof.

Numerical example
In this section, we give three numerical examples to verify the effectiveness of our results.
Example 1 Consider the following SKdVB equation The initial value is taken as The boundary conditions are We adopt the following boundary controller Let K 1 = −7, K 2 = −3, one can verify that condition (3) holds. By virtue of Theorem 1, it is obvious that system (33) achieves mean square exponential stability. Under the boundary controller (36), Ey 2 (x, t), as shown in Fig. 1 is mean square exponential stable. Moreover, Fig. 2 shows the mean square state norm of system (33).
Example 2 Consider the following uncertain SKdVB equation  (37) The initial value is taken as The boundary conditions are Adopt the following boundary controller Let K 1 = −17, K 2 = −6, it can be observed that condition (20) holds. By virtue of Theorem 2, one obtains that uncertain system (37) achieves mean square exponential stability.
For the numerical simulation, we choose the unknown parameter λ = sin(π xt). One can easily see that system (37) achieves robust mean square exponential stability. Figure 3 shows the system state in mean square sense (37). Moreover, Fig. 4 shows E y(·, t) 2 under the boundary controller (40).
We adopt the following boundary controller  Set the disturbed attenuation level γ = 0.9, and t f = 5. Under the initial value y(x, 0) = 0, the following ratio is calculated when taking 100 sample paths, Let K 1 = −12, K 2 = −38, we can verify that condition (29) holds. By virtue of Theorem 3, we know that disturbed system (41) achieves mean square H-infinity performance over the finite horizon [0, 5].

Conclusions
In this paper, boundary control for SKdVB equations has been addressed. First, we design a boundary controller, and a sufficient condition is obtained for mean square exponential stability by applying the Lyapunov functional method and inequality techniques. Then, the boundary controller is applied to achieve robust mean square exponential stability for uncertain SKdVB equations, and the corresponding sufficient condition is obtained. In addition, if there are also additive noises in the considered system, the H-infinity performance is investigated and a sufficient condition is obtained to ensure the H-infinity performance in the mean square sense. Finally, numerical simulations show the validity of our obtained results.