On the Control of the 2D Navier–Stokes Equations with Kolmogorov Forcing

(is paper is devoted to the control problem of a nonlinear dynamical system obtained by a truncation of the two-dimensional (2D) Navier–Stokes (N-S) equations with periodic boundary conditions and with a sinusoidal external force along the x-direction. (is special case of the 2D N-S equations is known as the 2D Kolmogorov flow. Firstly, the dynamics of the 2D Kolmogorov flow which is represented by a nonlinear dynamical system of seven ordinary differential equations (ODEs) of a laminar steady state flow regime and a periodic flow regime are analyzed; numerical simulations are given to illustrate the analysis. Secondly, an adaptive controller is designed for the system of seven ODEs representing the approximation of the dynamics of the 2D Kolmogorov flow to control its dynamics either to a steady-state regime or to a periodic regime; the value of the Reynolds number is determined using an update law. (en, a static sliding mode controller and a dynamic sliding mode controller are designed for the system of seven ODEs representing the approximation of the dynamics of the 2D Kolmogorov flow to control its dynamics either to a steady-state regime or to a periodic regime. Numerical simulations are presented to show the effectiveness of the proposed three control schemes. (e simulation results clearly show that the proposed controllers work well.


Introduction
In this work, we study the dynamics as well as the adaptive and the sliding mode control problem of seven-mode truncation system of the 2D Navier-Stokes equations with periodic boundary conditions and a sinusoidal external force along the x-direction. is type of forcing is known as the Kolmogorov forcing and the resulting flow is known as the 2D Kolmogorov flow.
In 1958, Kolmogorov [1] introduced the 2D Kolmogorov flow as an example to study transition to turbulence. e model has been successfully used to study 2D turbulent flows in atmospheric, oceanic, and astrophysical flows due to the weak dependence of the velocity field on the third dimension [2,3]. In the field of magnetohydrodynamics, Kolmogorov flow has been extensively used and was easily reproduced by suitably placed electrical and magnetic fields. Bondarenko et al. [4] observed this flow inside a specific electrically conducting fluid driven by an electromagnetic field [5]. In the literature, linear and nonlinear stability analysis of this flow was investigated for different domain sizes and forcing wave numbers [6][7][8][9]. Numerical simulations and investigations helped in the advancement of the understanding of Kolmogorov flow. In particular, it has been shown that the Kolmogorov flow dynamics exhibits complex structures transforming periodic states to chaotic attractors through a sequence of bifurcations including period doubling, period tripling and gluing bifurcations [10][11][12][13][14][15]. Some of these structures were realized in experimental laboratories [4,16].
In the last four decades, several reduced order models that approximate the dynamics of the 2D Kolmogorov flow were constructed using the Fourier Galerkin approach [10,11,[17][18][19][20][21][22][23][24][25]. Franceschini and Tebaldi [21] constructed a five-mode truncation ODE system of the 2D Kolmogorov flow when the external force acts on the mode (2, −1). In [21], a number of steady states and Hopf bifurcations were observed up to a Reynolds number equal to 50. Later on, in 1987, She [12] investigated the metastability and vortex pairing of Kolmogorov flow when the external force along the x-direction acts on the mode (0, 8). Also, Nicolaenko and She [14] studied the dynamics of the coherent structures, homoclinic cycles, and vorticity expolosion in Kolmogorov flow. In 1997, using the Karhunen-Loéve decomposition and symmetry, Smaoui and Armbruster [11] used a computationally effective method to construct a reduced order system of nonlinear ODEs that approximates the dynamics of Kolmogorov flow when the external force acts on the mode (0, 2). One decade later, Chen [23] and Chen et al. [24,25] obtained a reduced order system of ODEs when the force acts on the mode (0, 4).
Recently, the control problem of nonlinear PDEs with periodic boundary conditions has been the subject of many research studies [26][27][28][29][30][31][32][33][34][35][36]. Because of the infinite dimensional nature of these PDEs, the practical implementation of such controllers is a very difficult task. As a consequence, attempts were made to approximate these PDEs based on ODE approximations. e idea of inertial manifold to obtain such reduced systems of ODEs was introduced by Foias et al. [37]. Other efforts to construct systems of ODEs that capture the dynamics of the original PDEs were made by other researchers [26][27][28][29][38][39][40][41][42]. Smaoui and Zribi [26][27][28] constructed reduced order ODE systems that approximate the dynamics of the 2D Navier-Stokes equations using the truncated Fourier expansion method when the external force along the x-direction acts on the mode (0, k). Moreover, Smaoui [29] derived a seven-mode truncation system of ODEs and proposed controllers for its dynamics using Lyapunov based controllers. Extensive numerical simulations were presented to show the different behavior of Kolmogorov flow for Reynolds number range 0 < R e < 26.41, and Lyapunov-based controllers were designed to control the dynamics of the system of ODES to different attractors. Although the control problem of parabolic PDEs has been investigated, the control problem of the different finite dimensional approximations of the 2D Kolmogorov flow is not completely investigated. e main contribution of this paper is the design of an adaptive controller as well as a static and a dynamic sliding mode controllers to control the dynamics of the seven-mode truncation ODEs system of the 2D Navier-Stokes equations. e seven-mode truncation ODEs' system was completely derived by Smaoui [29]. is ODEs' system is the lowest dimensional system obtained so far that captures the dynamics of the 2D Navier-Stokes equations with sinusoidal external force f → � (α 3 ] sin αy, 0), where α � 2. We should emphasize here that that the design of such controllers for this well-known partial differential equation has not been treated elsewhere in the literature. First, the dynamics of this 2D Navier-Stokes equation described by a laminar steadystate regime and a periodic flow regime is briefly analyzed.
en, an adaptive control law and a static and a dynamic sliding mode control laws are designed and applied to the system of ODEs to control its dynamics either to a steady state or to a periodic state. It should be noted that other types of controllers for nonlinear systems such as observed-based finite-time tracking sliding mode control, output feedback active suspension control, robust H ∞ sliding mode control, and adaptive dynamic programming-based decentralized sliding mode control were explored by different investigators from various disciplines [43][44][45][46][47][48][49][50][51]. e paper is organized as follows. In Section 2, the 2D Kolmogorov flow equations and their Fourier Galerkin approximation described by a seventh-order nonlinear ODE system are presented. e dynamics of the reduced order ODE system described by a laminar steady-state regime and a periodic flow regime is described in Section 3. Section 4 presents the design of an adaptive control law which is used to regulate the states of the reduced order ODE system to a desired fixed state or to a periodic state without the knowledge of the Reynolds number. Section 5 presents sliding mode control laws to control the dynamics of the steady state and periodic regimes. e theoretical developments are verified by numerical simulations. Specifically, a static as well as a dynamic sliding mode controllers are proposed for the system of ODEs. Finally, some concluding remarks are given in Section 6.

A Seven-Mode Truncation O.D.E System of the 2D Kolmogorov Flow
e "basic 2D Kolmogorov flow" u → � (α sin αy, 0) was introduced by Kolmogorov [1] as an example to study transition to turbulence. is flow is the solution of the 2D Navier-Stokes equations: with force f → � (α 3 ] sin αy, 0) and with periodic boundary conditions in two directions 0 ≤ x, y ≤ 2π. e kinematic viscosity is ] � (1/R e ), where R e is the Reynolds number and the pressure is p.
A system of ODEs can be derived from the Navier-Stokes equations (1) and (2) for α � 2 by expanding , and f → using the following Fourier expansion forms: where k � (k 1 , k 2 ) is a wave vector with integer components, k ⊥ � (k 2 , −k 1 ), and the reality condition c k � −c −k must hold. e equation for c k k ≠ 0 is It can be easily checked that system (5) is invariant under the following symmetries: where r x , r y , and r o are reflection symmetries across the xaxis, the y-axis, and the origin, respectively. erefore, it can be concluded that r x , r y , r o with the identity transformation i form an Abelian group: G � r x , r y , r o , i .

The Laminar Regime of the Kolmogorov Flow
In this section, we analyze the dynamics of the seven-mode truncation ODE system given by (5) when the Reynolds number R e � 20 and R e � 24 corresponding to a steady-state laminar regime and a periodic regime, respectively.
At R e � 20, numerical simulations of system (5) shows that the system has seven fixed points. ese points are classified as four asymptotically stable points and three unstable points. Figure 1 presents the phase plane of the system for four different initial conditions, and Figure 2 depicts the vorticity ω � ∇ × u → of the corresponding four asymptotically stable fixed points. It can be verified that these fixed points can also be generated by applying the symmetries described in Section 2.
At R e � 24, the numerical simulations show the existence of four stable periodic orbits arising from a Hopf bifurcation at R e � 20.845 that remain stable up to R e � 26.323 (see Figure 3). Figure 4 presents the vorticity corresponding to one of the four periodic regimes at different times. It can be checked that the other stable periodic orbits Note the addition of controllers u 1 , . . . , u 4 in the dynamic model of the slave system. ese controllers will be designed to force the states of the slave system to follow the states of the master system.
Define the errors as the subtraction of the states of the master system from the states of the slave system, which can be written as follows: Using equations (7)-(9), the dynamical model of the error system can be written as follows:   √ e 3 e 6 − 3 where R � R e 2 − R e 1 .

An Adaptive Controller for the 2D Kolmogorov Flow.
In this section, an adaptive-based controller is designed to drive the states of the system in (8) to asymptotically converge to the states of the system in (7) without knowledge of the value of the Reynolds number.
Let the gains α i , (i � 1, . . . , 4) be positive scalars. Also, let the control gains b i (i � 1, . . . , 7) be positive scalars such that Also, define the estimate of the parameter R as R. e control scheme is given by the following theorem.

Theorem 1.
e application of the adaptive controller, with to the error model given by the set of ODEs (10) guarantees the convergence of the errors e i (i � 1, . . . , 7) to zero as t tends to infinity.
Proof. Define the parameter error e R as e R � R − R; then, Now, consider the Lyapunov function candidate V 1 such that Using the model of the error system given by (10), the control law given by (11), (12), and the update law of the parameter given by (13), the derivative of V 1 with respect to time is such that or Since the design parameters are positive scalars, then it is concluded that the Lyapunov function V 1 defined by equation (14) is positive definite and its derivative _ V is negative semidefinite. Also, since e 1 , . . . , e 7 and e R are bounded, then invoking Barbalat's lemma, the error functions in (9) asymptotically converge to zero as t tends to infinity. □ erefore, it can be concluded that the states of system (8) asymptotically converge to the states of system (7) as t tends to infinity.
Numerical simulations were carried out for the proposed adaptive controller. e values of the control gains α 1 , . . . , α 4 are taken to be α 1 � 50, α 2 � 50, α 3 � 50, and α 4 � 50. e values of the control gains are b 1  e simulation results of system (8) with the proposed control scheme given by (12) and (13) when the Reynolds number R e � 20 and R e � 24 are presented in Figures 5 and  6, respectively. Note that the dynamics at R e � 20 corresponds to a steady state regime and for R e � 24 corresponds to a periodic regime. Figures 5 and 6 depict the simulation results for the dynamics of the slave system before and after the control is switched on. Figure 5 shows how the control drags the dynamics of a periodic flow regime of the slave system to a steady state fixed point flow regime of the master system. On the contrary, Figure 6 shows how the dynamics of a steady state fixed point flow regime of the slave system is dragged into the dynamics of a periodic flow regime of the master system. Figure 7 presents the case when both the master system and the slave system are simulated with the same Reynolds number R e � 24 but with different initial conditions (i.e.,  Figure 7 shows how a periodic regime can be dragged into another symmetric periodic regime. erefore, it can be concluded that the numerical simulations clearly show that the proposed adaptive controller is able to force the states of the slave system to converge to the states of the master system even though the exact value of R e is not known.

Sliding Mode Controllers for the 2D Kolmogorov Flow
In this section, sliding mode controllers are proposed to control the dynamics of the seven-mode truncation ODE system presented in (5). e choice of this type of controllers is motivated by the fact that sliding mode controllers are known for their robustness and their insensitivity to modelling errors [43][44][45][46][47][48][49][50][51].
Recall from the previous section that the model of the error system is as follows: 6 Complexity where R � R e 2 − R e 1 . We will design a static as well as a dynamic sliding mode controllers to force the states of the slave system given by (8) to the states of the master system given by (7).

A Static Sliding Mode Controller.
Define the sliding surfaces S i (i � 1, . . . , 4) such as and define the signum function as follows: Let the control gains α i (i � 1, . . . , 4) and Γ i (i � 1, . . . , 4) be positive scalars. e control scheme is introduced by the following theorem.
when applied to the error system (17) guarantees the convergence of the errors e i (i � 1, . . . , 7) to zero.
Proof. Taking the time derivatives of S 1 , . . . , S 4 along the trajectories of the errors given by (17) and applying the controllers given by (20), we obtain Let the Lyapunov function candidate V 2 be such that Using the control law given by (20) and the error model given in (17), the derivative of V 2 with respect to time is such that erefore, _ V 2 < 0 for S i ≠ 0, for i � 1, . . . , 4. Hence, the trajectories associated with the discontinuous dynamics given by (21) converge to zero from any initial condition in a finite time given that α i and Γ i (i � 1, . . . , 4) are chosen to be Complexity 9 sufficiently large and positive scalars. It should be noted that since where Γ i (i � 1, . . . 4) are positive scalars; then, the above inequality is sufficient to ensure the finite time attractiveness of the sliding surfaces S i (i � 1, . . . , 4). e reaching time t reach is upper bounded by a function of S i (0) (i � 1, ..., 4) [52]. erefore, it can be concluded from (18) that the errors e 1 , e 3 , e 5 , and e 6 converge to zero in finite time.
After such a finite time, the errors' equations of , and e 7 can be written as Define the vector of reduced errors such that . e system of ODEs given by (25) can be written as or _ e r (t) � A r e r (t), where e characteristic polynomial is p(λ) � det (A r − λI) � (λ + 5)(λ 2 + 14λ + 45 + 21x 2 1 ). One of the roots of p(λ) is −5. e other two roots are located in the left-half of the complex plane since the coefficients of λ 2 + 14λ + 45 + 21x 2 1 are always positive. erefore, it can be concluded that the characteristic polynomial is Hurwitz and the matrix A r is a stable matrix. erefore, we can conclude that lim t⟶∞ e r (t) � 0.
Hence, it can be concluded that the errors e 1 , e 3 , e 5 , and e 6 converge to zero in finite time, while the errors e 2 , e 4 , and e 7 converge to zero asymptotically. □ erefore, the proposed static sliding mode controller forces the states of the slave system given by (8) to converge to the states of the master system given by (7). e performance of the proposed static sliding mode controller is simulated using the MATLAB software. Numerical simulations were carried out for the proposed sliding mode controller for three cases. e values of the control gains α 1 , . . . , α 4 are taken to be α 1 � 50, α 2 � 50, α 3 (7), and R e � 20 and the initial condition y(0) � [−1, 0.5, 0.7, 2.5, 0.1, 3.5, −0.4] T for the slave system in (8). e first state in this case corresponds to a periodic orbit, while the second state corresponds to an asymptotically stable orbit. For the second case, we choose R e � 20 and the initial conditions x(0) � [1, 0.5, 0.7, −2.5, 0.1, −3.5, 0.4] T for the master system in (7), and R e � 20 and the initial conditions y(0) � [−1, 0.5, 0.7, 2.5, 0.1, 3.5, −0.4] T for the slave system in (8) system. e two states in this case correspond to two symmetric asymptotically stable orbits.  (8). Note that the two states in this case correspond to two symmetric stable periodic orbits. Moreover, at the beginning of the simulations in each case, the controllers u 1 , u 2 , u 3 , and u 4 are set to zero for the first 10 seconds. en, the control law given by (20) is switched on to force system (8) to synchronize with system (7). Figure 8 presents the simulation results for Case 1. Figure 8(a) depicts the L 2 norm of the error ‖e‖ � ������������ 7 i�1 ‖x i − y i ‖ 2 versus time. Also, the states x 1 (t) and y 1 (t) versus time and x 7 (t) and y 7 (t) versus time are plotted in Figures 8(b) and 8(c), respectively. Figure 8(d) plots the state x 7 (t) and y 7 (t) versus x 1 (t) and y 1 (t); the figure shows the efficacy of the static sliding mode controller to drive the dynamics from one attractor to a different attractor. Figure 9 shows the simulation results for Case 2, and Figure 10 shows the results for Case 3. In the three cases, it is shown how the error converges to zero. Hence, it can be concluded that the designed static sliding mode control law in (20) is able to synchronize the ODE systems in (7) and (8) when these systems have the same or different Reynolds numbers, but they start from two different initial conditions. Clearly, the simulation results indicate that the proposed static sliding mode controller works well.

Remark 2.
Sliding mode controllers were implemented on many systems. However, most of the work on the control of the Navier-Stokes equations has been theoretical; numerical algorithms were developed to verify the theoretical results. e implementation of controllers on the Navier-Stokes equations is generally an open research area. It should be mentioned that a few works discussed some implementations issues related to the control of the Navier-Stokes equations. For example, Yan et al. [53] briefly described a practical control algorithm for these equation; Vazquez and Krstic [54] discussed some implementation issues related to a closed-form feedback controller for stabilization of the linearized 2D Navier-Stokes Poiseuille System.
In addition, it should be mentioned that constrained sliding-mode control is an active research area, and we are not considering it in the proposed work. However, for completeness, we refer the reader to [35,36,55,56]. x 1 (t) and y 1 (t) (d) Figure 8: A static sliding mode control of the asymptotically stable state regime at the Reynolds number R e � 20 in the slave system (8) to periodic regime at R e � 24 in the master system (7). e controller is switched on at time t � 10. (a) e L 2 norm of the error e versus time. (b) e states x 1 (t) and y 1 (t) versus time. (c) e states x 7 (t) and y 7 (t) versus time. (d) e states x 7 (t) and y 7 (t) versus x 1 (t) and y 1 (t) showing how the controller drives the states from one attractor to another attractor.  x 1 (t) and y 1 (t) (d) Figure 9: A static sliding mode control of an asymptotically stable orbit at the Reynolds number R e � 20 in the slave system (8) to a symmetrized asymptotically stable state regime at R e � 20 in the master system (7). e controller is switched on at time t � 10.   x 1 (t) and y 1 (t) (d) Figure 10: A static sliding mode control of a stable periodic regime at the Reynolds number R e � 24 in the slave system (8) to a symmetrized stable periodic state regime at R e � 24 in the master system (7). e controller is switched on at time t � 10. (a) e L 2 norm of the error e versus time. (b) e states x 1 (t) and y 1 (t) versus time. (c) e states x 7 (t) and y 7 (t) versus time. (d) e states x 7 (t) and y 7 (t) versus x 1 (t) and y 1 (t) showing how the controller drives the states from one attractor to another attractor. 12 Complexity

A Dynamic Sliding Mode Controller.
In this section, we design a dynamic sliding mode control to a system of seven ODEs derived from the 2D Navier-Stokes equation.
Let c i (i � 1, . . . , 4) be positive scalars and α i (i � 1, . . . , 4) and W i (i � 1, . . . , 4) be sufficiently large positive scalars. Define the sliding surfaces σ i (i � 1, . . . , 4) such as Theorem 3. e dynamic sliding mode control law, x 1 (t) and y 1 (t) (d) Figure 11: A dynamic sliding mode control of the asymptotically stable state regime at the Reynolds number R e � 20 in the slave system (8) to periodic regime at R e � 24 in the master system (7 showing how the controller drives the states from one attractor to another attractor. 14 Complexity when applied to the error system (17) guarantees the convergence of the errors e i (i � 1, . . . , 7) to zero in a finite time. Hence, the states of the slave system given by (8) converge to the states of the master system given by (7).
Proof. Taking the time derivatives of the sliding surfaces σ i (i � 1, . . . , 4) along the trajectories of the errors given by equation (17), we obtain Let the Lyapunov function candidate V 3 be such that Taking the time derivative of V 3 with respect to the trajectories given by (31), we obtain Since V 3 > 0 and _ V 3 < 0 for σ i ≠ 0 (i � 1, . . . , 4), hence σ i (i � 1, . . . , 4) converges to zero in finite time.
e trajectories associated with the unforced discontinuous dynamics equation (32) exhibit a finite-time reach ability to zero from any initial conditions provided the constants W i (i � 1, . . . , 4) are sufficiently large and positive.
Since σ i are driven to zero in finite time, the errors e i (t) (i � 1, 3, 5, 6) are governed after such a finite time by the following first-order dynamics: Since c i (i � 1, . . . , 4) are positive scalars, then the errors e i (t), (i � 1, 3, 5, 6) will then converge asymptotically to zero. Hence, the dynamic sliding mode controller guarantees the asymptotic convergence of the errors e i (t), (i � 1, 3, 5, 6) to zero. Using the same argument as in eorem 1 of Section 5, one can show that also converge asymptotically to zero. □ e performance of the proposed dynamic sliding mode controller is simulated using the MATLAB software. Numerical simulations were carried out for the proposed dynamic mode controller for three cases discussed in Section 5.1. e values of the control gains α 1 , . . . , α 4 are taken to be α 1 � 6, α 2 � 6, α 3 � 5, and α 4 � 5.
e values of the control gains c 1 , . . . , c 4 are chosen to be c 1 � 2, c 2 � 3, c 3 � 1, and c 4 � 3; the values of W 1 , . . . , W 4 are such that W 1 � 10, W 2 � 10, W 3 � 12, and W 4 � 12. Figure 11 presents the simulation results for Case 1. Figure 11(a) depicts the L 2 norm of the error ‖e‖ versus time. Also, the states x 1 (t) and y 1 (t) versus time and x 7 (t) and y 7 (t) versus time are plotted in Figures 11(b) and 11(c), respectively. Figure 11(d) plots the state x 7 (t) and y 7 (t) versus x 1 (t) and y 1 (t); the figure shows the efficacy of the dynamic sliding mode controller to drive the dynamics from one attractor to a different attractor. Figure 12 shows the simulation results for Case 2, and Figure 13 shows the results for Case 3. In the three cases, it is shown how the error converges to zero. Hence, it can be concluded that the designed dynamic sliding mode control law in (30) is able to synchronize the ODE systems in (7) and (8) when these systems have the same or different Reynolds numbers, but they start from two different initial conditions. Clearly, the simulation results indicate that the proposed dynamic sliding mode controller works well.

Concluding remarks
In this paper, the dynamics of a steady state flow regime and a periodic regime flow observed in a dynamical system of a nonlinear dynamical system of seventh-order nonlinear differential equations truncated from the 2D Navier-Stokes equations with periodic boundary conditions and a sinusoidal external force known as 2D Kolmogorov flow is analyzed. en, an adaptive controller is designed to drag the dynamics of Kolmogorov flow either to a steady state flow regime or to a periodic flow regime. Also, a static and a dynamic sliding mode controllers were designed to stabilize the dynamics of Kolmogorov flow. e presented numerical simulation results illustrate the effectiveness of the proposed adaptive controller. e dynamics and control of a reduced order system of less than seven ODEs whose dynamics are similar to the dynamics of the 2D Navier-Stokes equations will be the subject of future research work.
Finally, it should be mentioned that the proposed method can be extended to switched system [50,51,57].

Data Availability
e data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare that they have no conflicts of interest.