A numerical approach to an optimal boundary control of the viscous Burgers’ equation

Applied Mathematics and Computation 210 (2009) 126–135

Contents lists available at ScienceDirect

Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc

A numerical approach to an optimal boundary control of the viscous Burgers’ equation Ismail Kucuk *, Ibrahim Sadek Department of Mathematics and Statistics, American University of Sharjah, P.O. Box 26666, Sharjah, United Arab Emirates

a r t i c l e

i n f o

Keywords: Optimal control Nonlinear partial differential equation Burgers’ equation Modal expansion technique Control parametrization Boundary control

a b s t r a c t The dynamics of the forced Burgers’ equation subject to Dirichlet boundary conditions using boundary control is analyzed with the objective of minimizing the distance between the final state function and target profile along with the energy of the control. An efficient method is suggested to solve the optimal boundary control of the Burgers’ equation. The solution method involves the transformation of the original problem into one with homogeneous boundary conditions. This modifies the problem from one in which there are boundary controls to one in which there are distributed controls. The Modal space technique is applied on the distributed controls of the forced Burgers’ equation to generate a low-dimensional dynamical systems. The time-variant controls are approximated by a finite term of the Fourier series whose coefficients and frequencies giving optimal solutions are to be determined, thereby converting the optimal control problem into mathematical programming problem. The approximate solution space based on the control parameterization is obtained by using the Runge–Kutta method. Numerical simulations for the boundary controls are presented for various target functions to assess the efficiency of the proposed method. Ó 2008 Elsevier Inc. All rights reserved.

1. Introduction During the last two decades, considerable amount of work has been devoted to the area of active control of fluid flows (see e.g. reference [1]). The existence of controllers of a certain potential application can be achieved by using boundary and distributed control theory [2]. For a small viscosity in the dynamics of 1D nonlinear wave equation in fluids, an optimal control of the viscous Camassa–Holm equation is discussed in [3]. Exploitation of oil modeled by a parabolic partial differential equation (pde) is another application of optimal control of pde’s [4]. Fluid flow separation, combustion, and fluid-structure interactions are among the many applications for the use of control theory. In such applications, the Navier–Stokes equation played an essential role in modeling and in development of computational control algorithm [2]. A lot of attention from both the mathematical and control communities was focused on the 1D Burgers’ equation since the control of the Navier–Stokes equation is not easy problem to tackle numerically. More importantly, if a computational scheme does not work for the Burgers’ equation, it will certainly not work for the Navier–Stokes equation. An economical and convenient way of applying control on these applications is by means of actuators placed on the boundary of the domain. In this paper, the Burgers’ equation is examined within the context of optimal Dirichlet boundary control problem leading to a study of boundary value problems. The aim of this work is to provide an efficient approach for solving such class of boundary control problems. The optimal control problem is based on minimization of a given objective function which is determined by the distance between the final state at the terminal time and the target profile over the spatial domain, and the energy due to the boundary control actuators over a given * Corresponding author. E-mail address: [email protected] (I. Kucuk). 0096-3003/$ - see front matter Ó 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2008.12.056

I. Kucuk, I. Sadek / Applied Mathematics and Computation 210 (2009) 126–135

127

period of time. For the determination of the optimal boundary controls of the 1D Burgers’ equation, it is necessary to modify the problem from one in which there are boundary controls to one in which there are distributed controls. The method of modal expansion [5,6] is employed to convert the distributed control problem to that of the optimal control of lumpedparameter systems in a finite-dimensional space. In contrast to the standard optimal control or variational methods, the control parameterization method represents a distinct approach toward the solution of optimal control problems. In general, this technique approximates the control functions by finite term orthogonal (or non-orthogonal) functions with unknown coefficients, thereby converting an optimal control problem into a mathematical programming problem. The distributed controls of the Burgers’ equation are approximated by a finite term Fourier-type series where their unknown coefficients are sought giving a solution near the optimal (or sub-optimal) solutions. In the present paper, the solution method based on both the modal expansion and control parameterizations approach is called the ME-CP. The problem of distributed and boundary control of Burgers’ equation has received extensive attention recently [7–9]. The dynamics of the forced Burgers’ equation subject to both Neumann boundary conditions and periodic boundary conditions using boundary and distributed control is analyzed in [10]. The exponential stability in L2 has been proved when non-adaptive control is applied whereas asymptotic stability is proved when the adaptive control is applied. A backstepping boundary control law of Burgers’ equation with actuator dynamics is proposed in [11] where the system depends on the signals, first and second derivatives evaluated at the boundary points. With the use of Lyapunov analysis and a result from the classical theory of distributed parameter systems governed by parabolic equations, it is proven that the controlled closed-loop system is globally stable. Moreover, a unique global classical solution of the system is obtained in [11]. The stationary solutions of a one-parameter family of boundary control problems for a forced viscous Burgers’ equation is studied in [12]. In [13], the author derives nonlinear boundary control laws that achieve global asymptotic stability for both viscous and inviscid Burgers’ equation using Neumann and Dirichlet boundary controls. The numerical treatment of the present boundary control of Burgers’ equation was investigated in [14,15]. Different adjoint techniques for the optimal control of Burgers’ equation with Neumann boundary control are studied [14]. The viscous Burgers’ equation with a small viscosity coefficient is unstable with a standard Galerkin approximation [15]. The Galerkin-least-squares method is used in [15] to stabilize the problem. In [16], the problem is solved by employing the low dimensional model obtained by the KarhunenLoève Galerkin procedure (K-LGP) where the minimization of the objective function is performed using a conjugate gradient method [17]. The advantages of the use of present method (ME-CP) compared to the method (K-LGP) used in [16] are the amount of reduction in the computational work and improvement in its accuracy and convergency rate. This has been illustrated by means of numerical simulations of the boundary control of Burgers’ equation with various target functions. An efficient numerical scheme is presented to an optimal control of the Burgers’ equation by means of point-wise actuators in the spatial domain in [9] where the location of actuators can play a crucial role if they are not calculated optimally; on the contrary, the present paper investigates a boundary control problem of Burgers’ equation where actuator forces are placed along the boundary of the spatial domain and calculated optimally, and for which numerical scheme is easier to implement. Current results are more promising than the authors’ earlier work in [9]. 2. Statement of the boundary control problem In this paper, the boundary control of viscous Burgers’ equation with an external distributed force is considered. The governing equation of the viscous Burgers’ equation is given by

ut ðx; tÞ þ L½uðx; tÞ ¼ f ðx; tÞ for ðx; tÞ 2 Xx  Xt ;

ð1Þ

where

L½uðx; tÞ ¼ muxx ðx; tÞ þ uðx; tÞux ðx; tÞ: The variable uðx; tÞ is interpreted as the velocity of a fluid at a spatial point x at time t; f ðx; tÞ is the specific external force (force per unit mass) acting on the fluid and m is the kinematic velocity. The problem is subjected to the Dirichlet’s boundary conditions

uð0; tÞ ¼ g l ðtÞ;

uð1; tÞ ¼ g r ðtÞ;

ð2Þ

and initial condition

uðx; 0Þ ¼ u0 ðxÞ;

ð3Þ

in which g l ðtÞ and g r ðtÞ are the boundary control actuators applied along the boundaries of the domain. The superscripts l and r denote left and right end points of the spatial domain, respectively. In order to state the optimization problem, the Hilbert space is introduced

HðSÞ ¼ ff : R  S ! Rjkf k2 < 1g; where the norm kf k2 ¼< f ; f > and for all f ; g 2 HðSÞ,

< f ; g >¼

Z

f ðrÞgðrÞ dr S

in which S can be either Xx ¼ ð0; 1Þ or Xt ¼ ð0; TÞ.

128

I. Kucuk, I. Sadek / Applied Mathematics and Computation 210 (2009) 126–135

The Cartesian product of two copies of HðSÞ is denoted by H2 ðSÞ and the following definition is necessary for the statement of the control problem. Definition 1. A vector ~ gðtÞ is said to be admissible, if for a given T > 0; ~ gðtÞ ¼ ðg l ðtÞ; g r ðtÞÞ 2 H2 ðXt Þ. For a given target function uT ðxÞ and a terminal time T, the optimal control problem to be considered can be stated as follows: go ðtÞ :¼ ðg lo ðtÞ; g ro ðtÞÞ 2 H2 ðXt Þ such that Find an optimal boundary control actuator ~ go ¼ ~

Jð~ go Þ ¼ min Jð~ gÞ;

ð4Þ

~ g2H2 ðXt Þ

where

Jð~ gÞ ¼

1  n kuðx; TÞ  uT ðxÞk2HðXx Þ þ kg l ðtÞk2HðXt Þ þ kg r ðtÞk2HðXt Þ ; 2 2 2

ð5Þ

subject to (1)–(3). The first term measures the distance of the final state, uðx; TÞ, from the target function uT ðxÞ, and the rest denotes the energy due to boundary control actuators. The state function uðx; tÞ 2 HðXx  Xt Þ is the solution of (1), and the parameters  > 0 and n > 0 reflect the relative weight attached to a cost expenditure of control efforts. For example, the cost of the control can be reduced by taking larger values of  and n. For the problem described in (4), the following transformation is considered

uðx; tÞ ¼ ð1  xÞg l ðtÞ þ xg r ðtÞ þ wðx; tÞ:

ð6Þ

where wðx; tÞ is an auxiliary state function. Then, the viscous Burgers’ Eq. (1) becomes

 tÞw þ hðtÞðw þ hðx;  tÞÞ ¼ f ðx; tÞ; wt  mwxx þ wwx þ hðx; x

ð7Þ

 tÞ ¼ ð1  xÞg l ðtÞ þ xg r ðtÞ and hðtÞ ¼ g r ðtÞ  g l ðtÞ hðx;

ð8Þ

where

with homogeneous boundary conditions

wð0; tÞ ¼ 0; wð1; tÞ ¼ 0 for t 2 Xt ;

ð9Þ

and initial conditions

wðx; 0Þ ¼ u0 ðxÞ  ð1  xÞg l ð0Þ  xg r ð0Þ for x 2 Xx :

ð10Þ

3. Modal control space problem In this section, we transform the distributed parameter control problem (1)–(3) into a modal control lumped-parameter problem by means of eigenfunction expansion technique. Before applying the eigenfunction procedure to reduce the degrees of freedom of the system, a set of eigenfunctions needs to be specified that satisfies the homogeneous boundary conditions in (9). The set of orthonormal eigenfunctions

fwn ðxÞg1 n¼1 ¼

npffiffiffi o1 2 sinðnpxÞ

ð11Þ

n¼1 2

are the eigenfunctions of the operator L½w ¼ @@xw2 and satisfy the boundary conditions. Since the set of eigenfunctions fwn ðxÞg1 n¼1 is a complete orthonormal basis for HðXx Þ, any wðx; tÞ 2 HðXx Þ has a unique representation [18]

wðx; tÞ ¼

1 X

wn ðxÞan ðtÞ:

ð12Þ

n¼1

Although the state variable expansion yields an infinite-dimensional system theoretically, it is necessary to limit the dimension since actively controlling a large number of modes is a physically infeasible problem. This is achieved by retaining the eigenfunctions associated with the largest eigenvalues. Thus, Eq. (12) can be written as a truncated Fourier series expansion

wN ðx; tÞ ¼

N X

wn ðxÞan ðtÞ

ð13Þ

n¼1

where wn ðxÞ is defined in (11) and an ðtÞ, n ¼ 1; 2; . . . ; N, are Fourier coefficients of wN ðx; tÞ obtained by

an ðtÞ ¼ hwN ðx; tÞ; wn ðxÞiHðXx Þ

ð14Þ

Substituting (13) into (1), integrating over Xx , and taking the inner product of both sides of (1) which corresponds to Galerkin projector, result in a finite system of the modal functions in am ðtÞ, that is,

a_ m ðtÞ þ ðmm2 þ hðtÞÞam ðtÞ þ Im ðtÞ ¼ ~f ðtÞ;

m ¼ 1; 2; 3; . . . N

ð15Þ

129

I. Kucuk, I. Sadek / Applied Mathematics and Computation 210 (2009) 126–135

where hðtÞ is given by (8), the dot denotes the derivative with respect to time t,

pffiffiffi N 2 2m X g l ðtÞ þ g r ðtÞð1Þ1þmþn Im ðtÞ ¼ n an ðtÞ þ hðtÞðg l ðtÞ þ ð1Þmþ1 g r ðtÞÞ; mp p n¼1 m2  n2

ð16Þ

n–m

and

~f ðtÞ ¼ hf ðx; tÞ; w ðxÞi m HðXx Þ : The modal equations in (15) form a system of N first-order nonlinear coupled differential equations subjected to the modal initial conditions

am ð0Þ ¼ hu0 ðxÞ; wm ðxÞiHðXx Þ þ

pffiffiffi 2 r ðg ð0Þð1Þm  g l ð0ÞÞ: mp

ð17Þ

where u0 ðxÞ is the initial velocity given in (3). In view of the transformation (6) and the expansion (13), the objective function (5) in the first N modes becomes

J N ð~ gÞ ¼

1  n kwN ðx; TÞ  uT ðxÞk2HðXx Þ þ kg l ðtÞk2HðXt Þ þ kg r ðtÞk2HðXt Þ ; 2 2 2

ð18Þ

and the control problem (4) is reduced to the following modal control problem

min~g2H2 ðXt Þ J N ð~ gÞ:

ð19Þ

The necessary optimality condition for the control ~ gðtÞ 2 H2 ðXt Þ is that the first variation of J N ð~ gðtÞÞ in problem (19) with respect to ~ gðtÞ vanishes, i.e.,

d~g J N ,

d J ð~ gðtÞ þ gD~ gðtÞÞjg¼0 ¼ 0; dg N

ð20Þ

gðtÞ 2 H2 ðXt Þ is an arbitrary function vanishing at the end points of the domain Xt . for sufficiently small g > 0 and where D~ 0.35

0.2 0.18

0.3

Performance Index

0.2 0.15 0.1

0.14 0.12 0.1 0.08 0.06 0.04

0.05

0.02 0

0 0

1

2

3

4

5

6

7

8

0

9

0.5

1

Iteration Number

1.5

2

2.5

3

3.5

4

4.5

Iteration Number

0.5

Performance Index

Performance Index

0.16 0.25

0.4 0.3 0.2 0.1 0 0

0.5

1

1.5

2

2.5

3

Iteration Number

Fig. 1. Convergence history of the performance index function for different target functions at each loop in the Algorithm.

5

130

I. Kucuk, I. Sadek / Applied Mathematics and Computation 210 (2009) 126–135

4. Parameterization of the control problem In this section, a direct method for solving the modal control problem by parameterizing the boundary control actuator is developed. This control parameterization procedure is formulated for the variational problem (19) which is then reduced to the solution of a mathematical programming problem. Let the finite-dimensional subspace U m  H2 ðXt Þ be the linear space spanned by P i ðtÞ for i ¼ 1; 2; . . . ; m that can be taken as standard families of polynomials or functions such as orthogonal polynomials [19], trigonometric functions [20] or polynomial splines [21]:

1.5

1

1

0.8 0.6 0.4

g (t)

0

1

g0(t)

0.5

0.2 −0.5 0 −1 −1.5 0

−0.2 −0.4 0.05 0.1

0.15 0.2

0.25 0.3

t

0.35 0.4

0.45 0.5

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

t

1.5

0.5

1

0.4

0.5 0.3 0 0.2 g (t)

1

0

g (t)

−0.5 −1

0.1

−1.5 0 −2 −0.1

−2.5 −3

−0.2 0

0.05 0.1

0.15 0.2

0.25 0.3

0.35 0.4

0.45 0.5

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

t

t

2

3 2.5

1.5

2

0

g1(t)

g (t)

1 1.5

0.5 1 0

−0.5 0

0.5

0.05 0.1

0.15 0.2

0.25 0.3

t

0.35 0.4

0.45 0.5

0

0

0.05

0.1

0.15

0.2

0.25

t

Fig. 2. Boundary controls for different target functions.

0.3

0.35

0.4

0.45

0.5

131

I. Kucuk, I. Sadek / Applied Mathematics and Computation 210 (2009) 126–135

( Um ¼ ~ gm ¼ ðg lm ; g rm Þ 2 H2 ðXt Þjg jm ðtÞ ¼

m X

)

aji Pji ðtÞ; aji 2 R; j ¼ l; r :

i¼1

Assume that the set

n o ~m ðtÞ ~ðtÞj~ ~ðtÞ ¼ limm!1~ g U1 ¼ ~ g g

ð21Þ

~ ðtÞ 2 U 1 such that is dense in H2 ðXt Þ, in the sense that for each admissible control ~ gðtÞ 2 H2 ðXt Þ and d > 0, there exists ~ g

~ðtÞkH2 ðX Þ 6 d: k~ gðtÞ  ~ g t

ð22Þ

The normed linear space H2 ðXt Þ is a uniformly convex Banach space. Since the boundary optimal control ~ go ðtÞ 2 H2 ðXt Þ satgm 2 U m to ~ go 2 HðXt Þ, isfies (20), it follows from the approximation theory [22] that there exists a unique L2  approximation ~ i.e.,

k~ g ~ go kH2 ðXt Þ ¼ inffk~ gm  ~ go kH2 ðXt Þ j~ gm 2 U m g:

ð23Þ 2

In the control parameterization approach (CPA), the original problem of minimizing J N in (19) over ~ gðtÞ 2 H ðXt Þ is replaced by the finite-dimensional problem of finding ~ gm ðtÞ 2 U m that minimizes J N over U m , i.e.,

J N ð~ gom ðtÞÞ ¼ inf fJ N ð~ gðtÞÞg;

ð24Þ

~ gm 2U m

and the components of parameterization of ~ gm ðtÞ ¼ ðg lm ðtÞ; g rm ðtÞÞ is chosen to be of the form

g jm ðtÞ ¼

m X ~ ½aji cosðlji tÞ þ bji sinðlji tÞ ¼ ~ aTrj cosð~ lj tÞ þ ~ bTr j sinðlj tÞ;

ð25Þ

i¼1

2

1.4

1.5

1.2 1

u(x,T)

u(x,T)

1

0.5

0.8 0.6

0 0.4 −0.5

0.2

−1

0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0

1

0.1

0.2

0.3

0.4

1.2

1

1

0.8

0.8

0.6

0.6

u(x,T)

u(x,T)

1.2

0.4

0.4

0.2

0.2

0

0

−0.2

−0.2 0

0.1

0.2

0.3

0.4

0.5

x

0.5

0.6

0.7

0.8

0.9

1

0.6

0.7

0.8

0.9

1

x

x

0.6

0.7

0.8

0.9

1

0

0.1

0.2

0.3

0.4

0.5

x

Fig. 3. The comparison of the final state, uðx; TÞ (dotted line), and the target function, uT ðxÞ ¼ 1  x3 (solid line).

132

I. Kucuk, I. Sadek / Applied Mathematics and Computation 210 (2009) 126–135

where j ¼ l or r, the superscript Tr represents vector transpose, and ~ aj ¼ ðaj1 ; . . . ; ajm ÞTr ; ~ bj ¼ ðbj1 ; . . . ; bjm ÞTr , and ~ lj ¼ ðlj1 ; . . . ; ljm ÞTr are to be determined optimally. In view of Eqs. (18)–(24), one has the following programming problem: aoj ; ~ boj ; ~ loj Þ such that Find the optimal parameters ð~

J N ð~ aoj ; ~ boj ; ~ loj Þ ¼

inf

ð~ aj ;~ bj ;~ lj Þ2R3m

J N ð~ aj ; ~ bj ; ~ lj Þ:

ð26Þ

The necessary conditions of optimality can now be obtained by differentiating the performance index J N with respect to the aj ; ~ bj and ~ lj , i.e. unknown vectors ~

@J N ~ ¼ 0; @~ aj

@J N ~ ¼ 0; @~ bj

@J N ~ ¼ 0: @~ lj

ð27Þ

Solution of the latter system leads to a nonlinear system of equations for ~ aj ; ~ lj and ~ bj . The optimal control ~ go ðtÞ is obtained through (25) and (27). Finally, the optimal state function wðx; tÞ can be evaluated from Eq. (13) and consequently, the original optimal state function uðx; tÞ is obtained from Eq. (6). The fourth order Runge–Kutta method with fifth order error (RK45) [23] is used to solve the coupled system of differential equations in (15). Optimal boundary control actuator ~ go ðtÞ in (4) is computed by Nelder–Mead Simplex method (N-MSM) whose convergence is thoroughly discussed in [24]. The convergence to an optimal ~ gðtÞ in (19) is improved by an Algorithm, using a sub-optimal solution of the optimization problem as an initial guess for the same optimization problem in a set of loops of optimizations, introduced in [9]. 5. Numerical results and analysis In this section, we present numerical results for three different target functions in (5). The low-dimensional model is obtained by the eigenfunction expansion approach to solve the boundary control problem of Burgers’ equation. The fundamental 1

2

0.8

1.5

0.6

u(x,T)

u(x,T)

1

0.5

0

−0.2

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

−0.4 0

1

x

1.2

1.2

1

1

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0

0

−0.2

0

0.1

0.2

0.3

0.4

0.5

x

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

u(x,T)

u(x,T)

0.2 0

−0.5

−1

0.4

0.6

0.7

0.8

0.9

1

−0.2 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

x

Fig. 4. The comparison of the final state, uðx; TÞ (dotted line), and the target function, uT ðxÞ ¼ sinðpxÞ (solid line).

0.9

1

133

I. Kucuk, I. Sadek / Applied Mathematics and Computation 210 (2009) 126–135

issue is to reduce the computational complexity while obtaining robust results. Therefore, we focus on the results rather than the computational cost that is within the limits of mathematical methods introduced in Matlab. 2 Three target functions used to assess the present the solution method are uT ðxÞ ¼ 1  x3 ; uT ðxÞ ¼ sinðpxÞ and uT ðxÞ ¼ ex (highly nonlinear). The parameters used in the numerical simulations are m ¼ 0:01;  ¼ 0:001; b ¼ 0:01; N ¼ 4 in Eq. (13), m ¼ 2 in Eq. (25), and T ¼ 0:5 and the initial condition is u0 ðxÞ ¼ w1 ðxÞ. We set the unknowns in Eq. (25) as a vector for each boundary control actuator, j ¼ l or r

~ nj ¼ ½aj1 ; aj2 ; bj1 ; bj2 ; lj1 ; lj2 T :

ð28Þ

The numerical procedure starts with the initial guess ~ n0j ¼ ~ 061 ; j ¼ l; r. The initial guess ~ n0j is then mutated with the result of an optimization done in the process until the stopping criterion, a convergence in the control energy, given in the Algorithm is achieved. In Fig. 1, we show the history of performance index function versus the number of mutations. In each step, 1700 iterations are required to find the optimal value in Matlab that uses Nelder–Mead simplex direct method. The parameters in (28) in optimal boundary control actuators for different target functions are obtained as:

3

2.8

2.5

2.6

2

2.4 2.2

u(x,T)

u(x,T)

1.5 1

2 1.8 1.6

0.5

1.4

0

1.2

−0.5

1

−1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0.8

1

0

0.1

0.2

0.3

0.4

2.8

2.6

2.6

2.4

2.4

2.2

2.2

u(x,T)

u(x,T)

2.8

2 1.8

1.4

1.4

1.2

1.2 0.1

0.2

0.3

0.4

0.5

0.7

0.8

0.9

1

0.6

0.7

0.8

0.9

1

1.8 1.6

0

0.6

2

1.6

1

0.5

x

x

0.6

0.7

0.8

0.9

1

1 0

0.1

0.2

0.3

0.4

x

0.5

x

2

Fig. 5. The comparison of the final state, uðx; TÞ (dotted line), and the target function, uT ðxÞ ¼ ex (solid line).

Table 1 L2 norm of the difference between the target function U T ðxÞ ¼ 1  x3 and the state function at the terminal time. Iteration no.

ku  uT k2

Iteration no.

ku  uT k2

1 2 3 4

3.064177840093215 0.965297951399405 0.942921935724987 0.536957578292994

5 6 7 8

0.302766993206769 0.289428069686138 0.287113966026636 0.280247122597731

134

I. Kucuk, I. Sadek / Applied Mathematics and Computation 210 (2009) 126–135

0.5

1 st Itr 4 th Itr th

9 Itr

u(x,T) − uT(x)

0

−0.5

−1 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x Fig. 6. The difference between the state function, uðx; TÞ and the target function, uT ðxÞ ¼ 1  x3 .

(1) when uT ðxÞ ¼ 1  x3 ,

~ nl ¼ ½26:22; 167:6; 24:95; 7:930; 0:2414; 2:155T ; ~ nr ¼ ½2:247; 0:331; 1:872; 5:629; 6:382; 1:414T (2) when uT ðxÞ ¼ sinðpxÞ,

~ nl ¼ ½0:03813; 1:940; 1:159; 1:274; 12:80; 7:981T ; ~ nr ¼ ½1:213; 3:256; 0:8450; 3:502; 4:989; 5:248T 2

(3) when uT ðxÞ ¼ ex ,

~ nl ¼ ½6:837; 0:6816; 8:406; 13:74; 6:102; 1:821T ; ~ nr ¼ ½0:9498; 59:88; 1:259; 108:3; 0:06816; 0:004479T and the optimal boundary actuators are presented in Fig. 2. The comparison of the target functions with the un/controlled ones at the final time through the evolution of the control 2 process in Algorithm are shown in Figs. 3–5. Although the target functions 1  x3 and ex appear to be completely off from the uncontrolled state function at the terminal time, the perfect matching of the controlled state functions and the target functions seen in Figs. 3 and 5 is of great advantage of the technique presented in this paper. The L2 -convergence of the technique for the polynomial target function is shown in Table 1 as an example. Moreover, the difference between the target function and controlled state function at the terminal time is presented in Fig. 6. 6. Conclusion In this paper, we develop a solution technique for the optimal boundary control problem of one-dimensional Burgers’ equation. Our purpose is to minimize the objective function which is determined by the distance between the final state uðx; TÞ and the target profile uT ðxÞ, along with the boundary control effort. Upon transforming the boundary control problem into distributed one and applying the modal space approach, the basic control problem is then reduced to a system of coupled nonlinear differential equations whose dynamics is equivalent to the dynamics of original Burgers’ equation. The parametrization method is then implemented to approximate the control by a finite terms of a Fourier series type. The optimization problem is thereby reduced to a finite-dimensional minimization problem which can be solved by using N-MSM. Numerical results for the Dirichlet boundary control problem of the Burgers’ equation are presented to support the accuracy and efficiency of the method ME-CP. It is interesting to note that the numerical results produce excellent approximations for various target profiles and this in return it supports the advantage of using ME-CP compared to other techniques such as KLGP. Finally, applying the boundary control to the Burgers’ equations produced stronger results than the authors’ earlier work [9] where a point-wise control is applied.

I. Kucuk, I. Sadek / Applied Mathematics and Computation 210 (2009) 126–135

135

References [1] B.B. King, D.A. Kruger, Burgers’ equation: Galerkin least-squares approximation and feedback control, Mathematical and Computer Modeling 38 (2003) 1075–1085. [2] N. Smaoui, Nonlinear boundary control of the generalized Burgers equation, Nonlinear Dynamics 37 (2004) 75–86. [3] L. Tian, C. Shena, D. Dinga, Optimal control of the viscouse Camassa–Holm equation, Nonlinear Analysis: Real World Applications 10 (1) (2009) 519– 530. [4] S. Effati, M. Janfada, M. Esmaeili, H. Roohparvar, Solving of optimal control problem of parabolic pdes in exploitation of oil by iterative dynamic programming, Applied Mathematics and Computation 181 (1) (2006) 1505–1512. [5] M.J. Balas, Active control of flexible systems, J. Optim. Theory 25 (9) (1978) 415–436. [6] I.S. Sadek, J. Feng, Modelling techniques for optimal control of distributed parameter systems, Mathematical and Computer Modelling 18 (7) (1993) 41–58. [7] E.J. Dean, P. Gubernatis, Pointwise control of Burger’s equation – a numerical approach, Computers and Mathematics with Applications 22 (7) (1991) 93–100. [8] H.M. Park, Y.D. Jang, Control of Burgers equation by means of mode reduction, International Journal of Engineering Science 38 (7) (2000) 785–805. [9] I. Kucuk, I. Sadek, An efficient computational method for the optimal control problem for the Burgers’ equation, Mathematical and Computer Modeling 44 (11–12) (2006) 973–982. [10] N. Smaoui, Boundary and distributed control of the viscous Burgers equation, Journal of Computational and Applied Mathematics 182 (2005) 91–104. [11] W. Liu, M. Krstic, Backstepping boundary control of Burgers’ equation with actuator dynamics, Systems and Control Letters 41 (2000) 291–303. [12] A. Balogh, D.S. Gilliam, V.I. Shubov, Stationary solutions for a boundary controlled Burgers’ equation, Mathematical and Computer Modeling 33 (2001) 21–37. [13] M. Krstic, On global stabilization of Burgers’ equation by boundary control, Systems and Control Letters 37 (3) (1999) 123–141. [14] A. Noack, A. Walther, Adjoint concepts for the optimal control of Burgers’ equation, Comput. Optim. Appl. 36 (1) (2007) 109–133. [15] J. Atwell, B. King, Stabilized finite methods and feedback control for Burgers’ equations, in: Proceedings of the American Control Conference, vol. 4, 2000, pp. 2745–2749. [16] H.M. Park, M.W. Lee, Y.D. Jang, An efficient computational method of boundary optimal control problems for the Burgers equation, Computer Methods in Applied Mechanics and Engineering 166 (1998) 289–308. [17] R. Fletcher, Reeves, Function minimization by conjugate gradients, Computer Journal 7 (1964) 144–160. [18] A.W. Naylor, G.R. Sell, Linear Operator Theory in Engineering and Sciences, of Applied mathematical sciences, vol. 40, Springer-Verlag, New York, 1982. [19] Y. Chang, T. Lee, Application of general orthogonal polynomials to the optimal control of time-varying linear systems, International Journal of Control 43 (4) (1986) 1283–1304. [20] Y. Endow, Optimal control via Fourier series of operational matrix of integration, IEEE Transactions Automatic Control 34 (7) (1989) 770–773. [21] H.R. Sirisena, F.S. Chou, State parameterization approach to the solution of optimal control problems, Optimal Control Applications and Methods 2 (1981) 289–298. [22] R.F. Curtain, A.J. Pritchard, Functional Analysis in Modern Applied Mathematics, Academic Press, New York, 1977. [23] J.R. Dormand, P.J. Prince, A family of embedded Runge–Kutta formulae, Journal of Computer Applied Mathematics 6 (1980) 19–26. [24] J.C. Lagarias, J.A. Reeds, M.H. Wright, P.E. Wright, Convergence properties of the Nelder–Mead simplex algorithm in low dimensions, SIAM Journal on Optimization 9 (1998) 112–147.