Pointwise control of burgers' equation — A numerical approach

Compltterm Math. Applic. Vol. 22, No. 7, pp. 93-100, 1991 Printed in Great Britain. All rights reserved

0097-4943/91 $3.00 + 0.00 Copyrlght~ 1991 Pergamon Press plc

POINTW'ISE CONTROL OF B U R G E R S ' E Q U A T I O N A NUMERICAL APPROACH E. J. DEAN AND P. GUBERNATIS Department of Mathematics, University of Houston Houston, Texas 77204

(Received FebreaW 1991) A b s t r a c t - - W e describe a methodology for munerically solving pointwise control problems for the viscous Burgers' equatkm in one space dimension. The solution methods are based on a combination of finite difference and finite element techniques for the time and space dlscretizations with iterative algorithms such as quasi-Newton methods and GMRES. Numerical experiments validate the methods.

I. I N T R O D U C T I O N Motivated by applications in engineering and the environmental sciences, many issues associated with the control of the Navier-Stokes equations are currently under investigation (see [1,2], for example). Concentrating on the numerical aspects, it seems reasonable to first investigate a simpler model problem. An obvious choice is the ~iscous Burgers' eguation, since it retains many of the interesting features of the Navier-Stokes equations. In this paper, we address exact and approximate controllability problems for the Burgers' equation using pointwise controls associated with Dirac measures. This class of controls corresponds closely to the situation in many environmental science models. Of particular interest is the influence of the viscosity parameter on the exact controllability of the system. Numerical experiments show that the difficulty in controlling the system increases with decreasing values of the viscosity parameter, though the methods work very well when the final state is reachable. From a methodological point of view, we have been using classical but effective techniques such as evaluation of the gradient of the cost function by adjoint techniques, finite difference discretizations, and the iterative solution of the discrete control problems by either quasi-Newton (see [3]) or GMRES (see [4]) algorithms. 2. T H E

CONTROL

PROBLEM

One of the simplest models of nonlinear advection-diffusionis given by the viscous Burgem' equation in one space dimension. Ifwe interpretthe scalarfunction y(z, t) as modelling a velocity, at a point z and at time t, then the governing equation is given by

y,-

=/,

in

(0,I)x (0,r).

(1)

Here the parameter v plays the role of a viscosity. Suitable boundary conditions can be given, for example, by y=(O,t)=O, y(1,t) = 0, for t E (0,T), (2) while the initialdata is specifiedby y (z, 0) = y0 (z),

for z E (0, I).

This work was supported in part by the National Science Foundation under Grant RII-8917691. The authors are very ~'atdul to Professor Roland Glowinski for his invaluable sussestlous and contrlbuticm.

93

(3)

94

E.J. DEAN, P. GUBBRNATlS

We are interested in the problem of confrolling this system to produce, at a given time T, a desired state Yr. One possibility is to introduce boundary controk. A more reasonable choice, in the context of environments] science modelling, is the specification of controls at a finite set of points in the spatial domain. We shall ms]niy consider the case where we control at a single point in (0, 1). Given a fixed point a E (0, I) and a fins] state YT satisfying the boundary conditions in (2), we want to determine a control v E L~(O,T) so that y E L~((0, T); HZ(O, I)), the solution of the state equation y¢ - v Y~x + YY~ = f + v (0 6(z - a), in

( 0 , 1 ) x (0, T ) ,

y (~, o) = uo(~),

for ~ ~ (o, z),

y=(O,/)=O,

for t E (O, T),

YCl,t)=O,

(4)

satisfies y(z,T) = Fr(z) on (0, 1). Here, 6 is the Dirac measure at a. A major concern is to minimize the cost of the control v, leading to the following least squares formulation as an apprmimstion to the above problem: r~

J(v)

(5)

veL~(O,T) where

1

e

a(,,) = ~ II~'(T)- ~llL,(o,~> 2 2 + ~ lit'IlL,co,r>,

(6)

and ¢ is a small positive parameter. 3. EVALUATION OF T H E G R A D I E N T Almost any descent method for the numerics] solution of problem (5) will require the gradient of the cost function. Since VJ(v) E L2(O,T), we must have

6J(,,) =

va(,,) 6v e ,

(7)

where $J is the differential of J. It also follows from (4) and (6) that the di|ferentis] satisfies

6a(~) = ~

//

~6~dt +

/01

(yCx, T ) - ~rCx))6vC~,T)d~,

(81

where Sy solves the linearized Burgers' equation 6y~ -

v

6y~ + 6y y~ + y 6y~ = 6 ~ s 6(z - a), 6y (z, 0) = 0, for z • (0, 1),

6~ (0, t) = 0,

6y(1,t)=0,

(9)

for t e (0, T).

We introduce the a~ljoint state p = p (z,t); by multiplying both sides of the differential equation in (9) by p and then integrating in both space and time we get

~T~I

~ytpdzdt +

+

~T~I

6yympdzdt

y6ycpdzdt - v

6y~pdzdt

= / r 6,,(Op(a,O at. Jo

(10)

Pointwlse control of Bu~6~s' equation

95

If we now perform a suitable integration by parts, in both space and time, and use the initial and boundary conditions for y and 6y, we find 0T 6~ (t)p (~, t) dt =

o*P (z, T) 6y (z, T) dz

+

6y-~+y.p-(yp).--vp..

dzdt

(11)

-v

@. (1,t)p(1,t)dt

- f

6y(o,t)[~p.(O,t) + p ( o , o y(o,t)] dr.

do

By requiring the adjoint state function p to satisfy the adjoint equation --p,--vpx.--yp.=O, in (0,1) x (0,T), p (z, T) = y (z, T) - Yr (z), for z 6 (0, 1), vpz(O,t)+p(O,t)y(O,t)=O, p(1,t) = 0, for t e (0,T),

(12)

and comparing (8) to (11), we find the linear dependence of 6J on 6v to be

6J(~) =

0 T (,~

(t) + p (a,t)) 6~ (t) dr,

so that the gradient is identified as VJ(v)(t) = ev (t) + p (a, t),

for t E (0, T).

(13)

We notice that each evaluation of the cost function J requires the solution of the nonlinear state equation (4) while the evaluation of the gradient V J also requires the solution of the linear adjoint equation (12). If we let V = {z I z E HI(0, 1), z(1) = 0}, then the variational forms of (4) and (12), respectively, are

y (t) e v,

y (o) = yo,

/o1( y ~ z + v y x z z + y y x z )

dz=

/o' f z d z + v ( t ) z ( a ) ,

VzEV,

(14)

and p(t) E V, p(T) = y(T)- yT, o ( - p , z + v p , z= +py= z + p y z , ) dz = O, Vz 6 V.

(15)

4. T H E D I S C R E T E C O N T R O L P R O B L E M Let the time interval (0, T) be divided into N subintervals, each of length At = ~ . The discrete version of problem (5) is now

rain Ja,(v),

~ER N

(16)

96

E.J.

DEAN,P. GUBEP,NATIS

where JAr(V) = 1/2 f l ly N - YTl2dz+e/2 At )'~,= N 1 It]hi2, l / " - - {lTn}nN__l,and ~ mation to y ( T ) we get, by integrating (14) using the backward Euler scheme:

is the approxi-

yO : Y O For n = O , . . . , N -

1:

~n+l G ]7,

~ -~

1 1 (y"+~ - y") z dx + ~ fO 1 y~+~ z . d z

/o

"4-

yn+Xyn+lzdz:

/o

fn+lzdz+vn+lz(a),

VzEV.

(17)

By using a discrete version of the method used to find V J, we find that the gradient of the discrete functional is given by V Jar (v) = A t {eva A- F~n/a~'tN ~ ]In=l, where pn(a) is the approximation to p(a, nAt) we get, by integrating (15)backward in time by the scheme: pN+l _ yN -- YT. For n = N , . . . , h pn E V, "-~

.{.

/01

n - pn+ l

yn p,~ zx dz +

I

)) /01 z dz + v

p~ z• dz

y~ pn z dz = O,

V z E g.

(18)

Notice that (18) is nothing more than the backward Euler scheme for the system in (15). It is important to notice, however, that the terminal condition p (T) = y (T) - let has to be specified for pN+X and not for pN. We also use a simple space discretization for problems (17) and (18). Let the space interval (0, 1) be divided into I subintervals of equal length h = }. The space of trial and test functions, V, is now approximated by Vh, the space of continuous functions, linear on each subinterval ei = [(i - 1)h, ih], i = 1,... I, and vanishing on the right boundary. Computational]y, we solve (17), where y and z are replaced by Yh, zh E Vh and f is replaced by its piecewise linear interpolate. Similarly, we solve (18) by replacing p, y, and z by Ph, Yh, and zh E Vh, where V~ = {zhlzh E C°[0,1], zh(1) = 0, zh le~ E P1, i = 1,... I} and Px is the space of polynomials of degree one. Notice, finally, that the size of the discrete problem is determined by N, the size of the time discretization, while the cost of evaluating JAr and V J 4 t is determined by I, the size of the space discretisation. 5. N U M E R I C A L

RESULTS

The numerical methods used to solve the discrete control problem (16) were both descent techniques that used second derivative information on the cost function. If N wee not too large, a quasi-Newton method, using secant approximations to the Hessian of J4t, was feasible. For larger N, second derivativeinformation wee approximated by finitedifferencesand G M R E S was used to solve the linear systems. Initial test problems were generated by specifying a control v and integrating (4) so that a reachable state ~r -= yv (T) could be found. Using yr as data, we then solved (16) for ~ and compared it to v.

Pointwise conm)l of B ~ '

equation

97

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98

E.J. D~ts, P. {~IIIm~ATlS

Problem I v - 10-2, e = 10 -8, a = 2/3,

T=I, N = 128,

f(x,t) = 1,

for (x,t) E (0,1) × (0, T), for z E (0,1),

yo(x)=l-z,

YT - yu(T),

where

~(~) = j" t,

l (1 - t ) ,

if 0 ~ t ~ 1/2; i f l / 2 ~ t ~ 1.

Figure I shows both v and the computed #. Notice that # is less expensive than the control that generated the data Zfr. Figure 2 shows both let and the computed final state ~ T ) ; the two are virtually indistinguishable. These results were typical for all test problems of this type. A more realistic situation arises when the target state Ztr is not known to be reachable. In the following problem, Fr is only assumed to satisfy the boundary conditions.

Problem u ---- 10-~, e = .125, T=I, N = 60,

f(z,t)-

f l ,2 ( 1 - z ) ,

if 0 _< z < 1/2; if 1/2 _< z _< 1,

Yo(x) - O, YT (z) = 1 -- z s,

for t ¢ (O,T).

for x ¢ (0, 1), for z E (0, I).

Figure 3 compares yr and y (T) when the control is placed at a = .6. We are clearly having difficulty controlling the solution upstream of the developing front, with the situation deteriorating as the parameter v decreases, due to the increasing hyperbolic nature of the problem. Placement of the control point further upstream, at a -" .2, has little effect on the solution clue_=to the right boundary. Surprisingly, for either control point, we did not notice an improvement when T was increased. Figure 4 shows better results when controls are placed at both a = .2 and a = .6. To keep the size of the problem enma, the controls were not solved for sinadtaneous~. We first solved for the control at a - .2. Making this control part of the forcing term in (I), we then solved the problem for the control at a = .6. This expedient improves y ( T ) as an approximation to yr. We see that the method seems to be very effective for reachable states. For an arbitrary target state, it seems that multiple control points will be required due to the local effect of each.

Pointwi~ control of Bwrllm' equation

99

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100

E.J. DSAN, P..GuIwiUlATlS P~PE~NCES

~put/ou~, XCASB report no. 90-45, Institute for ComlmtCr Applications in Science and Ensimmr~, NASA Lamslq Emma,all Center, Hampton, Virginia, (1990). 2. M.D. Gunzbm'xer, L.S. Hou and T.P. Svobodny, Numerical apprn~,,,aioa of am opClmal cm~rol pmMma associated with the N&vim,-Stok~ equations, Appl. McstK. Left. g (1), 29-31 ( 1 ~ ) . 3. J.E. Dennis, Jr. msd R.B. Scl~bel, N~merical Methode ]or Uneos+~ismi Op|imimsllou n d N ~ r Eq.atlonm Preatlce-H~ll, (1983). 4. Y. Saad and M.H. S~-,]tz, GMRES: A Sma'alhed ,,,;,;,,,d residual allro~thm for solvi~ n o a ~ , . , , s t ~ linear systems, SIAM J. 5¢L SJatiaf. Compsf. Y, 856-869 (1986). 1. J.A. Burns and S. KartS, A control la'O~__m~_for Burgm~' equation with boumimd