Optimal control problems related to the navigation channel engineering

Vol. 40 No. 1 SCIENCE IN CHINA (Series E) Feb~ua~ 1997

Optimal control problems related to the navigation channel engineering"

ZHU Jiang ( ~ ~ ) , ZENG Qingcun (~F)~q~f), GUO Dongjian (]]t]~:~r

and LIU Zhuo (~J -~-)

( Institute of Atmospheric Physics, Chinese Academy of Sciences, Beijing 100029, China)

Received July 12, 1996

Abstract The navigation channel engineering poses optimal control problems of how to find the optimal way of engineering such that the water depth of the channel is maximum under certain budget constraint, or the cost of the engineering is minimum while certain goals are achieved. These are typical control problems of distributed system gov- erned by hydraulic/sedimentation models. The problems and methods of solutions are discussed. Since the models, usually complicated, are nonlinear, they can be solved by solving a series of linear problems. For linear problems the solutions are given. Thus the algorithms are simplified.

Keywords: navigation channel engineering, hydraulic/sedimentation model, optimal control.

The sedimentation in dams, rivers, mouth estuaries and coastal regions has important influ- ences on environment, economy and even daily activities, especially in China. For example, the sedimentation in the Changjiang River estuary forms mouth bars in all its four channels leading to the sea. The mouth bar in Nancao is the biggest one, where the water area is 25 km long and the water depth is less that 7 m. This makes it a big natural obstacle to navigation and shipping since the minimal water depth for the 3rd- and 4th-generation container ships is 12.5 m. Due to the construction of big dams, the long term sedimentation in reservoirs will cause the rising of water level at upperstream, and the effects will bring about some environmental problems to the upper region and dams themselves. To solve these problems one of the methods is artifical sand taking- out. Therefore, it is natural to ask when and where to take sand out in order to make the maximum use of the limited budget. This is an optimal control problem. Another example is the artificial grain feeding. In rivers with firm solid boundaries like Rhine, the clear water flushes the river bed. This causes the going-down of water levels, and consequently the ground-water horizon is reduced which possibly results in desiccation of the landscape and makes the navigation channel worse. Among the methods to solve the problem, the artificial grain feeding is good and econom- ic [1] . How to feed grain to have the best control effect is another optimal control problem. This kind of control problem can be generalized as problems of natural cybernetics. The natural cybernetics was formalized by Zeng [2' 3]. The natural cybernetics addresses self control behavior and the mechanism of artificial comrol of natural environment. It investigates the theory, methods and techniques of artifical control. Discussed here is an application of the optimal control method to the problems of optimal engineering of navigation channel, which is also an example of natural cybernetics.

In recent years the computer techniques have developed rapidly. As a result, more and more

attention is paid to the mathematical modeling methods in studying the interaction of water flow

* Project supported by the National Natural Science Foundation of China.

No. l OPTIMAL CONTROL ON NAVIGATION CHANNEL ENGINEERING 83

and sedimentation. The mathematical models now can simulate quite well the main features of

flow structure, sedimentation and changes of bed E4'51 . These models provide the necessary condi-

tions and feasibility of solving the problems like optimal engineering of navigation channel through

the optimal control approach. Now it is the right t ime to solve optimal control problems related to

this kind of engineering. This paper, based on Zeng [2], discussed the problem further.

1 Optimal control problems related to the navigation channel engineering

First we give the governing equations of water flow and sedimentation. Here is only a two-

dimensional model. But our approach remains valid for three-dimensional ones.

3v l v l v - - + v �9 ~T v + f ( k x v ) = - g V z + A A v - C D - - , (1) 3 t h

3h a T + V �9 ( v h ) = 0, (2)

OhS a-~- + v �9 ( v h S ) : e V �9 ( h V S ) + , ~ ( S * - S ) , ( 3 )

Oz b t~ ~ - t + V �9 gb = aco(S -- S * ) , (4)

where v is the vector of velocity, h = z - zb is the water depth, z is the surface height, zb is

height of bed, f is Coriolis constant, g the accelerate rate of gravity, A the coefficient of viscosi-

ty, CD coefficient of bottom drag, S is concentration of sediment in water, r is diffusion coeffi-

cient of sand, oJ is the falling speed Of suspended sediment particles, Pl is density of dry sand.

There are various formulas of S *, the capability of sediment transport of flow, and gb, the local

rate of sediment transport per unit width (see Tang[6]for details). Please refer to ref. [4] for de-

tails on boundary conditions.

Now we pose the optimal control problems related to the navigation channel engineering (also

see ref. [2 ] ) . Here the term "navigation channel" bears a general meaning. I t can be a band-

shaped region, or a block or even a pointwise region. So the problems include those related to en-

gineering of artificial port.

1.1 Problem 1.1

Denote the rate of taking sand out by ~:b(a function of x , y , t ) . Assume the unit cost is giv-

en and denoted by K l ( x , y , t ) , the total budget is given and denoted by c. Then

T

| t[ y, y,t)dadt (5) 0 1"/

where gl is the considered region. We are looking for an optimal design of ~b which is constrained

by budget such that the water depth of navigation channel is maximum, or the objective function

where K2 ( x , y ,

T

m a x f f f K 2 ( x , y , t ) h ( x , y , t ) d /2dt , (6) 0 1~

t ) is a given weighting function which may equal 1 in navigation channel and

84 SCIENCE IN CHINA (Series E) Vol. 40

zero elsewhere, and may also depend on time, for example take a large value in the season short of

water.

Considering the artificial control, eq. (4) changes to

O~b . p l y + V �9 gu = a w ( S - S ) - pleb. (4) '

Here we have defined an optimal control problem of a distributed system. The system is described

by eqs. ( 1 ) - - (3 ) , (4) ' and the constraint of control is given by inequality (5) . The objective

function is defined by (6). An additional constraint is

z b ) 0 , (7)

which means that the sand which is taken out will not be thrown into water at another location.

The method is commonly used today.

In ref. [2], an approach to the above problem is given. We can solve the nonlinear problem

by solving a series of linear problems and approximate the solution of the original problem. For

convenience write the above problem in an abstract form. The governing equations are

Oqv Ot - L ( r + F + gt,

Aqb = G,

L,=0 = q5~

x , y E O , t E ( O , T ) ,

( x , y ) E OO,

(x , y) E O,

(8)

where 3/] is the boundary, and A is the operator that acts on boundary which represents the

boundary conditions in the original problem. We consider the control space as L 2, and the inner

product is defined below. Rewrite (5), (6) as

T

0 n

T

m a q f f K 2 q~dDdt, (6)' 0 f]

where q5 -- (v, h, S, Zb) T , and gr is the control, the components of K2~ corresponding to v,

S, zb are zero. An additional constraint is ~ ~ 0 (taking sand out, not putting in), K1 > 0,

K2 ~ 0. Use the iterative method, assume the n-th iteration is known, write

On+l = ~ + ~ , grn+l = ~n + ~ ,

Ln+l = Ln + 3L, An+l ='An + 8A,

and write the first order variation of eq. (8).0 Then one has

38~ - Ln3q5 + ~ , 3t

A,ScP = O,

~r = 0,

( x , y ) E ~ , t E (0, T),

(x, y) E 3~Q,

(x, y) E ~.

(9)

No. 1 OPTIMAL CONTROL ON NAVIGATION CHANNEL ENGINEERING 85

The constraint is

The objective function is

T T

f ~ f K , Sgrdg2dt ~ c - f ; f K l gt.dg2dt, O 0 O n

6'gt+ gq, >~ 0.

( 1 0 )

(11)

Here 8L, 8A depend on 8~, L.

T

maxf 2 Od dt 0 0

= L.8@ + ( S L ) @ ~ , a n d A n = A n d @ - ( 8 A ) ~ .

(12)

Problems ( 9 ) - - ( 1 2 ) can be solved by the adjoint method. Define the inner product of the two functions on s by

( A , B ) = I[-ABdO. n

The adjoint operator F* of F is defined by (FA, B) = (A, F* B ) , V A, B. Thus we can define the adjoint equation of eq. (9) as

t OSq~* ""

--5-7--t - LZr - K2,

X28~* = 0, (13)

#~* [ t = r = 0,

where L,~, A,~ are adjoint operators of L, , A, . We can obtain ~q'" by solving the adjoint eq. (13). From (9), (13) we have

T T

f ( 8 ~ ' , 8 ~ ' ) d t = f(K2, d~)dt. (14) 0 0

So the objective functional equals the right-hand side of (14), and constraints are given by (10), (11). The problem is simply a linear programming problem.

The straightforward method to solve the linear programming problem is to use the available software package. But the approach needs huge computer memory since we need define a matrix which has dimension of ( number of control variable + 1 ) times (number of constraints + 2). As- suming the mathematical model has grid size of 20 times 20, and 500 time levels, then 160 Gb of memory is needed. This is too big for most, if not all, current computers. Therefore, we need another approach which we postpone to next section.

1.2 Problem 1.2

This problem is finding the optimal way of engineering in order to achieve the specific water depth along the navigation channel which is suitable for navigation with the minimum cost, i .e. minimizing the following objective functional

T

! f f W 1 " ( h ( x , y , t ) - h . ) 2 d x d y d t J = J1 + od2 -- O1Na2(t)


T

+ a f f f w2 o ~b(x, y , t ) 2 d x d y d t , (15) o

where h~ is the water depth suitable for navigation, [21 denotes the area of navigation channel,

122( t ) = { (x , y ) E 12;h ( x , y , t ) ~ h~ t, w l, w2 ( > 0) are given weighting functions which are

similar to K2, KI in Problem 1. l , and the weighting a > 0 . Constraint (7) must be satisfied.

Problem 1.2 can be solved as Problem 1.1 . The gradient of J can be obtained by solving the

adjoint equations. The adjoint equations read

where

---Yit - L : a a , "

X ; a e = o,

~ [ t=T = O,

- G ,

G = (0 ,2 w l ( h - ha)Zalnaz(t),O,2w2 a ~b) T.

If write 5q~ ~ = (v ~, h ~, S " , zb )T, then V J ( ~ b ( x , y , t )) = - z b ( x , y , t ) .

(16)

and Y i s j u s t gt n+t.

or

2 Linear problems

2.1 Problem 2.1

In fact it is pointed out in ref. [ 2 ] that the linear problem can be solved explicitly. For sim-

plicity, write

A ( x , y , t ) = K l ( x , y , t ) , B ( x , y , t ) = ~ ( x , y , t )

and define the inner product < X, Y > by

fr ff x(x ' Y ( x , y , t )d12d t .

B y ( 1 0 ) - - ( 1 2 ) , (14) , the linear problem is

m a x ( B , 8a/r), (17) s. t. (A, ~ ) ~ c , , ~ ~ - g z ,

m a x ( B , Y) , (18) s . t . Y ~ 0 , ( A , Y ) = c,

We can use the projected gradient method to solve Problem 1 . 2 after the gradient is ob-

tained. We will give the admittable descent direction by projecting the gradient in the next sec-

tion.

No. 1 ( ) P T I M A L C O N T R O L ON N A V I G A T I O N C H A N N E L E N G I N E E R I N G 87

To avoid introducing additional notations and assumptions, we only consider the finite-di-

mensional case. And it will facilitate the numeric calculation. So thereinafter, a function is in fact

an m-dimensional vector, an integration is in fact a summation, and a differential equation is in

fact a finite difference equation. Let

C = (ci)i~l, Y = ( y i ) m l , A = (ai)i'=l.

It is easy to see (through Z = YA, transfer (18) to a problem about Z) that the solution of (18) is

(i) If c , ~ 0 , V i = 1, " - , m , then Y ~ 0 .

( i i ) I f there exists an i such that ci > 0 , let i0 be the index such that c i / a i is the largest.

Then

[ c / a i o , i = i O,

Y i = ~ L O, i =/: i o .

Thus we obtain g r ~ 1. Note that it is meaningless if ( i ) occurs. In fact case ( i ) will never

appear. If it does appear, then

B ~ (A'~[)2A II A "

R e c a l l A = K i , B = 8 ~ " One has

T T

( f f f K 2 t d l ' 2 d t ) 3 ~ " ~ ( f ~ K 1 3 ~ " d ~ d t ) K t . 0 .0 O n

By assumption of K t > 0 , let 8~ " = Klg . Then

T

g ( x , y , t ) ~ ~ , g x , y , t . I~K2dg2dt 0 rl

The right-hand side is a weighting average of left-hand side, and this can only happen when g is

a constant q. That is

~ " ( x , y , t ) ~ q K l ( x , y , t ) .

From the "initial condition" of the adjoint equation (13) , we have q = 0, and hence k2 = 0. So if

we assume K2 is not always zero, case (i) never appears.

2.2 Problem 2 .2

Gradient ~TJ can be obtained by integrating the adjoint equation (13) . Then using Rosen ' s

projected gradient method we can find an admittable descent direction. Also consider the finite-di-

mensional case. Let x be an m-dimensional vector ( x l , "" , x = ) T . If X is an admittable point,

assume, without losing generality,


xl . . . . . x . = O, X.§ > O, "" , x. . > O,

and write the gradient at x as ( g l , "" , g , . )T ' Then the admittable descent direction is (0, "" , O,

- g . + 1, " " , - g i n ) T " In addition, the admittable interval of line search is (0, i ) , where

,,1<~i<,, gj ;g) > 0 "

It is noted that the approach presented here is only one of many alternatives. It is hoped that a

good method is chosen by numerical experiments and comparison.

3 Summary

In the present paper we described the optimal control problems related to the engineering of

navigation channel and the ways of solving them. Due to the nonlinearity of the problems the

main ideal approach is to decompose problems into a series of linear problems. And we give the

explicit solutions of the linear problems and this greatly reduced the requirement on the memory of

computers and the computing time. It is possible to solve the problems on current computers.

We still cannot prove the convergence of the solutions of linear problems because the dynam-

ics and processes of model are too complicated. The key to applying the approach to' realistic prob-

lems is to develop a good mathematical model which can well reflect the main realistic features of

flow, sedimentation and their interactions, and the adjoint model. We have developed a 2-dimen-

sional model [a] and its adjoint model 1) , and are ready to carry out some numerical experiments.

More investigations to problems like convergence can be done by the numerical experiments.

The problems addressed in ref. [2] and this paper are a class of typical control problems of

distributed systems and are worth further investigations.

References

1 Nestmann, F . , Improvement of the upper Rhine, tail water iffezheim, in Proceedings o f the 5 th International Symposium on River Sedimentation, Karlsruhe, Germany, 1992, 3, 1130--1152.

2 Zeng, Q. C. , Silt sedimentation and relevant engineering problem: An example of natural cybernetics, in ISIAM95 Procetd-

ings of the Invited Lectures, Berlin: Akademic Verlag, 1995. 3 Zeng, Q. , Natural cybernetics, Climate and Environment research (in Chinese), 1996, 1(1) :11. 4 Zeng, Q. , Guo, D., Li, R. , Numerical simulations of sedimentation and evolution of delta, Progress in Natural Sciences,

1995, 5(3):309. 5 Li, Y. , Xie, J . , Wu, W. , Studies and applications of two dimensional mathematical models of sedimentation, in Proceedings

of the 2 nd National Symposium on Basic Theory of Sedimentation, Beijlng (in Chinese), Beijing: China Construction Ma- terial Press, 1995.

6 Tang, R. (ed . ) , Sedimentation Studies (in Chinese), Beijing: Hydraulic Power and Electric Power Press, 1990.

1) Zhu, J . , Zeng, Q. , Guo, D. et a l . , Adjoint sensitivity analysis of a mathematical model of sedimentation.

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Optimal control problems related to the navigation channel engineering