Nhom2_A. Bogobowicz, Theoretical aspects of modeling and control of water quality in a river section, Appl. Math. Comp. 41 (1991) 35–60

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Theoretical Aspects of Modeling and

Control of Water Quality in a River Section

Agnieszka Bogobowicz *

Department of Civil Engineering

University of W aterloo

W aterloo, N2L 3G1, Canada

Transmitted by John Casti

ABSTRACT

This paper is concerned with applying control theory to a problem of water quali ty

in a river. The problem concerns improvements in the dissolved-oxygen balance and

chlorides balanced by treating the effluent from particular sources and by flow

regulation by means of multipurpose reservoirs. Numerical solutions for the optimal

control of water quality have already been shown and applied in research on a water

management system [17]. In this paper some theoretical results on control in

coefficients are presented. They consist in determining the appropriate functional

spaces of control variables and necessary conditions for control-problem solution.

NOTATION

A = area of cross section [m],

A,, = steady-state river cross-section area [ml,

B = width of channel [ml,

C, = dissolved-oxygen concentration in saturation [mg/],

K 1 = biodegradation and sedimentation coeff icient [I/h],

K 2 = atmospheric reaeration coeff icient [I/h],

K,, = BOD coef ficient [l/h],

Q =flow rate [m3/s],

*Visiting Assistant Professor. Permanent address: Institute of Geophysics of Polish Academy of

Sciences, Warsaw, Poland.

APPLIED MATHEMATICS AND COMPUTATION 41:35-60 (1991)

0 Elsevier Scien ce Publishing Co., Inc., 1991

35

655 Avenue of the Americas, New York, NY 10010 0096-3003/91/$03.50



36 AGNIESZKA BOGOBOWICZ

Q. = steady-state river flow [m/s],

S,, S,, S,, S, =lateral sources of pollution [(mg/l)/hl,

t = time

u = control variable [ml,0 =river flow velocity [m/s],

w = concentration of chlorides [mg/ll,

x = longitudinal river dimension [ml,

y =concentration of BOD [mg/l],

z = concentration of dissolved oxygen [mg/ll,

6 = increment of flow rate [m”/sl,

I) = increment of cross-section area [m2 1,

77 , 72 = control variables,

5 = control variable,

5r, 5a, la, l4 = constants,+r, 4a, +a, b4, C& = adjoint state variables.

Function Spaces

Sobolev spaces H’(R) and H’(n):

where

Banach space E(a) c H’(n) with the norm

Linear spaces @a(R), @r(fi>:



Mo deling a River Section 37

with the norms

Il44D,c n, (II4I&n, + II4( *30) lZyo.L)+I4@ *>ltdaJ’*

Spaces W (fl,), @,,(fl,), @l(fl,), R , =(x0, L)X(O , T), x0 E(Q LX are

defined in exactly the same way.

INTRODUCTION

The research concerns the application of control theory to an environmen-tal problem. The problem to be addressed is in the field of water-quality

improvement. What is analyzed is a method of controlling the effluent from

particular sources and getting the most benefit with the least expenditure of

control energy. The control problem is formulated within the framework of a

water management system (Figure 1).

SM

x-

FE. 1. Operation of the multireservoir system on the upper Vi&da.




WATER-QUALITY PROBLEM

In the consideration of water pollution, dissolved oxygen (DO) plays a key

role in many processes and has become accepted as an indicator of waterquality. The biochemical oxygen demand (BOD), which is measured in

eflluent discharges and which is connected with DO, is also used as an

indicator. The third indicator concerned in this research is the chlorides

content.

The obvious approach to improving water quality is treating the eMuent at

the source, and that method is considered in this paper. The second possibil-

ity considered here involves varying the system parameters, such as flow

regulation by means of reservoirs. During periods of natural low-flow condi-

tions a common practice is to increase the flow by controlled releases from

reservoirs. The effects of the reservoir storage itself on water quality must becarefully taken into account, but low-flow augmentation may be a reasonable

alternative to treatment in the case of a multipurpose reservoir.

OPTIMAL CONTROL OF WATER QUALITY

The specific problem to be considered is the application of distributed-

parameter systems control theory to optimal releases and flow augmentation.

uncontrolled

lateral discharge

IFactory

If@J

I I/d -- , v

River

FIG. 2. The section of the river studied here.



Modd ing a Riv er Section 39

The open-channel flow is modeled by two one-dimensional equations (lin-

earized equations of B. Saint-Venant). The concentration of DO, BOD, and

chlorides in the stream is modeled by three one-dimensional equations of the

hyperbolic type. The control that minimizes an appropriate criterion func-tional is sought. The criterion functional to be minimized should penalize

deviation of the DO and chlorides distribution from standard values, devia-

tion of the flow from an assumed value, and also large controls. In order to

optimize the criterion functional it is necessary to specify the relation

between system control and system state. The system model will be formu-

lated in the next section.

The control inputs were chosen as follow (see Figure 2):

u(t) = increment of flow rate at point of inflow from reservoir (e.g. x0)

l(t) =rate of water treatment at controlled input point (e.g. x,),

vr(t) = boundary condition at point x = 0 for BOD,

q&t) = boundary condition at point x = 0 for chlorides.

For inputs so chosen one of the possible functionals determined for the

section (0, L) and the time interval (0, T) is the following:

where

(f-a)+= i-”i :;;z;>

u(t) =[u(t), 7~Jt), ~2G),i(t)l = control inputs,

w * = standard concentration of chlorides [mg/l],

u* =standard flow rate [m3/s],

[r, 12, 13, & = constant coefficients,

T = time duration over which control will be applied,

L = section length.

In the control system it is also necessary to consider the system bound-

aries. This paper includes the constraints for the control variables u,[, but

the numerical solution was obtained with respect to the constraints for the

state variables (y,z, w) also. The constraints for the control variables pre-




sumed here are in the form of an admissible set

and for state variables in the form

0 < w(x,t) < w,

where V,, jj, g, W are constants.

SYSTEM MODEL

Open-Channel Flow Model

W e begin by recalling the Saint-Venant equation, which we consider to be

valid in this paper. Unsteady flow in the channel, assuming that the density

is constant, can be described by the dependent variables, for example, the

flow rate and the cross-section area at any given river cross section. We

consider two equations-the continuity equations and dynamical

equation-as a departure point:

aA JQ-at +~=CI(“&

(3)

2Q aQ Q2 aA 1 aA+A----#--g +~~+sf-s,=o,

where

4 = lateral inflow,

Sr = frictional slope,

S, = channel bottom slope.



Mod eling a Riv er Section 41

For a description of the frictional slope we use the following empirical

formula of Manning:

s

f- Q' Q'

A”k ’

where n, k are real numbers. It is assumed then that river flow rate Q and

cross-section area A take the form

QOP (x,t) E (O,%J) (O>T)>Qo+6(x,t), (x,t)~(x,,L)x(O,T),

where Qo, A, denote the steady-state tlow and cross-section area of the open

channel respectively. The functions 6, $ can be determined by solving the

system of linear hyperbolic equations if the lateral inflow and bottom slope

are constant:

a* as

dt+dx=O’

where

2lQol QlQolCl==' Cz=-A”+‘k,

in 51 i = (x0, Ll X (0, T > with appropriate boundary and homogeneous initial

conditions. Boundary conditions for the equations are assumed as follows:




Water-@u&y Model

The model includes transport and sedimentation but neglects other phe-

nomena such as photosynthesis. The model is also one-dimensional. Homo-

geneity over the width and depth is assumed. Concentrations, denoted by the

symbols y (BOD), z (DO), and w (chlorides), are assumed to be functions of

a space variable x E (0, L) and a time variable t E (0, T). The functions

y(*, *>,z(*, *>,w( *, *> can be determined by solving the following system of

hyperbolic equations:

I + I

at- + K, yA = S,A

axin R,

a(h) + a(@>at - + K,zA + &,A = K,C,A + S,ax in fi, (6)

a(Aw) + a(@>-=

at

s A

a x3 in Ln

with the boundary conditions

y(O,t) =771(t)> z(O,t) = q(t), 40,t) =772(t) (7)

and the initial conditions

Y(X>O) = YOW z(x,O) =&j(T), w(x,o) = w,(x), (8)

where qr(-), qs(*), z,(*), ~a(.), z,,(*>, w,,(*) are given hnctions.We assume also that a jump of the function y(*, .) occurs at a given point

x,E(O,L):

Y(C t) = y(q, t) + AS& t(t)]. (9)

The condition (9) describes the presence of a source of pollution at the point

X =x1.

Some Aspects of Syst em Modeling

Formulating the open-channel model (OCM) and water-quality model

(WQM) in the form of partial differential equation leads to some questions of

solution existence and numerical approximation.



Modeling a River Section 43

Classical methods for hyperbolic equations show the existence of a soiu-

tion for sufficiently smooth coefficients. In the present problem, however, the

coefficients of the water quality model are solutions of the flow model and as

such can be discontinuous. The properties of these solutions govern theexistence, uniqueness, and regularity of the water-quality model and of the

entire control problem. This topic is discussed in the next section.

CONTROL-PROBLEM ANALYSIS

In the formulation of the optimal-control problem we are interested in

seeking the functions (controls) that minimize the performance index and are

solutions of the model equations on the admissible set of control variables.

The subsequent analysis establishes conditions on control variables under

which a solution for the water-quality model exists.To be able to rely on several well-known approaches and results of

functional analysis and prove the existence of a minimum of a functional over

the admissible set, it is convenient to assume for a while that this set is a unit

sphere in a space with the norm I[*[[. If the unit sphere {u E X: Ilull < 1) were

compact, the existence of the element minimizing a functional would be a

result of the continuity of functionals in X = C (Weierstrass theorem [ll]).

However, for infinite spaces this sphere is not compact. In choosing an

appropriate metric space it is necessary to assure the compactness of a unit

sphere as well as the continuity of the functional J describing the perfor-

mance index. By changing only the metric in the space we cannot generallyensure simultaneously that the closed unit sphere is compact and that the

functionals are continuous.

Let then us think about a new, more general, definition of the metric

space to achieve the existence of elements minimizing the functional defined

over the space. We are reminded of definition of weak convergence, i.e.

convergence in the adjoint space, i.e. in the space of linear functionals

defined over the given space X. The notion of weak convergence is closely

associated with weak solutions of differential equations.

To explain the idea of weak solution, suppose model equations are

determined by an operator M = C, d a,(f)d~/~x~cting on y(f), where

x’ = (x, t), and suppose we are given a function f defined on R c X. et us

assume (but it is an assumption) that the equation

My=_f (10)

has a classical solution. Let 4 be any function in C”(X) which is zero in the

vicinity of aR and for large If 1.We call it a test function. If we introduce the



44

notation

AGNIESZKA BOGOBOWICZ

(11)

our assumption (that a solution exists) implies that

(f>4) = (m/,4) = (?4>M”4), (12)

where

provided that the coefficients of M are in C(G). The following inequality

also holds true if the solution exists:

l(f,4) 1 GdlM*4ll (13)

for all 4 in R and some constant c.

Forgetting for a while these last considerations, let G be a set of thoseg E L’(n) for which there is a 4 E C”(R) satisfying

M*4=g. (14)

We set for g E G the functional

%=(4,f), (15)

where g is given in Equation (14) and 4 is a function in C”(KI>. Of course

we should check that this definition makes sense, namely, that S depends on

g and not on the particular 4, and also that we can extend the functionals to

be bounded on the entire t”(n). But we want here only to present the idea,

without detailed proofs.

We know from Riesz’s Theorem [ll] that there is y E L’(n) such that

lly ll = IISII Gc (16)



Modeling a Riv er Section 4s

and sg = (y, g) for all g E _~~(a). If 4 E C”(fi), then M*4 E G and

(Y,M*4) = (f>4). (17)

Now we would like to integrate (17) by parts to obtain

(My-f,4)=0 (18)

for all 4 E C”(n), and it would follow that My = f. But we do not know that

y E c”(n) and that My is meaningful; we know only that y E L2(0).

However, let us try to use the identity of (12) and (17) and simply define:

DEFINITION. y E L2(fi) satisfying (17) is a weak solution of the equation

My=f.

This new definition of a weak solution is sufficient for considering the

optimal control problem, which we do below.

As an example we can mention that the space C(a) is not adjoint with

any other space. That is why we do not assume that boundary conditions are

continuous. The technical reason is that in many systems the control variablechanges in piecewise continuous way.

Weak solutions of the open-channel model and the water-quality model

are defined here by the following integral identities determined for test

functions f#~r @r(Ln,), &2 E @Jar):

j/iL T-s(*,t)f$$(x,t)f-$tdx=~Tu,~I(xo,t)dt,

x0 0 1

( 194




and for test functions C&E @I(fi ), 44 E &(fi ), &J E @I:

’// Ty(x,t)0 0 A%+yf$ - K&4

=iLj)(w)(K,C, +S,)4,(x,t) dtdx

’//0 0

f-$- + Q%) d td x = kLiTS,A( x, t>&(x,t) dtdx

+ j Lwo ( x ) @x , o ) c # &, o ) dx + j T A( o , O~ z ( W&t )t. (lgb)0 0

The open-channel model weak solution is characterized by

LEMM A 1. Assume that in (4) A, > 0, Q. > 0, and Qi B - gAi # 0 (see

Friedrichs [8]). Then there exists one weak solution of the open-channel model

and there exists a constant C such that

IlSllw(q) + 119llw(n,, mIlL~(o,T) (20)



Modeling a River Section

Furthermore, i f

u EFP( O, T ) , u ( O) = ' ( O) =O,

then

PROOF. Let us write the flow model in matrix form:

aY

-=.g+,y,at

where Y = [I), 8]r, and A, B are matrices:

47

(21)

(22)

(23)

bt K = aY/&X, 5’ = aY/at.

The following equations hold true:

a y-=A:+,,,at

Y(O,x)=Y,(r), Y(U)=Y,(t),

aK

-=A~+BK,at

K(o,X) =Ko(X), K(t,O) =Kl(t)> (24)

aT-=A;+Ba,at

7T(O,r) =lT,(x), ?r(t,O) =571(t).

After the transformation of the initial condition for K the formulas become

K(o,X) =:(0,X) - Ye(X)= jTKo(Z)dZ +Y(o,o),0

(25)aY- = AK + BYI,=o *at

A&( s> d.s = Y,(t) - Y(O,O) - Bk’Y,( 7) do,




and the boundary conditions for w give the following relations:

- @z) dz = Ydt> - Y(O,O),

(26)a y

P=Az+By =c. /xq,(~)ds=AIY,,(r)-Y(O,O)]+B~xYo(d)ds.0 0

These equations imply that if

Y,(-) E [L2(OJ)12 and Y,( -) E [L2(0,T)]’

then

REMARK. We also show that if

Y, E [ Lrn(O, L)12 and Y, E [L”(0,T)12

then

YE [Lrn(0)12>

(27)

. (28)

(29)

(30)

where L”(a) = {C#Jess supn 4 <m}.

Let us take the parametric form of Equation (23) using the method of

characteristics:

a y

-=A;+,,,at

Yi =/o~i~(t,s)*+Yi(t,O),

Yi( t,o) = “b”’[ I

given as a boundary condition.




This yields

(31)

Also, via the same considerations but for JY, /ax we obtain

G c(Il&~(o,T) + IIu~IIL-(o,T)). (32)

The estimate (32) of the control variables shows that the control u-the

boundary condition for the water-flow model-should be a bounded function

with a square integrable first derivative.

For the WQM we find the existence of weak solutions. We formulate this

result in the following theorem.

THEOREM 1. 1f the a.ssumpt ions of Lemma 1 are fulfil led, there exist s a

w eak solution (y, z, w ) E [W(fI)13 of the model (6).

PROOF. Let us consider one equation of the WQM:

a(&) + a(@/)at

ax+kyA=f

with boundary condition

and initial condition

(33)

(34)

t/It=0= yo(x). (35)




We see then that if we introduce the bilinear form

E(y&=/y -A$-Qg+k&QR [ I

(36)

and the operator

then (19b) becomes

To prove the existence of a solution for that equation we have to show that

E(4J,4) 2 (Yll4ll& V4=@1

(see Lions’s Theorem [13]). We can easily see that

and

-]j;d fi = - f/ d”“(x,O) dx if we take 41t=r = 0,

which yields

E(0,0)=~~L 4z(x,t)dr+~~*Q(0,t)q52(0,t)dt+/ dln




We will show that there exists (Y for which

51

That requires proving that

1. Q(0, t) > y > 0,

2. (k - aQ/axXx, t) > y > 0.

Condition 1 is satisfied due to the assumptions of Lemma 1. To prove

condition 2 we replace the variable y by ZJ= ye-“. For the new variable we

have the equation

a(&?) + a(Q*i)at

T+(k+S)ij=e-6tf=f.

We can always find k, = k + 6 that gives

( 1l-z (r,t)>y>O

where cr = min(y/2, i). n

CONTROL-PROBLEM SOLUTION

For the control problem the necessary optimality conditions can be

obtained by applying the abstract results in 1151.As done in 141, the following




theorem can be proved:

THEOREM 2. There exists an optimal solution (ii, fl,,q,,[) minimizingth e perfomnance index (1) ov er th e set of admissible controls U.., (2). The

optimal control sat isfies the follow ing syst em:

1. State equations: for S,(I E W (a),

a * a sjy+-g=o>

2Q0 a s ~ ; a * 1 a*-----A, a x A; ax

+---+C’6+C&=o,

I + a- + KlyA =&A,

a t a x(38)

a(h) + a(Qz)

at- + K,zA + K,,yA = K,C,A + &A,

a x

a(Aw) + a(@> = s- ~ A

at ax3 >

w ith boundary and initial conditions.

2. Adjoint sta te equat ions: V+,, $z E &(fil ) and t /&, 44, & E 4,_,(fl>,

a41 Q$ a4, 1 a4,-- ---_-at gA3, a x B ax f $42 = (K&s + s,)4, + $34.5,

1 a4, a4, 2Qo a42--------go, at ax gA2, ax + “‘~2 =O’

-A Z - Qf$ + K,A& + K21A44 = 0, (39)

_A!$-Qf.$=-&(w-w *)+.




3. etim lit y condition: V(I-J,~1,772,5) E u8d,

53

T4f#J,(x,,t)f~(u)- f$+$-$i 0 0 I

2(xo>t)_f‘i(U)

+[~~45+A(xl,t)S,~,(x,,t)](5-i) (40)

A simple derivation of the above theorem is based on the variational

formulation of the problem (see [4]). To derive the variation of J we will use

LEMMAS. The variational form of the W QM is

a(Aay) a(Aay >-=

at- ~ - K, ayA - SoAt,

ax

a(A6z) a(Qaz)-=_--

at axK,azA - K,,ayA,

a(Aaw) a(Qaw)

at = - ax ’

w here ay ,az,aw are the variations of y,z,w .

For simplicity we denote 8 y, LLz, w as Y, 2, W below.

PROOF OFTHEOREM 2. The variation of J has the form

(41)

(42)




We can simplify the derivation of a] by introducing new equations called

adjoint equations. They are strictly connected with the Lagrange functional,

but we will introduce them directly. Let us find functions +r, #~aE @r(Q,>,

&s, +4, C& E @Jn) to make the following expression independent of statevariables:

jLjT~I(c~-z)+~drdt+jLjT~2(w-~*)+Wd~dt. (43)0 0 0 0

From the evident identities

//T L&(C,-z)+Zdxdt

0 0

a(fw a(Qz)= (c,-~)+~+4~ at+yy- K,ZA - K,,AY dxdt,

~T&(u-.*)u~dt=~T[13(u-u*)~‘+~Lc$~(;+;)d~]dt~

(44)

and from the integral identities for weak solutions

- T41(~0mxm)t,/0

+ c,S + c& dxdt1

dxdt

1 T%?O--

gA / ~4dxodfi(u(t)) dt00 0




/+ K,YA - &,A( x1, t)s dxdt

a, 1= -A~-Qf$+K,&A dxdt1

- TsoA(w)5(t)dt+jTA(0,t)~3(0,t)~,(t)dt,/0 0

ax+K&+K,,YA dxdt

= -i,(c,-~)+-A~-~~+K,ae, dxdt1+

/YAK,&, dxdt,

n

--Lb-w*)+W+d&t)

= {;(u;-~*)+-Af$Qf%x dxdtI

-/‘Q(o,t)~,(O,t)?,(t)t,0

(45)

it is clear that if we take 4, as absolutely continuous solutions of the adjoint

state equations (39) we obtain the gradient of J in a form independent of the

state variables:

2Qo Qo gA2,7+r--

0 o QoB

Qo~3(0,t),Qo~5(0,t),~~~+ A(w)S,,&(x,,t) I (46)




1 2 6 74 5

TIME (days)

FIG. 3. Optimal flow increment at the point x = x0.

0 1 2 3 4 5 6 7 8

TIME (days)

FIG. 4. Optimal values of BOD discharges at the point r = 0.



Modeling a River Section 57

a= 100

C0

0

g 1.0

!z

$0,c .

8

g 0.6

F

2 0.4

?I

g 0.2

0

0 1 2 3 4 5 6 7 8

TIME (days)

FIG. 5. Optimal values of chlorides discharges at the point r = 0.

0 1 2 3 4 5 6 7 6

TIME (days)

FIG. 6. Optimal values of the rate of BOD treatment at the point r = r,.



58

and the optimality condition becomes

AGNIESZKA BOGOBOWICZ

+(~~~(t>+A(x,,t)So~,(x,,t))[~(t)-~(o] n (47)

NUMERICAL RESULTS

A problem of water quality control was

some of the results were interpreted for

(control variables) were not obtained. The

formulated for a real system, and

that system. Analytical solutions

discrete form using the finite-dif-

ference method was formulated, and the minimum of a discrete form of the

1.0

0 10 20 30 40 50 60 70 80

ITERATION

FIG. 7. Decrease of the cost functional.



Modeling a Rioer Section 59

performance index was found subj ect to constraints. The minimization was

achieved by applying the penalty-shift method [7] combined with the conju-

gate-gradient method. As shown in [4, 51 we used the following data:

L = 23.0 km, A, = 25.0 rn’, g = 12.0 mg/, x,, = 1.8 km,

T = 3 days, Qa = 20.0 m3/s, W = 400 mg/, X1 = 10.7 km,

and the estimated coefficients

K, = 0.19exp(0.029 s) ( s = initial BOD at point x = 0))

K,, = K,,

K, = 1.72Q,/A,.

Selecting the start point as u = 0, 5 = 0, v1 = 8 mg/, 772 = 800 mg/, theresults in Figures 3-7 were obtained.

REFERENCES

1 N. U. Ahmed and K. L. Teo, Opt imal Control of Dist ributed Parameter Syst ems,

Elsevier North Holland, New York, 1981.

2 M. Benahmed, Identif ication de Non-IinCaritk ou des Parametres Repartis dans

Deux Equations aux D&i&es Partielles, Non-linkair, These d’Etat, Univ. Paris 9,

1978.

3 A. Bogobowicz and J. Sokolowski, Optimal Water Quality Control by Treating the

Eflluent at the Polhltion Source and by Flow Regulation, Hydrol. Sci. Bill. 129,

1981.

4 A. Bogobowicz, Problems of Modelling and Optimal Control of Water Quality in

River, PH.D. Thesis, Inst. of Systems Research, Polish Academy of Sciences,

Warsaw, 1986.

5 A. Bogobowicz and J. Sokolowski, Model& and Control of W ater Quality in a

Rioer Section, Lecture Notes in Control and Inform. Sci. 59, Springer-Verlag,

1983.

6 J. A. Cunge and Wagner, Integration Num&rique des Equations d’Ecoulement de

Bar& Saint-Venant par un Schema Implicite de Differences Finies, La Houille

Blanche, No. 1.

7 W. Findeisen, J. Szymanowski, and A. Wierzbicki, Teeoria i Metody Obliczeniowe

Optymalizacji (in Polish), PWN, Warsaw, 1976.

8 K. 0. Friedrichs, Symmetric hyperbolic linear differential equations, Comm. Pure.

Appl. Math. 7:345-392 (1954).




9 J. G. I. Dooge, Linear Theory of Hydrologic Systems, Tech. Bull. 1468, Agricul-

tural Research Service, USDA, Washington, 1973.

10 W. Hullet, Optimal estuary aeration an application of distributed parameter

systems theory, Appl. Math. Optim., No. l(1974).11 W. Kolodziej, Some Chapters of Mathematical Analysis (in Polish), PWN,

Warsaw, 1970.

12 Z. Kundzewicz, Parameter Approximation of Hydrologic Models, Ph.D. Thesis,

Inst. of Geophysics, Polish Academy of Sciences, Warsaw, 1979.

13 J. L. Lions, Equations different ielles operutionelles et prohlkm es uux limit es,

Springer-Verlag, Berlin, 1981.

14 D. L. Russel, Quadratic performance criteria in boundary control of linear

symmetric hyperbolic systems, SZAM J. Control 11, No. 3 (1973).

I5 J, Sokolowski, Optimal control in coefficients for weak variational problems in

Hilbert space, Appl. M&h. Optim. 7 (1983).

16 J. Sokolowski. Control in coefficients of PDE, Abh. Akud. Wiss. DDR, Jahrgang,

No. 2N, 1981.

17 K. A. Salewicz, A. Bogobowicz, A. Kozlowski. and T. Terlikowski, Modelling and

Control of the Multireservoir Water System, Report, Environmental Protection

Department, Katowice, Poland, 1986.

Documents

Nhom2_A. Bogobowicz, Theoretical aspects of modeling and control of water quality in a river section, Appl. Math. Comp. 41 (1991) 35–60