On optimal control of processes in distributed objects

ON OPTIMAL CXNTROL OF PROCESSES DISTRIRUTED ORJECTS

IN

(OB OPTIMAL'NOM UPRAVLENII PROTSRSSAMI RASPREDELENNYKH OB'EKTAKA) PMM Vo1.27,

No.4, A.

1963,

I.

V

pp.688-696

EBOROV

(Frunze) (Received

A number of trolled

[d

and Lerner optimal

practical

processes

problems

in systems formulated

control

of

April

have with

case

in which

integral

the

is

case,

is

is

general

studying

con-

Butkovskii

form the problem and showed

that

of in certain

Pontriagin’s maximum the optimum conditions

is

described

note of

the $u.

Li [pi] _- ~

characterizing function

by the

for

nonlinear

which

is

control

state

of

sufficient

controlled

of

process Bu;

+ bi (z, 1) az

linear

con-

in Q and u in

function,

and D is

one type are

of

problem

described

concerning

by systems

Rozonoer’s method [51 differential equations. The article also necessary optimum conditions. by a system

the

nonlinear

a

space.

we formulate

processes

the

which

an admissible

Euclidean

optimum conditions, described Let

for

parameters.

systems

process

function a vector

u(S)

present

control

linear partial obtaining the the

m is

in m-dimensional

In the optimal

a need

by applying [3,41 found

controlled

a vector

system,

general

region

to

equation

where Q(P) trolled

the

most

in such

cases this problem may be solved principle 121. Later Butkovskii the

let

1963)

distributed

in its

processes

23,

in the

local

sense.

of

the

quasi-

is used in indicates

when the

process

equations.

be described

aui

+ Ci (;e, t) at’

by the

equations

= ri (z, ‘, ‘1, ’ . , ‘,,

1045

‘) (f = 1.. ..n)

(1)

1046

A. I.

Egorov

where the functions bi and ci have continuous second respect to x and t in the region G (O\

derivatives v is

with a con-

trolling

or closed)

convex

Parameter

which

assumes

values

in the

(open

domain V of some r-dimensional Euclidean space. The functions f i are assumed to be continuous in x and t and twice continuously differentiable

with

respect

We shall tions

to

~1,

assume that

(Goursat

. . . , un and U. the

functions

ui satisfy

the

satisfy

functions

0;

the matching

For our class of

bounded

domain If

the

of

the

v (z,

t)

problem

of

the

sequentially.

are

functions

continuous

V(X,

and

we shall

take

the

in x and t and defined

t) has a discontinuity

coordinate Vl

t) =

axes,

continuous (1)

We first

to

for

set

in the

if

O
solve

in the

the

on the

line

t = a

O
then

functions, (2)

on a line

example

if

b, 4

va (2, t)

tion

differentiable

in V.

function

where vi(x,

(2)

= ~~(0).

control

piecewise

one of

continuously

~~(0)

admissible

values

control to

condi-

ui to* t, = $,i (t)

and yti are

conditions

functions

G with

parallel

additional

conditions) “i (2V O) = ‘Pi (x)7

where

the

the

corresponding

solu-

G may be constructed

region

problem

L, [“il = f (x* t* ulr * * *I II*, v1 tx9 l))t

ui (I, 0)

(n
This with

we find L,

has a unique

respect the

to

solution

x and t:

solution

which

is

ui = uil(z, t)

Ui = ‘i’(‘,

of

continuously

twice

t)

[6. pp.63-671. the problem

ui (z, a) = uil (2, a),

IaJ = f (5 t, U1r* * -9 u,, vt (5 m,

differentiable In like

manner

ui (OP t, = lcli (t)

(a
Consequently,

we have u (I,

corresponding However,

to the on the

t) =

of

solution

u1 (2, 4

if

O
2

if

a
discontinuous

line

do not have continuous

a continuous

(I,

t)

control

discontinuity

function. t = a the

functions

ui(r,

t)

derivatives. If the function V(X, t) is regular to this function we have a unique soluL?, P. 191, then corresponding ui(x, t) have tion u(r, t) = {ul, . . . . u,), such that the functions

Optimal

continuous second is not

control

first

of

processes

derivatives

with

in

distributed

respect

to

reason

we may always

function V(X, t) there cobtinuous derivatives and is

continuous

Let Ai

(i

if

= 1,

V(X,

tion,

and consider

where

1 and T are constants

Among all

the

the

not

each

admissible

solution

x and t if

u(x, v(x,

control

t) ‘which

t)

is

(2) 2).

hi%3

regular

regular. system

functions

of

real

continuous

numbers;

in the

let

region

ai(

G. We

function V(X, t), denote by 11(x, t) to (2) corresponding to this control

the func-

functional

appearing

admissible

function

to

. . . . n) be a given

t) be given Pi(‘) and yi(X, take the admissible control solution of the problem (1)

control

is

to

a unique

respect

t)

of the problem (1) to functions (see Example

assume that

corresponds with

1047

x and t and integrable

derivatives. In this case the solution unique within the class of continuous

For this

objects

t)

V(X,

control such

that

in the

definition

functions, the

it

functional

of

is

the

required

S attains

region to

its

G.

a

find

minimum

(maximum) value. The admissible functional respect It this

is

should

be noted,

for

which

be called

by the

kind

is

of

It

is

encountered

processes

[81,

u in equations cases to select is

number of

(1)

function

will

a minimum (maximum) of

min-optimal

(max-optimal)

this

with

to S.

tions.

point)

control

attained

great drying

from the

in the

of

[91,

etc.

to minimizing

cases

the

problem

For example,

let

the

controlled

(2)

and let

with

study

processes

(1) makes it possible the best operation,

equivalent

conditions

that

way,

interest

is

a boundary

to

and desorption

x, t, u, v) dzdt

of

the

parameter

some functional. of

be described

tional

of

applica-

the process and in many the mathematical view-

a study

be required

problem

physical

The presence

or maximizing

process

of

gas sorption

to control which (from

reducible

it

value

viewpoint

the

In a

functional

by equations

to minimize

the

func-

S.

1048

A. I.

Be introduce

the

auxiliary

Bgorov

variable

a,, by means of

the

equation (3)

and the

supplementary

conditions u (0, t) =

u (x, 0) = 0

(4)

Then the problem reduces to minimizing the functional defined on the functions uo, ,. ., u,, given by equations the supplementary conditions (2) introduce the auxiliary functions &u. Mi [WJ = Q& with

the

$

@iWJ -

supplementary

and (4). To obtain a solution, we wi(x, t) by means of the equations

$

(C+Wi)= jj !J=l

(5)

conditions

X

wi (x, I’) =

S = us{ 1I T), (1) and (3) and

.2.

is

s

ai (E)exp

Ci (Evr)

dE df - Ai exp ((Ci (Ev1”) de) )

wi(l* $1 = SR,(rl:I,(Z(bi(L.:)h~-~iex*(Sb((1,-~~~!,

(i=l,...,n)

5

T

(6)

T

the constants Ai and the functions a;(x), Pi(t) and Yi(r, t) are from the functional S. The system of equations (5) is linear. and to each admissible control funcso are the conditions (6). Therefore,

where taken

tion there corresponds a unique vector function which is defined in the region C and satisfies (5)

and the

We

shall

condition

supplementary

say that

the

conditions

control

(6).

function

set

V(Z,

t)

satisfies

a maximum

if

H (u (2, t), w (z, t), v (2, 21,2,

it=))

tf

sup H (u (x, t), w (2, 0, v, x, f)(v~)(H),

where the symbol (( = )) represents of the reg.ion G except possibly for number of

lines

and the

The minimum condition Theorem function

We

10(x, t) = {VI, . . . . w,} the system of equations

zero is

Y(X,

that is valid at all points points that lie on a finite

plane. determined

1 (Maximum principle). t)

equality a set of

be win-optimal

in a similar

In order (max-optimal)

that

manner.

an admissible

with

respect

control to S. it

Optimal

must satisfy

control

of

processes

a maximum (minimum)

To prove

this,

we consider

distributed

the

functional

1

--H (U, W, UpX, t) dr dt

G

If ing

u = u(x,

to

the

t)

any arbitrary Let

to

We shall which

admissible

control

it

are

function equations

Li [A$]

the

by u(x,

U(X,

with

problem

correspondto

zero

for

solution

U(X,

t)

We shall

denote

(6)

corresponding

to

the

the

t)

and

W(X,

same problems

t)

+

Aw(x,

t)

but correspond

func-

to

the

t)

+ AU, where Au(x, t) is some increment of the increments Aui and Awi will t). Evidently

u(x,

= A2

boundary

(2)

equal

to S.

to

M, [Awi] = Ag

, 1

and the

to

is

and the

t)

respect (5)

+ Au(x,

t) of

~(2,

(1)

t),

solutions

function

the control satisfy the

of

problem functional

t).

function

and u(x,

denote

the the

be min-optimal

solution t)

of

V, then

w = w(x,

an admissible

by w(x, t) the functions u(x,

solution

function

function

corresponding

tions

the

is

control

1049

objects

condition.

WiLi [UiI SSD i=l

[u,w, v] =

J

in

(i = l,..,, n)

t

conditions

Aui (5, 0) = Aldi (0, t) = Awi (2, 2’) = Au+ (2, t) = 0

(6)

where tIH (z + AZ, v + Au, z, t) azi

aH -= A azi We shall the

denote

operators

by AJ an increment

Li are

AJ = J [z +

aH (2, V, x, t) azi , 2 = (Q,. . *, WJ

linear.

AZ, v +

of

the

functional

J.

(9)

Then,

since

we find

Au] -J

[z, v] = i\i

(AwiLi

AwiLi [Aui] +

[ui] +

G i=l + wiLi

It

is

[ Aui]] dxdt

known [lo,

t) which are q(x, spect to x and t,

-

ss G

p.1961 twice the

Cs IpLi 1~1 "G

is

valid

in the

[H (z +

region

AZ, v +

that

for

piecewise following QMi

[~ll

Au, x, t) -

arbitrary

continuously equation

dxdt

= SP,i a

G where a is

H (z, v, z, t)] dxdt

functions

(Green’s (x, t) dt-‘P,,

a contour

p(z,

differentiable

=

t)

0

(IO)

and

with

re-

theorem) (x, t) dx

enclosing

the

region

G,

1050

A.I.

Since

the

region

C

is

Egorov

a rectangle,

we find

T

ss PLi IqIdx’t zIPli

(2, f) -

=I

G

Integrating

of this

by parts

equation,

In the

last (5)

(0, t)] dt +

the

one-dimensional

integrals

on the

right

side

we obtain

T [P (1, t) q (2, t)lt+, -

pLi 191dxdt = 1s G

equations

P,i

0

equation

we set

and conditions

p = Au;, (6)

[P

T w qW)l~=O -

p = Aw,.

and (8).

it

Then,

follows

by virtue

of

that

n

SD

[AuJ dxdt

wiLi

G

+

s’ pi

=

-

$

[AiAui(l,

T) +

i=l

i=l

i ai (2) AUK (xv T) dx + 0

(t) AI+ (I, t) dt + is yi (cc, t) Aui (x, t) dxdt ] + is i G

G

0

{=I

Au, (2, t) dx dt

‘l’ a?

or

n-m

?Z

[Aui] dxdt = -

wiLi

A1s + fS 2

G f=l

G $=I

where ds is

the

increment

change

from the control Moreover. we have

added function

to

the

v(x,

Aui (x, z) dx dt

%

functional s as t) to the control

8

rSSUlt

of

function

the v + Av.

7%

AwiLi [ui] dxdt G

tion

In equation (11) we set q = AU;, (7) and the boundary conditions

$S i: G

=

(13)

i=l

i=l

AwiLj

IAu~I dxdt

p = fLwi. Then, by virtue (8). it follows that

= Ss ~ G

i=l

A ~

Auidx dt t

of

equa-

Optimal

On the

control

of

processes

in distributed

1051

objects

hand, equation

other

(1$J AwiLi IAuil dsdt

Ag

= 1s i

AWidx dl

G i=l

G i=l

al so holds. From the

where

last

two

t denotes

formulas

we find

a %-dimensional

that

vector

“1’ ..*. mn and Aaff/azi is determined formula to the function H, neglecting second in the increments AZ, v + Av, x, t) -

H (, +

H (2, v, 2, t) = H (2, v -t- Av, 2, t) -

2n 8H( ‘7 va+zAu~x7 t, Azi+ +I2 t

$

i==l

where it

0 d

follows

6 <

with components ul, . . . , u,, from formula (9). We apply Taylor’s all terms of order higher than the

1. From equation

2n aaH (2 + QAz, V .+$ az,azj i, J=l by virtue

(lo),

+

of

Av,

H (2, v, 2, 4 + X, t)

formulas

A2 A2 * j

(12)

to

(I51

(151,

that

aH ‘“;;* x* ‘) Azi dx & +

AJ = G

i

i=l 2n

aH

“;,u’

x* t,

Azi dxdt - 1s 2

1

G id --

1 2n 2 5s I2.l G i. j=l

-

im (2, “at; Av, ‘J, t) A~, dr dt _ i

G i=l @ff(z

+

eA& v + dZi azj

ss

H (2, v -I- Av, x, t) -

Av,

I,

t)

AZ ~~ &,dt i j

_

H (z, v, 8, #)I dzdt = 0

G

Collecting terms we obtain

and applying

Taylor’s

formula

to

the

function

aH/& i,

AS = -

ss

[H (z; v f

Au, z, t) -

H (2, v, x, 41 dzdt + ‘1

(‘1 = rll +

G

aH (2, v + Av, x, a$

t) -

aH

‘z;zv’

i

=* t,

1

A+

&g &

rlr)

W)

1052

A.I.

Egorov

PH (z + 8 AZ, v + Au, 2,

t) _

(0

az,azi

<

81 d

1)

- PH (z + OIA,e, v + Av, x, t) AziAzj dx dt az,azj 3 Formula troi

(16)

function

remainder auxiliary

defines

the

increment

q, variables

oi;

equations

may be represented

From this.

b,q

+

functional

S when

the eonthe the

ciAui = oi

aq at

the

setting C?Atl. --&,f

Then these

of

to v + Au. In order to estimate n equations of (7) and introduce

changes from v(x, t) we consider the first

dH

=

by virtue

in the ri = bici

‘iA5 + A q

of

conditions

(6).

it

form dCi -g-

$

follows

that

(17) 1

q(z,

5

ri (z:,2) Aui (2, T) +

t) =

A g

(18) 1

Since

the

functions ments

functions

aH/bi

uk and vi,

bi and

we find

from

0

where P

is

respect

(17)

in the with

region

G. and the

respect

to

4 I + iv, i

(i =

I Av, (~9 4 l] dz

positive

constants.

positive 0 to

constant.

Using

Integrating

1 and summing over

I,....nf

this

estimate,

all

this i,

inequality

we find

(B = nlPNJ

(t) < A + B i U fz) dz 0

P

i=l

i [hi 0

argu-

k=l

specified

to x from

U (t) = i

the

(18)

we find

a specified

U

I AQz,

bounded condition

j=l

where N, and N, are

with

are

a Lipschitz

I oi (5, t) I < i [~~ i

from equation

ri

satisfy

(x, t) 1dt,

A z

nlN2

52

G k=l

1Av, (x, t) 1dx dt

Optimal

We make use tion

U(t)

control

of

of Lemma I of

will

satisfy

the

in

processes

[Il,

p. 191,

distributed

objects

according

to

which

1053

the

func-

inequality U (t) < AeB’

From this

it

follows

that

t) 1dxdt Q nlPN,e BT

] Auj (x,

7 j Au,, (z, ss 22

Consequently,

from the

inequalities

(19)

I Aui (2, 8) I Q RI # $I where Rl

is

a specified

A similar

t) 1dxdt

G li=l

j=I 0

estimate

positive

we find

/ Auk (z, 4 I dx dt

constant.

can be found

for

t) 1. Consequently

I~wi(x,

r

1s2

1Azi (x,t){I
1 Auk (2, t) 1 dr dt

(i = I,,..,

2n)

(20)

G k==l

where R is

a positive

Formula

(16)

ities (20) sequently,

for

are the

Although

the

similar validity

readily proved that article.

for

increment

theorem the

that of

Rlf

the

R.

functional

S and the

inequal-

to the corresponding formulas in [51. Conof the maximum principle we have formulated

by repeating

the

conditions

number such

almost

literally

we have proved

existence

of

the

does

optimal

not

control

proof

of

provide

Theorem

is 1 of

any sufficient

functions,

it

enables

us to

find, among all the solutions of the boundary value problem (1) individual isolated solutions which satisfy the maximum condito (2), tions. In fact, to solve the problem by the maximum principle, we must determine (3)

2n f

and (H).

finding

ui,

ential trary

equations.

of

the

tions

of

the

some of

them,

therefore,

We thereby

case

and it

optimum control

determine

solutions where the

to

is

clear

2n + 1 equations system

are

of

we shall the

set

of

isolated

exist,

conditions

then

we should

be optimal.

system

of

equations

arbi-

supplementary solu-

(2) which satisfy the condithat there are a finite

from physical

functions

for

differ-

encounter

the

(l),

equations

second-order

by means of

value

solutions

that

these

In the

2n equations

problem (1) to If we find maximum principle.

such

problem

first

a ncomplete”

can be eliminated

and (6).

boundary

wi and v from the

we have

In solving which

(2)

tions

number of

ui,

mi and ~1. The

functions

conditions

the

1 unknowns

Consequently,

(1)

is

linear

of expect

1054

Egorou

A.I.

uk + ‘pi

tv)

fi =

I,...,

w

7%)

k=l

the following

theorem holds.

Theorem 2. In order that the control function U(X, t) in the system of equations (21) be min-optinal (aax-optimal~ with respect to S in the local sense, it is necessary and sufficient that it satisfy a maximum (minimum) condition.

The proof of this of the corresponding Example i.

theorem is almost literally theorem in [51.

Let a controlled

process

the same as the proof

be described

by the equation

where v i he control1 ing parameter. with IV/ < 1. It is required to find a contAo function tt(.z, t), 0 f x < 2. 0 f t Q 7’. such that for the solution of equation (22) that corresponds to this control function and satisfies the conditions U (z, 0) = u (0, t) = 0

(23)

the functional

ss

s=

(z-

1) u (z, t) dx dt

00

should

reach a minimum.

We define

the function

M(X, t)

SW

m-2 and the snpplementary will be of the form

aw -I-

ax

conditions

w (x, t) = ;

We construct

by means of the equation

aw at

=-2w-b-

S(X, T) = w(2,

a- -

1) tf = 0. This function

4 (1 _ e2(f-T))

the function H = u?(-

2u + vf

By the maximumprinciple, the optimal termined in accordance with the formula

control

function

will

be de-

Optima2

v (2, t) =

control

of

processes

1

sign -i_ (2eJCF2 - z) (i -

in distributed

ez(‘-*))

objects

1055

(O
or v (G 8) = sign (2e”+ -

z)

Here the expression in parentheses is equsl to zero at only one point y of the intervsl (0. 2). where y < 1; we hsve 2~*-* - x > 0 if x < y and 2eXs2 - x < 0 if z > y. Therefore

snd consequently the solution of the problem (22) for P = u(x, t) the form u(2,t)= On

% (1- e-')(1- eTst) { l/% (1 - emet) (&-('-Y) - eex

if

O\(ts;T

y
this solution S = lj4I2T + e-”

or,

ifO
is of

-

i] [ys -

4ev+

+

2e”]

(25)

if we tslceinto 8CCOUnt the fact that y =2 2@

W)

we find S =

I/(; [2T +

e-ZT -

11 [ya -

2y +

2&-s]

A direct check will convince us that if in formula (25) the quantity S is regarded 8s 8 function of the Variable y, varying over the interval (0, 21, then S will reach its minimum st the point y at which equstion (26) is S8tiSfied. This means that, 8mong 811 the control functions of the form (24), the control function 11(x, t) for which condition (26) is satisfied at the point y will give the function81 its minimum value. If we apply Theorem 2. we find that this control function is min-optimal. Example

2.

Let a controlled 8% --z.z

axat

v,

process

IvlGlr

be described

O
Odt
by the equation

(27)

with the supplementary conditions a (2,0) -_ u(0, t)= 0

(W

It is required to determine the admissible control function for which the functional

Egorov

A. I.

1056

1

s= reaches

its

minimum

To find axat

the

= 0 and

value,

function the

s 0

1

u (2, 1) dx -

and W(X,

to

t)

supplementary

(I, t)

u s 0

find

that

dt

minimum

value

of

S.

a%

we can make use of the equation conditions W(X, 1) = x - 1, ~(1.

t)

=

t.

1 -

Consequently,

W(X,

t)

=

x

t,

-

and

from

the

maximum

condition

we find

that V (x, t) = Thus,

we obtain

two

a2u

axat with

the

where

supplementary

T(X)

is

corresponding Thus,

in

the

Therefore,

I), for

(28).

From

is the the

the

1

of

additional condition. For example, to have continuous derivatives au/&

this

-

t)

u(x,

w-4

(t-x>@ we find

that

with

q1(0)

= 0.

The

l/3.

solution

corresponding

t) that u(x, the functional

a function value of

value

function

function, S is

functional

considered

function 9, while

the

d!?f-- -

differentiable the

example

keeping

finding

axat -

conditions

of

v (x, t) = 1 (t < 2)

x
an arbitrary value

optimum control trary function q.

equations (t-

I

=

1 (t>

to

depends S is

independent

S fixed on U(X, t). we can we may require the function and &/at

on the

the

on an arbi-

switching

line

.%= t. Then

In the arbitrary sponds to case apply control

it

example

considered

function q~ appearing the regular optimal

may depend the maximum functions

the

functional

in the control

on arbitrary functions. principle proved above, only

continuously

S is

solution function

independent

U(X, V(X,

Therefore. we should

differentiable

t), t).

of

the

which correIn the general

if we wish to assing to regular solutions

of

impose an u(x, t)

of

the

Optimal control

of

processes

in distributed

objects

1051

Goursat problem with predetermined conditions on the continuity of the derivatives. The above method of investigation may be applied to the study of optimal processes that can be described by hyperbolic equations with supplementary initial conditions and some forms of boundary conditions. Example

3. Let a controlled L

[u] z

process

aau T@

-

be described

2 = f (2,

Q2

by the equation

t, u, v)

where a2 is a positive constant, v is a controlling parameter, and f is continuous in x and t and is twice continuously differentiable with respect to u and V. Let the function U, determined by this equation, satisfy the supplementary conditions

where (pi(x) and yi(t) are continuously differentiable functions satisfying the matching conditions. The class of optimal control functions is determined in the same manner as in the problem considered above. We define the functional S by the formula T S =

{ a (z) a (z, 7’) dz i- 5 fi (f) u f 1, t) dt -I- 5 s 7 (r, t) u (2, t) dx dt 0

In this

c

0

case the role

of the system (3)

L [WI= w af I and the supplementary in the form w

(3, T) = 0,

The function

conditions

for

aw(2, T) / at=

a (4,

au-

is 7

played by the equation

(q t)

the function

W(X, t) must be taken

w (0, 1) = 0,

w (4 t) = P 0)

H is of the form

By the same reasoning as we applied to the study of the Goursat problem, we can readily establish the validity of Theorems 1 and 2.

BIBLIOGRAPHY 1.

sisteButkovskii. A-G. and Lerner, A. Ia., Ob optimal’nom upravlenii mami s raspredelennymi parametrami (On optimal control in systems with distributed parameters). Avtoarat ika i Te Zerekhanika, Vol. 21, 1960.

1058

2.

A. I.

Egorov

Pontriagin. L.S., Boftianskii, V.S., M~shchenko, E. I?., Matemotichezkaia sov

(MathematicaE

theory

of

Gamkrelidze, teoriia

optimal

R.B. and

optimal~nyk~

protses-

Fizmatgiz,

processes).

1961.

3.

Butkovskii, A. G., Gptimal’nye protsessy v sistemakh s raspredelennymi parametrami (Optimal Processes in systems with distributed parameters). Avtomatika i Telemekhanika, Vol. 22, No. 1, 1961.

4.

Butkovskii, A. 0,. Printsip maksimuma dlia optimal~nykh sistem s raspredelennymi parametrami (The maximumprinciple for optimal systems with distributed parameters). Automat ika i Teleaekhanika, Vol. 22, No. IO, 1961.

5.

Rozonoer, L. I., Printsip maks~mumaL.S. Pontriagina v teorii optfmal’nykh sistem (Pontriagin’s maximum principle in the theory of I-III. Autonatika i Telenekhanika, Vol. 29, optimal systems), Nos. 10-12. 1959.

6.

Sobolev,

S.L.,

mathenat

ical

Uravneni phys

7.

Privalov,

8.

Tikbonov, A.N,, shehenie gaza absorption of material). Zh.

9.

Tikhonov, fiziki

10.

I. I.,

ia

matenat

Gostekhizdat.

its).

Integral’nye

uravneniia

f iziki

(Equations

of

1954. (IntegraE

equations).

1935.

Zhukovskii, A. A. and Zabezhinskii, 1a.L.. Pogloiz toka vozdukha slowe zernistogo materials (The gas from a current of air by a layer of granular Fiz. Khiaii, Vol. 20, No. IO, 1946.

A.N. and Samarskii, (Squat

icheskoi

ions

of

A-A.,

mathenat

ical

Tricomi, F., Lektsii po uravneniiam (Lectures on partial differential

Uravneniia

natemat

icheskogo

Gostekhtzdat,

physics).

v chastnykh equations).

1953.

proizvodnykh Izd-vo

inostr.

fit.,

1957.

11. Nemytskii, rentsial’nykh equations).

V.V.

and Stepanov, uravnenii

Gostekhizdat,

V.V.,

(The

Kachestuennaia

qualitative

theory

teoriia of

diffe-

differential

1949.

Translated

by

A.S.