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The optimization of some linear systems with distributed parameters
THE OPTIMIZATIONOF SOlMELINEAR SYSTE!lS WITH DISTRiBUTEDPARAMETERS* A. R. RA6rNOVrCH Iv(oscow (Received
13 December
1966)
1. Introduction COPIT390L
systems
equation
(or set
meters.
In these
Banach set
space
ment
systems
We shall
some
maximum
cases
[61 the maximum paper
of the elliptic
principle
was
principle which
formulated
original
and conjugate
since
we do not wish
changes
or problems
obtained
are,
to consider for which
were
in a fairly
be obtained
problems
general
is a fairly
for which
changes
that
its
studied,
but it particular the
formulated.
In
formulation.
In this
the motion
of which
a natural assume
natural
data
is
generalizathat
the solutions
it is known
in the initial
attain
In them
R’e shalt
POand
requirea way
In [Z-S1
principle
of the author,
This
were
considered.
equations.
are correct.
small
type
for systems,
for any (r) E
this
in such
- will
function.
and the maximum
and hyperbolic
systems
IJsually
WOE Q,,
of this
the Green’s
in the opinion
a solution
compact.
on some
may be con-
the
of the
requirement
that
there
cause
are no
large
in the result.
In this in Section
*
has
problems
parato some
bounded
which
of the trajectory
parameters down
will
parabolic
problem
solutions,
distributed written
by equations
functions
the control
criterion
general
were
the maximum
defined
set
to select
- optimality
with
of measurable
is weakly
In [II fairly
systems
with distributed
that
this
differential
of as belonging
S? a set QO is defined
to know how to calculate
of systems
conjugate
systems
w may be thought
the space
It is required
value.
is necessary
called
by a partial
In the space
assume
functional
is specified
are usually
the control
St (for example,
is satisfied.
that
of which
of equations)
with norm vrai max).
trolled.
tion
the motion
paper
the maximum
3 and the conjugate
Zh. qchisl.
principle
will be obtained
equation
will
swat. mat. Fiz. 8, 1, 195-202,
274
be extracted. 1968.
for the systems The maximum
considered principle
275
A. B. Rabinovich
in terms
of the conjugate
in terms
of the Green’s
fairly
troublesome
may be carried
has
function
thing,
that
while
this
advantage
over the maximum
the calculation
the solution
of the Green’s
of the conjugate
principle
function
(and original)
is a system
out on a computer.
2. 1.
system
We consider
Examples
of the systems
considered
the problem div (A grad U) = o(z)
with
the boundary
conditions la(A grad U, v) -
Here
z belongs
boundary
to P a set
of the domain
? O. Here
In order sufficient
that that
below
normal
Euclidean
/? is a symmetric,
to an.
the control
It is required
ti,’ E ~3~ should
\ oo(5)Y(z)d5
Therefore,
in order
to find
not contain
must The
The maximum I,et
definite
n X n matrix,
it is necessary
and
(2.1)
D
conjugate
system (2.2)
= -C(r)
conditions
does
a’(z) obtained
2.
set
max tj o(r)Y(s)dz, ~EQO
(x) is th e so 1u t’Ion of the
,a(A grad Y, v) -
YO found.
positive
~EiQo
where
the boundary
which
.yn, a.0 is the from some
to find
= masI(o)=
. L!
be satisfied,
(2.3),
space is taken
be optimal,
div (A gradY) with
which
the condition I(oo)=
should
of the n-dimensional
II and w(x) is the control
and everywhere
and v is the inward
BClan = Y(S).
the the
optimal control
be substituted
control
principle
us consider
is proved
the following
control,
in this
it is necessary
cl,(x) and hence
in the integral
alO(x) found
(2.3)
fN’(a~ = 0.
(2.1)
way will
in a more problem
arising
and its maximum
be the optimal
general
to solve
may be solved;
formulation in some
heat
(2.2), then
the
on the set
one. in Section exchange
3.
276
Some linear systems
with distributed parameters
processes f= with the initial
div(A grad U)+(b,
grad V) = f(s)
conditions U(x, 0) = Uo(x)
and boundary
conditions (A grad V, Y) - pU[ me = o(z,
Here, as before, o(x,
t)
x E, D. The problem is considered
is the control,
which belongs a
grad = It is required
In this case
a
in the cylinder
nX 10, n;
to the domain RO: “1; .
Z~‘az,““&J
t).
aiv
to find
the necessary
the satisfaction
and sufficient
condition
of optimality
of the condition T
I(o”)=
of the control
T
(j dt S o”(2, t)Y(z, t)dx = mini(w) 0
OE%
BD
where Y(z, t) satisfies
o
i3D
dia(d grad Y)+ div(bY) = 0
conditions
Y (x, T) = and the boundary
me,
the equation - G-
with the initial
= min 5 df 1 o(z, t)Y(s, t)ds
c (x)
conditions (A grad Y, Y) - [B + (b, ~11 ‘PI a~ = 0.
The formulation
of the maximum principle
3. Let us consider
a problem
arising
is very similar
for both problems,
in the theory of oscillations,
is
211
A. 3. Rabinovich
with
the initial
conditions
autx,0) =
-
U(s, 0) = Uo(x), and the boundary
U,‘(x)
at
conditions
a;- BUl
BD= Y(‘%*I’
Here
3 is Laplace’s
some
Ranach
operator,
space
9.
and the control
It is required
the necessary condition J(Q)=
and sufficient
from the domain
C(x) -
D
condition
at
for optimality
is the satisfaction
T c Y 0
dl
s
tO(5,
t, oo)Y(s,
qdx
=wm3gi(o)=
D T G=
dt
s 0
where
3’(x,
t) satisfies
20(x,
s D
t, 0)Y
d2Y
a ---(c,
Y)-
at2 with
the initial
and the boundary
Let
Cauchy’s
0
problem
cW
= AU
conditions d”--w(x, ap--1
It is required
SZZ
conditions
us consider
the initial
A(aT) = 0
f=(x, T) at
T) = C(x),
WU with
t)dz,
conditions Y(x,
4.
(x,
the equation
--
to find
0)
RO of
au@, T) dx;
s
max *iSo
o is taken
to find
. = Im(x, w),
m = 1, 2, . . . , N.
of the
Some linear systems with distributed paramctcrs
278
The necessary
and sufficient
condition
for optimality
is the satisfaction
of the
condition N
2
f(oO)=
(-flN-”
0) shn(x, a*)aN-my(x* atN_m
m=i
max
--&*;
of the equation PY
(-‘)i’+=
AY+Co
conditions
a*-iY(x,T) atm_’
(-I)“-’
3. 1. The following the cylinder
_
D
where I!J!(x,6) is a solution
with the initial
dx
m = 1, 2, . . . , N.
* = CN_m+f,
Formulation of the maximum principle problem
is a natural
generalization
p X f 0, 71 let the motion of the system
of these
problems.
to be controlled
In
be specified
by the equation
with the initial
conditions
a+-w(z, at+ and boundary
operators, (0 beIongs
iV >, 1; 4 is a symmetric, system assumed
0)
J = &,a@,QJ),
m = 1, 2, . . . , N,
(3.2)
conditions
I n are prescribed The control
(34
div(A grad U) + (6 grad u)-l- lo(% 4 (4
&;= m-o
are assumed that either
in general
non-linear.
to the domain R. of the Ranach positive-definite
to be differentiable a f 0 everywhere
n X
R
matrix;
the required
space
9;
a!+~> 0, if
the coefficients
number of times;
of the it is also
in an X [0, ?‘j, or that a I 0, and then
279
A. B. Rabinovich
S f 0 everywhere
in ar? X [0,
s required
n is either
the d omain
71;
or P = Rn.
bounded,
to find
(3.4)
max S WE”0 where T
So(U)= .\ dt 0
s
1 G(z,
8+‘U(x,
Cm(x, t)
T)
dx,
ai?-’
D
tx,t,
‘N+,
t)U(z, qdx,
(3.5)
D
a
m = 1, 2, . . . , N,
u 6%t) ax,
(3.6)
a # 0;
if
(3.7) ‘N+L(“v t, p (AwdU, The
choice
of the form of the functional
in the case
a = 0 the value
that
choice
by this
system
Zemark. conditions
In the case (3.3)
0 = IZn we obtain
system.
We call
m &(a,,,‘P)=
with
the
initial
cauchy’s
firstly,
in the
conjugate
a = 0 or a 4 0. and the boundary
div(bY)+
Co
m = 1, 2,
. . . , N,
(3.8)
conditions
and the boundary
conditions
system.
form it is
(3.9)
t=T
a(~gradY, conjugate
problem,
that,
and secondly,
the system
= c,,
explicit
conditions
of whether
div(A gradY)-
k=m
the
surface,
(3.7) are absent.
and the functional
2. The conjugate
independently
a=O.
by the facts
on the lateral
S,v+ 1 the boundary
can be expressed
if
SN+1 is justified
of U(x, t) is given
of the functional
(3.8)~(3.10)
VI,
v) -@+
It is obvious
that
a(& ~)ly(a~ (3.8)
is always
(3.10)
= -CN+I solvable.
Indeed,
in
Some linear systems
UN\fC
=C N*
aN_ly - @NY)’
= 'N-1,
. . . . . . . . . alY -
(a‘uy
..I.......
+ . . . + (-l)N-‘(aNY)(N-l)
.
.
(3.11)
.
l=;lT = Cr.
in the expressions (ak?Y)(k-m) and collecting I \ for Ijrtrn’ (x, T) a set of equations which is obviously
we obtain
since
the matrix of this set of equations
stand
a,~ with the sign + or -.
obvious
.
the brackets
Removing terms,
parameters
w’th distti!wed
(9ere
that as soon as problem
coefficients
1.
are naturally Remark
Wm)(t,
solvable,
and along its diagonal
T) = PY(z,
(3.1) - (3.4) is given,
of (3.1) - (3.3) and of the functional9
we can solve first (3.11), Remark
is triangular
similar
(35)
T) /aP
). It is
that is, as soon as the - (3.7) have been given,
and then (3.8) - (3.10).
In the case of Cauchy’s
problem the boundary
conditions
(3.10)
absent. 2. Together
with !P(r, t) it is possible
. For example,
‘u@, r--t)
if the coefficients
to consider
-(z,
t) =
of (3.1) - (3.3) do not depend
on
t, ?((x, t) is of the form
with the initial
(3.12)
$ = div (A grad &) - div (b%) - &I
&$ m=lJ conditions
(3.23)
and the boundary
conditions (3.14)
o(A grad q, v) - [fi f o(b, v)] @!a~ = -CN+rTherefore, system 3.
if the original
is correct
system
The maximum principle.
necessary
was correct,
then in this case the conjugate
also.
and sufficient
For the control
that the relation
WOE 90 to be optimal,
it is
281
A. R. Rabinovich
should
be satisfied,
where T Jo(o)=
1
at
Jm(0)E
j
(i
Z,(o)
h-m
D
1
lo(o)Y
(3.15)
ds,
D
0
(--i)~-m(a*Y)(~-“)It--a) dz, ,
(3.16)
T [ JN+l
0s
(O) =
dt
s 8D
N+1(0) y dx,
s s o dt BD
i -
‘?(x, t) is the solution
Ln(*, 0) = L(i) of 0, since
_.!-
ZN+1 (co) [(A
J(o)
of the conjugate
(3.8) - (3.9).
An inhomogeneity
boundary conditions
It is not difficult
Y
0
0) is independent is also independent
l-4,
in the control
initial
of
subscripts
2).
the role of inhomogeneities
(3.8) itself
and inhomogeneous
permit functional6
Section
to those
permits functionals conditions
in of
and inhomo-
of the form (3.6) and (3.7)
Proof of the maximum principle to see that
N
N
u2
I,(*, fm(o)
of
to be maximized.
4.
T
may be simplified
where the summation extends
we have explained
the form (3.5) to be maximized,
1.
functional
on o (see paragraphs
Simultaneously,
Remark 2.
a = 0;
system.
of the maximum principle
= C, 1 J,(o),
depends
if
Y) - (b, v) Yl dx,
f>or some m, that is, if the expression
for which Z,(O)
geneous
grad I,
’
in this case the corresponding
Therefore,
respectively
a#O; (3.17)
1. The formulation
Remark
if
T
I
o.
+1
a,~~w
m=o
-
U
2 (-l)m(am~)(m) m-0
1
(4.1)
dt =
t=T i
(__l)k-m((lkyp--m)
II t=o CL=,
(here
Zcrn)5 ~2
easy to obtain
/ at”‘) . Indeed,
k=7?3
using the formula of integration
by parts, it is
282
Some linear systems
WiJ
distributed
m-1
T
s
__(_~)qJ(a,y)cqat
ppampL)
2
=
(-¶)m-l-~U(l)(a,y)(m-[-oI:~~*
1-o
0
from which
formula
Ye consider
(4.1)
easily
5 dx i[V D
If YJ satisfies the initial
follows.
the integral I-
&UWMr~
the conjugate
conditions
(-1)qQqqdt. ??I=.0
P&=0
0
system
(3.2)
(3.8-(3.101,
and (3.9)
2
I=
using
(3.6)
(4.1),
and (3.16)
by virtue
of
we obtain
N
2
S,(U)-
(4.21
I,(W).
~-1
m-1
Changing
then
and formulas
N
2.
parameters
the order of integration
and
using
(3.1)
a&
(3.8),
we
obtain
C,v]
dx.
T z= s 0
dt
[Y div(A
s D
grad U)-
U div(A
grad Y)]
dx + T
T +
1
dt
1 cfY(b,
0
Applying
Green’s
formula
to the first
u
div (A grad Y) div (A grad V)
and Gauss’
U div(bY)]
theorem
I
div(bYU)dz
D
(3.5)
5 dt 1 [lo(w)Y
and (3.1,5),
integral
ax = -
sl
Y
BD (A gradY,
=
-
s
YU(b,
v)dx
BD
we obtain
r I=
s
0
at
s
IU(AgradY,y)-Y(AgradU,y)-Y~(b,v)ldz+Jo(o)-S90(U).
BD
If Q f 0, we have
-
D
to the sticond
s and using
ds +
cl
Y
SI
D
grad U)+
D
u Y) (-4 grad u, v)
Iax,
A. B. Rabinovich
283
T
I=
s St&c(AgradU, at
v)-pq
+
In the case
>
1 J&4gradY,v) - [B +
dt
i
a(4 v)l ‘PI ds + Jo(o)- So(u).
BD
0
rising
dz+
a
aD
0
(x.= 0
(3.3),
(7.7),
(3.10!,
(3.17),
we obtain
1 = Jo(O) +I~+i(o) Comparing
(4.?)
and (a.?),
-SO(u)
we obtain X+1
N+1 So
2
S,(V)=
J(o)=
p”.(o). m-0
m-0
By (4.4)
the functional
S(U), whence
(4.3)
-sN-i(u).
I(w)
the maximum
attains
a maximum
principle
simultaneously
with the functional
follows.
by .T. Oerry
Translated
REFERENCES 1.
2.
BUTKOVSKII,
A. G.
distributed
parameters.
ECOROV,
A. I.
parameters. 3.
F.GOROV,
EGOROV,
Yu.
6.
conditions
On optimal
V.
PLOTNIKOV,
V. I.
distributed
parameters.
Mat. Sb. 64,
V.
principles Telemekh.
Some problems
22,
of optimality
of systems
10, 1288-1301, for systems
with
1%1. with
distributed
1%6. in systems 26,
containing
6, 977-994;
of the theory
7,
of optimal
objects 1188-11%.
with
distributed
1965. Zh. tfchisl.
control.
1963.
On a problem
of optimal
Dokl. Akod,
Necessary
1, 79-101,
for the optimization
Telemekh.
3, 371-421,
S, 887-904,
F,GORO\‘, Yu.
principle
Necessary
Automatika
mat. Fiz.3, 5.
maximum Avtomatika
#Mat. Sb. 69, A. 1.
parameters. 4.
The
condition 1961.
control
Nauk SSSR. of optimality
by stationary
systems
170,2, 290-293,
1966.
of control
in Banach
with
spaces.
Mat.
13 December
1966)
1. Introduction COPIT390L
systems
equation
(or set
meters.
In these
Banach set
space
ment
systems
We shall
some
maximum
cases
[61 the maximum paper
of the elliptic
principle
was
principle which
formulated
original
and conjugate
since
we do not wish
changes
or problems
obtained
are,
to consider for which
were
in a fairly
be obtained
problems
general
is a fairly
for which
changes
that
its
studied,
but it particular the
formulated.
In
formulation.
In this
the motion
of which
a natural assume
natural
data
is
generalizathat
the solutions
it is known
in the initial
attain
In them
R’e shalt
POand
requirea way
In [Z-S1
principle
of the author,
This
were
considered.
equations.
are correct.
small
type
for systems,
for any (r) E
this
in such
- will
function.
and the maximum
and hyperbolic
systems
IJsually
WOE Q,,
of this
the Green’s
in the opinion
a solution
compact.
on some
may be con-
the
of the
requirement
that
there
cause
are no
large
in the result.
In this in Section
*
has
problems
parato some
bounded
which
of the trajectory
parameters down
will
parabolic
problem
solutions,
distributed written
by equations
functions
the control
criterion
general
were
the maximum
defined
set
to select
- optimality
with
of measurable
is weakly
In [II fairly
systems
with distributed
that
this
differential
of as belonging
S? a set QO is defined
to know how to calculate
of systems
conjugate
systems
w may be thought
the space
It is required
value.
is necessary
called
by a partial
In the space
assume
functional
is specified
are usually
the control
St (for example,
is satisfied.
that
of which
of equations)
with norm vrai max).
trolled.
tion
the motion
paper
the maximum
3 and the conjugate
Zh. qchisl.
principle
will be obtained
equation
will
swat. mat. Fiz. 8, 1, 195-202,
274
be extracted. 1968.
for the systems The maximum
considered principle
275
A. B. Rabinovich
in terms
of the conjugate
in terms
of the Green’s
fairly
troublesome
may be carried
has
function
thing,
that
while
this
advantage
over the maximum
the calculation
the solution
of the Green’s
of the conjugate
principle
function
(and original)
is a system
out on a computer.
2. 1.
system
We consider
Examples
of the systems
considered
the problem div (A grad U) = o(z)
with
the boundary
conditions la(A grad U, v) -
Here
z belongs
boundary
to P a set
of the domain
? O. Here
In order sufficient
that that
below
normal
Euclidean
/? is a symmetric,
to an.
the control
It is required
ti,’ E ~3~ should
\ oo(5)Y(z)d5
Therefore,
in order
to find
not contain
must The
The maximum I,et
definite
n X n matrix,
it is necessary
and
(2.1)
D
conjugate
system (2.2)
= -C(r)
conditions
does
a’(z) obtained
2.
set
max tj o(r)Y(s)dz, ~EQO
(x) is th e so 1u t’Ion of the
,a(A grad Y, v) -
YO found.
positive
~EiQo
where
the boundary
which
.yn, a.0 is the from some
to find
= masI(o)=
. L!
be satisfied,
(2.3),
space is taken
be optimal,
div (A gradY) with
which
the condition I(oo)=
should
of the n-dimensional
II and w(x) is the control
and everywhere
and v is the inward
BClan = Y(S).
the the
optimal control
be substituted
control
principle
us consider
is proved
the following
control,
in this
it is necessary
cl,(x) and hence
in the integral
alO(x) found
(2.3)
fN’(a~ = 0.
(2.1)
way will
in a more problem
arising
and its maximum
be the optimal
general
to solve
may be solved;
formulation in some
heat
(2.2), then
the
on the set
one. in Section exchange
3.
276
Some linear systems
with distributed parameters
processes f= with the initial
div(A grad U)+(b,
grad V) = f(s)
conditions U(x, 0) = Uo(x)
and boundary
conditions (A grad V, Y) - pU[ me = o(z,
Here, as before, o(x,
t)
x E, D. The problem is considered
is the control,
which belongs a
grad = It is required
In this case
a
in the cylinder
nX 10, n;
to the domain RO: “1; .
Z~‘az,““&J
t).
aiv
to find
the necessary
the satisfaction
and sufficient
condition
of optimality
of the condition T
I(o”)=
of the control
T
(j dt S o”(2, t)Y(z, t)dx = mini(w) 0
OE%
BD
where Y(z, t) satisfies
o
i3D
dia(d grad Y)+ div(bY) = 0
conditions
Y (x, T) = and the boundary
me,
the equation - G-
with the initial
= min 5 df 1 o(z, t)Y(s, t)ds
c (x)
conditions (A grad Y, Y) - [B + (b, ~11 ‘PI a~ = 0.
The formulation
of the maximum principle
3. Let us consider
a problem
arising
is very similar
for both problems,
in the theory of oscillations,
is
211
A. 3. Rabinovich
with
the initial
conditions
autx,0) =
-
U(s, 0) = Uo(x), and the boundary
U,‘(x)
at
conditions
a;- BUl
BD= Y(‘%*I’
Here
3 is Laplace’s
some
Ranach
operator,
space
9.
and the control
It is required
the necessary condition J(Q)=
and sufficient
from the domain
C(x) -
D
condition
at
for optimality
is the satisfaction
T c Y 0
dl
s
tO(5,
t, oo)Y(s,
qdx
=wm3gi(o)=
D T G=
dt
s 0
where
3’(x,
t) satisfies
20(x,
s D
t, 0)Y
d2Y
a ---(c,
Y)-
at2 with
the initial
and the boundary
Let
Cauchy’s
0
problem
cW
= AU
conditions d”--w(x, ap--1
It is required
SZZ
conditions
us consider
the initial
A(aT) = 0
f=(x, T) at
T) = C(x),
WU with
t)dz,
conditions Y(x,
4.
(x,
the equation
--
to find
0)
RO of
au@, T) dx;
s
max *iSo
o is taken
to find
. = Im(x, w),
m = 1, 2, . . . , N.
of the
Some linear systems with distributed paramctcrs
278
The necessary
and sufficient
condition
for optimality
is the satisfaction
of the
condition N
2
f(oO)=
(-flN-”
0) shn(x, a*)aN-my(x* atN_m
m=i
max
--&*;
of the equation PY
(-‘)i’+=
AY+Co
conditions
a*-iY(x,T) atm_’
(-I)“-’
3. 1. The following the cylinder
_
D
where I!J!(x,6) is a solution
with the initial
dx
m = 1, 2, . . . , N.
* = CN_m+f,
Formulation of the maximum principle problem
is a natural
generalization
p X f 0, 71 let the motion of the system
of these
problems.
to be controlled
In
be specified
by the equation
with the initial
conditions
a+-w(z, at+ and boundary
operators, (0 beIongs
iV >, 1; 4 is a symmetric, system assumed
0)
J = &,a@,QJ),
m = 1, 2, . . . , N,
(3.2)
conditions
I n are prescribed The control
(34
div(A grad U) + (6 grad u)-l- lo(% 4 (4
&;= m-o
are assumed that either
in general
non-linear.
to the domain R. of the Ranach positive-definite
to be differentiable a f 0 everywhere
n X
R
matrix;
the required
space
9;
a!+~> 0, if
the coefficients
number of times;
of the it is also
in an X [0, ?‘j, or that a I 0, and then
279
A. B. Rabinovich
S f 0 everywhere
in ar? X [0,
s required
n is either
the d omain
71;
or P = Rn.
bounded,
to find
(3.4)
max S WE”0 where T
So(U)= .\ dt 0
s
1 G(z,
8+‘U(x,
Cm(x, t)
T)
dx,
ai?-’
D
tx,t,
‘N+,
t)U(z, qdx,
(3.5)
D
a
m = 1, 2, . . . , N,
u 6%t) ax,
(3.6)
a # 0;
if
(3.7) ‘N+L(“v t, p (AwdU, The
choice
of the form of the functional
in the case
a = 0 the value
that
choice
by this
system
Zemark. conditions
In the case (3.3)
0 = IZn we obtain
system.
We call
m &(a,,,‘P)=
with
the
initial
cauchy’s
firstly,
in the
conjugate
a = 0 or a 4 0. and the boundary
div(bY)+
Co
m = 1, 2,
. . . , N,
(3.8)
conditions
and the boundary
conditions
system.
form it is
(3.9)
t=T
a(~gradY, conjugate
problem,
that,
and secondly,
the system
= c,,
explicit
conditions
of whether
div(A gradY)-
k=m
the
surface,
(3.7) are absent.
and the functional
2. The conjugate
independently
a=O.
by the facts
on the lateral
S,v+ 1 the boundary
can be expressed
if
SN+1 is justified
of U(x, t) is given
of the functional
(3.8)~(3.10)
VI,
v) -@+
It is obvious
that
a(& ~)ly(a~ (3.8)
is always
(3.10)
= -CN+I solvable.
Indeed,
in
Some linear systems
UN\fC
=C N*
aN_ly - @NY)’
= 'N-1,
. . . . . . . . . alY -
(a‘uy
..I.......
+ . . . + (-l)N-‘(aNY)(N-l)
.
.
(3.11)
.
l=;lT = Cr.
in the expressions (ak?Y)(k-m) and collecting I \ for Ijrtrn’ (x, T) a set of equations which is obviously
we obtain
since
the matrix of this set of equations
stand
a,~ with the sign + or -.
obvious
.
the brackets
Removing terms,
parameters
w’th distti!wed
(9ere
that as soon as problem
coefficients
1.
are naturally Remark
Wm)(t,
solvable,
and along its diagonal
T) = PY(z,
(3.1) - (3.4) is given,
of (3.1) - (3.3) and of the functional9
we can solve first (3.11), Remark
is triangular
similar
(35)
T) /aP
). It is
that is, as soon as the - (3.7) have been given,
and then (3.8) - (3.10).
In the case of Cauchy’s
problem the boundary
conditions
(3.10)
absent. 2. Together
with !P(r, t) it is possible
. For example,
‘u@, r--t)
if the coefficients
to consider
-(z,
t) =
of (3.1) - (3.3) do not depend
on
t, ?((x, t) is of the form
with the initial
(3.12)
$ = div (A grad &) - div (b%) - &I
&$ m=lJ conditions
(3.23)
and the boundary
conditions (3.14)
o(A grad q, v) - [fi f o(b, v)] @!a~ = -CN+rTherefore, system 3.
if the original
is correct
system
The maximum principle.
necessary
was correct,
then in this case the conjugate
also.
and sufficient
For the control
that the relation
WOE 90 to be optimal,
it is
281
A. R. Rabinovich
should
be satisfied,
where T Jo(o)=
1
at
Jm(0)E
j
(i
Z,(o)
h-m
D
1
lo(o)Y
(3.15)
ds,
D
0
(--i)~-m(a*Y)(~-“)It--a) dz, ,
(3.16)
T [ JN+l
0s
(O) =
dt
s 8D
N+1(0) y dx,
s s o dt BD
i -
‘?(x, t) is the solution
Ln(*, 0) = L(i) of 0, since
_.!-
ZN+1 (co) [(A
J(o)
of the conjugate
(3.8) - (3.9).
An inhomogeneity
boundary conditions
It is not difficult
Y
0
0) is independent is also independent
l-4,
in the control
initial
of
subscripts
2).
the role of inhomogeneities
(3.8) itself
and inhomogeneous
permit functional6
Section
to those
permits functionals conditions
in of
and inhomo-
of the form (3.6) and (3.7)
Proof of the maximum principle to see that
N
N
u2
I,(*, fm(o)
of
to be maximized.
4.
T
may be simplified
where the summation extends
we have explained
the form (3.5) to be maximized,
1.
functional
on o (see paragraphs
Simultaneously,
Remark 2.
a = 0;
system.
of the maximum principle
= C, 1 J,(o),
depends
if
Y) - (b, v) Yl dx,
f>or some m, that is, if the expression
for which Z,(O)
geneous
grad I,
’
in this case the corresponding
Therefore,
respectively
a#O; (3.17)
1. The formulation
Remark
if
T
I
o.
+1
a,~~w
m=o
-
U
2 (-l)m(am~)(m) m-0
1
(4.1)
dt =
t=T i
(__l)k-m((lkyp--m)
II t=o CL=,
(here
Zcrn)5 ~2
easy to obtain
/ at”‘) . Indeed,
k=7?3
using the formula of integration
by parts, it is
282
Some linear systems
WiJ
distributed
m-1
T
s
__(_~)qJ(a,y)cqat
ppampL)
2
=
(-¶)m-l-~U(l)(a,y)(m-[-oI:~~*
1-o
0
from which
formula
Ye consider
(4.1)
easily
5 dx i[V D
If YJ satisfies the initial
follows.
the integral I-
&UWMr~
the conjugate
conditions
(-1)qQqqdt. ??I=.0
P&=0
0
system
(3.2)
(3.8-(3.101,
and (3.9)
2
I=
using
(3.6)
(4.1),
and (3.16)
by virtue
of
we obtain
N
2
S,(U)-
(4.21
I,(W).
~-1
m-1
Changing
then
and formulas
N
2.
parameters
the order of integration
and
using
(3.1)
a&
(3.8),
we
obtain
C,v]
dx.
T z= s 0
dt
[Y div(A
s D
grad U)-
U div(A
grad Y)]
dx + T
T +
1
dt
1 cfY(b,
0
Applying
Green’s
formula
to the first
u
div (A grad Y) div (A grad V)
and Gauss’
U div(bY)]
theorem
I
div(bYU)dz
D
(3.5)
5 dt 1 [lo(w)Y
and (3.1,5),
integral
ax = -
sl
Y
BD (A gradY,
=
-
s
YU(b,
v)dx
BD
we obtain
r I=
s
0
at
s
IU(AgradY,y)-Y(AgradU,y)-Y~(b,v)ldz+Jo(o)-S90(U).
BD
If Q f 0, we have
-
D
to the sticond
s and using
ds +
cl
Y
SI
D
grad U)+
D
u Y) (-4 grad u, v)
Iax,
A. B. Rabinovich
283
T
I=
s St&c(AgradU, at
v)-pq
+
In the case
>
1 J&4gradY,v) - [B +
dt
i
a(4 v)l ‘PI ds + Jo(o)- So(u).
BD
0
rising
dz+
a
aD
0
(x.= 0
(3.3),
(7.7),
(3.10!,
(3.17),
we obtain
1 = Jo(O) +I~+i(o) Comparing
(4.?)
and (a.?),
-SO(u)
we obtain X+1
N+1 So
2
S,(V)=
J(o)=
p”.(o). m-0
m-0
By (4.4)
the functional
S(U), whence
(4.3)
-sN-i(u).
I(w)
the maximum
attains
a maximum
principle
simultaneously
with the functional
follows.
by .T. Oerry
Translated
REFERENCES 1.
2.
BUTKOVSKII,
A. G.
distributed
parameters.
ECOROV,
A. I.
parameters. 3.
F.GOROV,
EGOROV,
Yu.
6.
conditions
On optimal
V.
PLOTNIKOV,
V. I.
distributed
parameters.
Mat. Sb. 64,
V.
principles Telemekh.
Some problems
22,
of optimality
of systems
10, 1288-1301, for systems
with
1%1. with
distributed
1%6. in systems 26,
containing
6, 977-994;
of the theory
7,
of optimal
objects 1188-11%.
with
distributed
1965. Zh. tfchisl.
control.
1963.
On a problem
of optimal
Dokl. Akod,
Necessary
1, 79-101,
for the optimization
Telemekh.
3, 371-421,
S, 887-904,
F,GORO\‘, Yu.
principle
Necessary
Automatika
mat. Fiz.3, 5.
maximum Avtomatika
#Mat. Sb. 69, A. 1.
parameters. 4.
The
condition 1961.
control
Nauk SSSR. of optimality
by stationary
systems
170,2, 290-293,
1966.
of control
in Banach
with
spaces.
Mat.