5 / 15
η
(
t
∗
) =
η
∗
, then, for functions
B
i
(
t
) =
b
i
(
t
) +
c
i
∗
d
i
(
t
)
,
i
= 1
, m
, all
conditions of theorem 2 are fulfilled and, thus, terminal problem (3), (4)
for system (2) has got a solution.
Let us assume that
ρ
6
m
. The Euclidean norm will be considered as the
vector norm from the space
R
ρ
and
ρ
×
ρ
-matrices. Let
r
= max
{
r
1
, . . . , r
ρ
}
.
For all pairs of the indices
l
и
j
, where
l
∈ {
2
, . . . , r
}
,
j
∈ {
1
, . . . , ρ
}
,
l > r
j
, we will introduce the formally additional variables
z
j
l
. Let us denote
z
l
= (
z
1
l
, . . . , z
ρ
l
)
T
,
l
= 1
, r
. According to the definition, let us assume that
if
l > r
j
, then
∂q
i
/∂z
j
l
= 0
for all
i
= 1
, ρ
. Let us take
∂q/∂z
l
for denoting
ρ
×
ρ
-matrices with elements
∂q
i
/∂z
j
l
,
i, j
= 1
, ρ
.
Irrespective of the number
i
, we specify the functions
d
i
(
t
)
with the
formula
d
i
(
t
)
≡
d
(
t
) =
t
r
(
t
∗
−
t
)
r
t
∗
R
0
t
r
(
t
∗
−
t
)
r
dt
.
(9)
We will denote
L
= max
[0
,t
∗
]
{
d
(
t
) +
|
d
0
(
t
)
|
+
|
d
00
(
t
)
|
+
. . .
+
|
d
(
r
−
1)
(
t
)
|}
.
Let us prove the following auxiliary statement.
Lemma 1.
Let
P
(
t
)
,
R
(
t
)
be
ρ
×
ρ
-matrices with elements
P
ij
(
t
)
,
R
ij
(
t
)
∈
C
[0
, t
∗
]
, and there exists such a number
λ
∈
R
, with all
y
∈
R
ρ
,
t
∈
[0
, t
∗
]
, the inequality is fulfilled
:
(
P
(
t
)
y, y
)
6
λ
k
y
k
2
.
(10)
Then
ρ
×
ρ
-matrix
W
(
t
)
,
which is the solution to the Cauchy problem
˙
W
=
P
(
t
)
W
+
R
(
t
)
,
W
(0) = 0
,
(11)
satisfies the inequality
k
W
(
t
∗
)
k
6
e
λt
∗
t
∗
Z
0
k
R
(
t
)
k
e
−
λt
dt.
(12)
J
We denote
j
-th matrix columns
W
(
t
)
and
R
(
t
)
with
W
j
(
t
)
and
R
j
(
t
)
,
respectively. Then the system
˙
W
=
P
(
t
)
W
+
R
(
t
)
can be written in the
following form:
˙
W
1
˙
W
2
. . .
˙
W
ρ
=
P
(
t
) 0
. . .
0
0
P
(
t
)
. . .
0
...
...
. . .
...
0 0
. . . P
(
t
)
W
1
W
2
. . .
W
ρ
+
R
1
(
t
)
R
2
(
t
)
. . .
R
ρ
(
t
)
.
20
ISSN 1812-3368. Herald of the BMSTU. Series “Natural Sciences”. 2014. No. 5