13 / 15
In connection with the fact that
γ
= (
k
M
−
1
k
εL
+
k
P
k
)
e
λt
∗
−
1
λ
≈
≈
0
.
947
<
1
, the condition of theorem 3 has been satisfied and the
terminal problem under consideration has got a solution.
Now we select the function
b
1
(
t
) =
−
t
3
+
t
2
as the function
b
1
(
t
)
,
which satisfies the conditions
b
1
(0) = 0
, b
0
1
(0) = 0
, b
1
(2) =
−
4
, b
0
1
(2) =
−
8
,
and the function
b
2
(
t
) =
t
3
−
2
t
2
as the function
b
2
(
t
)
, which satisfies the
conditions
b
2
(0) = 0
, b
0
2
(0) = 0
, b
2
(2) = 0
, b
0
2
(2) = 4
.
Let us specify the initial approximation for the vector of parameters
c
(0)
= (0; 0)
T
and the accuracy
σ
= 0
.
001
. Let us build the sequence
of approximations
{
c
(
j
)
}
using formula (25), assuming that
η
∗
= (
−
5; 4)
T
,
Ψ(
c
(
j
)
) =
η
(
t
∗
, c
(
j
)
)
, where
η
(
t, c
(
j
)
) = (
η
1
(
t, c
(
j
)
)
, η
2
(
t, c
(
j
)
))
T
, which is
the solution to the Cauchy problem:
˙
η
1
=
−
0
.
1
η
2
+
b
1
(
t
) +
c
(
j
)
1
d
(
t
) +
b
0
2
(
t
)+
+
c
(
j
)
2
d
0
(
t
) + 0
.
08 cos(
b
0
1
(
t
) +
c
(
j
)
1
d
0
(
t
);
˙
η
2
= 0
.
1
η
1
+
b
2
(
t
) +
c
(
j
)
2
d
(
t
) +
b
1
0
(
t
)+
+
c
(
j
)
1
d
0
(
t
)
−
0
.
08 sin(
b
0
2
(
t
) +
c
(
j
)
2
d
0
(
t
);
η
1
(0) = 0
, η
2
(0) = 0
,
being determined on each iteration using the Runge – Kutta method of the
fourth order. The calculations showed that inequality (27) can be fulfilled
with
J
= 6
, hence the point
c
(6)
= (
−
3
.
287; 8
.
933)
T
is the fixed point of
the mapping v with the accuracy
σ
. The functions
z
1
1
=
b
1
(
t
) +
c
(6)
1
d
(
t
)
, z
1
2
=
b
0
1
(
t
) +
c
(6)
1
d
0
(
t
)
, z
2
1
=
b
2
(
t
) +
c
(6)
2
d
(
t
)
,
z
2
2
=
b
0
2
(
t
) +
c
(6)
2
d
0
(
t
);
η
1
=
η
1
(
t, c
(6)
)
, η
2
=
η
2
(
t, c
(6)
)
specify the
t
-parameter curve in the range of system (28) conditions,
connecting the initial and final system statuses. The controls
u
1
=
b
00
1
(
t
) +
+
c
(6)
1
d
00
(
t
)
,
u
2
=
b
00
2
(
t
) +
c
(6)
2
d
00
(
t
)
realize this curve as the trajectory of
system (28) and are a solution of the terminal problem under consideration.
Functional relations
z
1
1
(
t
)
,
z
2
1
(
t
)
,
z
1
2
(
t
)
,
z
2
2
(
t
)
,
η
1
(
t
)
,
η
2
(
t
)
,
u
1
(
t
)
,
u
2
(
t
)
are
shown in the picture.
Conclusion.
The terminal problem for the affine systems, which are
not linearizable by a feedback, is considered. It is supposed that using
a smooth nondegenerate change of variables within the range of states,
28
ISSN 1812-3368. Herald of the BMSTU. Series “Natural Sciences”. 2014. No. 5