3 / 15
and mapping
Φ : Ω
→
Φ(Ω)
, prescribed by the system of functions
z
i
k
=
F
k
−
1
ϕ
i
(
x
)
, k
= 1
, r
i
, i
= 1
, m
;
η
l
=
ϕ
n
−
ρ
+
l
(
x
)
, l
= 1
, ρ,
was a diffeomorphism.
In variables
z
1
,
. . .
,
z
m
,
η
, system (1) has a quasicanonical form (2). If
the matrix of the coefficients in system controls (2)
g
(
z
1
, . . . , z
m
, η
) =
g
11
(
z
1
, . . . , z
m
, η
)
. . . g
1
m
(
z
1
, . . . , z
m
, η
)
...
. . .
...
g
m
1
(
z
1
, . . . , z
m
, η
)
. . . g
mm
(
z
1
, . . . , z
m
, η
)
is nondegenerated on the set
Φ(Ω)
, then system (2) is called regular on the
set
Φ(Ω)
.
We will assume that system (1) satisfies the conditions of theorem 1,
while
Φ(Ω) =
R
n
. Then system (1) will be transformed to an equivalent
of a quasicanonical form (2), which is determined on the whole range of
states, and the terminal problem for system (1) — to the equivalent terminal
problem for system (2): to find continuous controls
u
1
=
u
1
(
t
)
, . . . , u
m
=
=
u
m
(
t
)
,
t
∈
[0
, t
∗
]
, transforming system (2) for time
t
∗
from the initial
state
Φ(
x
0
) = (
z
1
0
, . . . , z
m
0
, η
0
)
(3)
to the final state
Φ(
x
∗
) = (
z
1
∗
, . . . , z
m
∗
, η
∗
)
.
(4)
Controls
u
1
=
u
1
(
t
)
, . . . , u
m
=
u
m
(
t
)
, which are the solution to problem
(3), (4) for system (2), simultaneously constitute the solution to the initial
terminal problem for system (1). In this connection, we will consider
terminal problem (3), (4) for system (2).
The solution to the terminal problem for the system of a quasicano-
nical form.
Paper [9] describes the following necessary and sufficient
condition for the existence of the terminal problem solution for a regular
system of a quasicanonical form.
Theorem 2.
For the continuous controls
u
1
=
u
1
(
t
)
, . . . , u
m
=
u
m
(
t
)
,
t
∈
[0
, t
∗
]
to exist, which are the solution of terminal problem
(3)
,
(4)
for regular system
(2)
, it is necessary and sufficient for the functions
B
i
(
t
)
∈
C
r
i
([0
, t
∗
])
,
i
= 1
, m
to exist, and
:
1)
vector-functions
B
i
(
t
) =
B
i
(
t
)
, B
0
i
(
t
)
, . . . , B
(
r
i
−
1)
i
(
t
)
T
satisfies the
conditions
B
i
(0) =
z
i
0
, B
i
(
t
∗
) =
z
i
∗
;
18
ISSN 1812-3368. Herald of the BMSTU. Series “Natural Sciences”. 2014. No. 5