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39
By the trajectories of the differential inclusion (1) we mean every absolutely
continuous
n-vector function
)
(
t
x
x
,
]
,
[
1
0
t
t
T
t
, satisfying almost everywhere on
]
,
[
1
0
t
t
T
the relation
))
(
,
(
)
(
t
x
t
F
dt
t
dx
.
The differential inclusion (1) is called controllable from the initial state
0
x
to the final
state
1
x
("pointwise" controllable) if there exists a trajectory
)
(
t
x
defined on some segment
]
,
[
1
0
t
t
T
, such that
0
0
)
(
x
t
x
,
1
1
)
(
x
t
x
.
Definition 1. The sets of zero-controllability of the differential inclusion are called the set of
all those points
n
R
x
0
from which the origin of coordinates is achievable along the
trajectories (
0
0
)
(
x
t
x
,
0
)
(
1
t
x
) of the differential inclusion (1).
Let be a mobile, i.e. time-dependent terminal set
0
),
(
t
t
t
M
M
. By analogy with the
concept of a zero-controllability set, we can introduce the concept of
M
controllability for the
case of a mobile terminal set as follows.
Definition 2. Point
)
(
,
0
0
0
t
M
x
R
x
n
, we call the point of
M
controllability of the
differential inclusion (1) for a given mobile terminal set
)
(
t
M
M
, if there exist a absolutely
continuous trajectory
)
(
t
x
defined on some segment
]
,
[
1
0
t
t
T
such that
,
)
(
0
0
x
t
x
)
(
)
(
1
1
t
M
t
x
.
We denote by
)
,
(
F
M
W
the set of all points of
M
controllability of the differential
inclusion (1). The main goal of this paper is to study such properties of the set
)
,
(
F
M
W
that
clarify its topological structure.
Let
)
,
,
,
(
0
1
0
F
x
t
t
X
T
be the set of reachability of the differential inclusion (1) from the
starting point
n
R
x
0
at time
0
1
t
t
, i.e. the set of possible points
n
R
x
1
for which there
exist trajectories
)
(
t
x
x
,
]
,
[
1
0
t
t
T
t
, such that
0
0
)
(
x
t
x
and
1
1
)
(
x
t
x
. From definition 2, it
is clear that point
n
R
x
0
is the point of
M
controllability of the differential inclusion (1) if
and only if there exists
0
1
t
t
such that
)
(
)
,
,
,
(
1
0
1
0
t
M
F
x
t
t
X
T
, where
)
(
],
,
[
0
0
1
0
t
M
x
t
t
T
.
So, it is clear that, in order to study the properties of the controllability set of the
differential inclusion (1), it is necessary to study the structure of the set
)
(
)
,
,
,
(
:
)
,
,
,
(
1
1
0
1
0
t
M
F
t
t
X
R
F
M
t
t
K
T
n
at
0
1
t
t
, taking into account properties
)
(
t
M
M
and
)
,
(
x
t
F
F
.
From the definition of sets
)
,
(
F
M
W
and
)
,
,
,
(
1
0
F
M
t
t
K
, the validity of the following
equality easily follows
)
(
\
))
,
,
,
(
(
)
,
(
0
1
0
0
1
t
M
F
M
t
t
K
F
M
W
t
t
.
(2)
Obviously,
if
)
,
(
)
,
(
2
1
x
t
F
x
t
F
,
then
)
,
,
,
(
)
,
,
,
(
2
1
0
1
1
0
F
t
t
X
F
t
t
X
T
T
and
at
0
2
1
),
(
)
(
t
t
t
M
t
M
,
from
the
relation
)
(
)
,
,
,
(
1
1
1
1
0
t
M
F
t
t
X
T
follows
)
(
)
,
,
,
(
1
2
2
1
0
t
M
F
t
t
X
T
. Therefore,
)
,
,
,
(
)
,
,
,
(
2
2
1
0
1
1
1
0
F
M
t
t
K
F
M
t
t
K
,
)
,
(
)
,
(
2
2
1
1
F
M
W
F
M
W
.
Hence, in particular, we get that if there are maps
),
(
:
,
:
1
1
n
nxn
R
R
B
R
R
A
such that
n
R
R
x
t
x
t
F
t
B
x
t
A
1
)
,
(
)
,
(
)
(
)
(
, then to check the property of
M
controllability of the
5th Global Congress on Contemporary Sciences & Advancements
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10th May 2021
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40
differential inclusion (1), it is sufficient to check the
M
controllability of the differential
inclusion
)
(
)
(
t
B
x
t
A
x
. (3)
3. Main results.
Let us study the structural properties of the set of
M
controllability of the differential
inclusion (3). According to the accepted notation
)
,
,
(
B
A
M
W
there is a set of all points of
M
controllability of the differential inclusion (3) for a given terminal set
0
),
(
t
t
t
M
M
.
Further, denoting
)
,
,
,
,
(
1
0
B
A
t
t
X
T
the set of reachability of the differential inclusion (3),
)
,
,
,
,
(
1
0
B
A
M
t
t
K
the set is defined similarly to the set
)
,
,
,
(
1
0
F
M
t
t
K
, i.e.
)
(
)
,
,
,
,
(
:
)
,
,
,
,
(
1
1
0
1
0
t
M
B
A
t
t
X
R
B
A
M
t
t
K
T
n
.
Since, according to (2)
)
(
\
))
,
,
,
,
(
(
)
,
,
(
0
1
0
0
1
t
M
B
A
M
t
t
K
B
A
M
W
t
t
,
then the structural properties of set
)
,
,
(
B
A
M
W
are expressed in terms of similar properties of
sets of the form
)
,
,
,
,
(
1
0
B
A
M
t
t
K
.
In the future, we will assume that the elements of the matrix
)
(
t
A
are measurable on
any
]
,
[
]
,
[
0
1
0
t
t
t
T
and
)
(
||
)
(
||
t
a
t
A
, where
)
(
)
(
1
T
L
a
, and the multi-valued map
)
(
)
(
n
R
t
B
t
is measurable on any segment
]
,
[
]
,
[
0
1
0
t
t
t
T
and
)
(
||
)
(
||
t
b
t
B
, where
)
(
)
(
1
T
L
b
.
It is well known [8] that for every integrable function
n
R
T
b
:
, the absolutely
continuous solution of equation
)
(
,
),
(
)
(
0
t
x
T
t
t
b
x
t
A
x
is represented by the Cauchy
formulas
t
t
A
A
T
t
d
b
t
Ф
t
t
Ф
t
x
0
,
)
(
)
,
(
)
,
(
)
(
0
.
(4)
where
)
,
(
t
Ф
A
is the fundamental matrix of solutions to equation
.
,
)
(
T
t
x
t
A
x
The relation of
)
(
)
,
,
,
,
(
1
1
0
t
M
B
A
t
t
X
T
is equal to the inclusion of
)
(
)
,
,
,
,
(
0
1
1
0
t
M
B
A
t
t
X
T
. Therefore,
)
(
)
,
,
,
,
(
0
:
)
,
,
,
,
(
1
1
0
1
0
t
M
B
A
t
t
X
R
B
A
t
t
K
T
n
.
Now, using the last equality and formula (4), we can get the following result.
Theorem 1. The set
)
,
,
,
,
(
1
0
B
A
M
t
t
K
is represented by the formula
1
0
)
(
)
,
(
)
(
)
,
(
)
,
,
,
,
(
1
1
0
0
1
0
t
t
A
A
t
M
t
t
Ф
dt
t
B
t
t
Ф
B
A
M
t
t
K
(5)
Corollary 1. If 1
)
(
1
t
M
is a convex compact, then
)
,
,
,
,
(
1
0
B
A
M
t
t
K
is also a convex compact of
n
R
. If
)
(
1
t
M
and
)
(
t
convB
are strictly convex at
]
,
[
1
0
t
t
T
t
, then
)
,
,
,
,
(
1
0
B
A
M
t
t
K
is strictly
convex.
Let's say:
)
,
,
(
1
0
B
A
t
K
)
,
,
0
,
,
(
1
0
B
A
t
t
K
. Then it is clear from formula (5) that
1
0
)
(
)
,
(
)
,
,
,
(
0
1
0
0
t
t
A
dt
t
B
t
t
Ф
B
A
t
t
K
.
(6)
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