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x

= (

x

1

, . . . , x

n

)

T

R

n

, u

= (

u

1

, . . . , u

m

)

T

R

m

;

F

(

x

) = (

F

1

(

x

)

, . . . , F

n

(

x

))

T

, G

j

(

x

) = (

G

1

j

(

x

)

, . . . , G

nj

(

x

))

T

;

F

i

(

x

)

, G

ij

(

x

)

C

(

R

n

)

, i

= 1

, n, j

= 1

, m,

which is not linearizable by a feedback, it is required to find such continuous

controls

u

1

=

u

1

(

t

)

, . . . , u

m

=

u

m

(

t

)

,

t

[0

, t

]

that for given time

t

can

transform system (1) from the initial state

x

(0) =

x

0

to the final state

x

(

t

) =

x

.

Transformation of the system to a quasicanonical form.

The

following theorem [11] sets the necessary and sufficient conditions under

which system (1) is transformed to a quasicanonical form

˙

z

i

1

=

z

i

2

;

. . . . . . . . . . . . . . . .

˙

z

i

r

i

1

=

z

i

r

i

;

˙

z

i

r

i

=

f

i

(

z

1

, . . . , z

m

, η

) +

m

X

j

=1

g

ij

(

z

1

, . . . , z

m

, η

)

u

j

, i

= 1

, m

;

˙

η

=

q

(

z

1

, . . . , z

m

, η

);

(2)

r

1

+

. . .

+

r

m

=

n

ρ, z

i

= (

z

i

1

, . . . , z

i

r

i

)

T

, η

= (

η

1

, . . . , η

ρ

)

T

;

q

(

z

1

, . . . , z

m

, η

) = (

q

1

(

z

1

, . . . , z

m

, η

)

, . . . , q

ρ

(

z

1

, . . . , z

m

, η

))

T

.

In the formulation of the theorem, vector fields are used

F

=

n

X

i

=1

F

i

(

x

)

∂x

i

, G

j

=

n

X

i

=1

G

ji

(

x

)

∂x

i

, j

= 1

, m,

which one-to-one correspond to system (1) in the range of states

R

n

and

the vector fields

ad

0

F

G

j

=

G

j

,

ad

k

F

G

j

= [

F,

ad

k

1

F

G

j

]

,

k

= 1

,

2

, . . .

, where

[

X, Y

]

is a commutator of the vector fields

X

and

Y

.

Theorem 1.

For the transformation of the affine system

(1)

on the set

Ω

R

n

to a quasicanonical form

(2)

it is necessary and sufficient to have

the following features

:

1)

functions

ϕ

i

(

x

)

C

(Ω)

,

i

= 1

, m

,

satisfying the system of the

first-order partial differential equations in the set

Ω

ad

k

F

G

j

ϕ

i

(

x

) = 0

, k

= 0

, r

i

2

, i, j

= 1

, m, x

Ω;

2)

functions

ϕ

n

ρ

+

l

(

x

)

C

(Ω)

,

l

= 1

, ρ

that for all

x

Ω

G

j

ϕ

n

ρ

+

l

(

x

) = 0

, j

= 1

, m, l

= 1

, ρ

ISSN 1812-3368. Herald of the BMSTU. Series “Natural Sciences”. 2014. No. 5

17