Time-optimal guaranteed search of mobile object in the plane

Ի Ի ՈՒԹ ՈՒ Ի Ի Ի Ի Ղ Ի Ц

68, №1, 2015

62-50

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ч : ,

Keywords: guaranteed search, optimal control

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Avetisyan V.V., Stepanyan V.S.

Time-optimi guaranteed search of mobile object in the plane

A problem of time-optimal guaranteed search of moving object with controllable speed in plane is observed, where in the initial moment only the circular of the sought object is known. The sought object is a object with

controllable speed. The discovery, i.e. the identification of the precise coordinates of the object, is implemented through the moving information basis of the cone, the apex of which is connected to the current coordinates of the sough object. It is required to locate the object as quickly as possible. A control algorithm is proposed, in which r a guaranteed search of the sought object in minimal time is completed.

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(1.1) - (1.5), 1 .

ё ,

, , . . 2.

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, (1.5),

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(2.2) :

0 0 0

(

1 3

)

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l

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

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Т

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0 0 0

(

1 3

)

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    

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,

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– ,

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0

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(2.4)

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(

3

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[

,

]

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  .

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(2.6)

, . . ..

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₀

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₀

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( ,

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:

0,

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2

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0,

(

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(

)

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





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Cu

V

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V

l

r u

H

u u

R

u

Cu

V

Cu

V

u

u

U

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,

1, 2, 3

i

H i



( .2) – ,

, ,

1 3

(

,

)

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u

u

u

,

u

K

(

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₁K

,

u

₃K

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;

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(

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,

u

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,

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P

(

u

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,

u

₃P

)

;

u

M

(

u

₁M

,

u

₃M

)

,

u

N

(

u

₁N

,

u

₃N

)

P K

u u

;

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N P;

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M N, ,

u

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(

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,

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H

₁, –

2

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1 2

(1

)

0

1

P

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C

C U

V

u

C













,

 

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3 1

0

P P

u



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_

U



u



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

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0

Q

u



,

u

₃Q

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0

C





, (2.14)

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₂ 1/ 2 2

1 3

0

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u

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_

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0

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u

C

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, (2.15)

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1 2

(1

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0

1

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C U

V

u

C









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,

 

1/ 2 2 2

3 1

0

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u





_

U



u



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



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0 0 0 0 0 0

1 0 2 2

0 0 0

[(

) /

]

{[(

) /

]

}

(

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0

[(

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М

l

r

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C

l

r

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u

R

l

r

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



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R

l

r

R

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u

R

l

r

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C





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.

. 2

(2.12) – (2.17) .2,

u

S

 

(

U

, 0)

u

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(0, )

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,

0

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 

u

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(

R u

₀

)

Q

u

Q

N Q

u u

(

u

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(

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)(

u

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(

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u

₁N

(

u

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.

0 0 0

R

 

l

r

u

M

(

R u

₀

)

Q

u

Q

J Q

u u

(

1

(

0

)(

3

(

0

))

1

(

3

)

М М J J

u

R

u

R



u u

,

u

J _–

u

M

(

R u

₀

)

Q

u

12



u

32

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U

2

R

0

 

l

0

r

0

:

2 2 2 2

0 0 0 0 0 0 0 0

1 2 2

0 0 0 0

2 2 2 2

0 0 0 0 0 0 0 0

3 2 2

0 0 0 0

[(

) / (

)]

{[(

) / (

)]

}

0,

[(

) / (

)]

[(

) / (

)] {[(

) / (

)]

}

.

[(

) / (

)]

J

J

l

r

l

r

V

C

l

r

l

r

C U

V

u

l

r

l

r

C

CV

l

r

l

r

l

r

l

r

C U

V

u

l

r

l

r

C

















































(2.18) 2

H

J

u

S

u

M

u

N

u

P

u

u

L K

u

Q

u

0

u

₁

3

u

1

H

3

H

0 0

3 1

0

l

r

V

u

u

R C

C







3

V

u

C



1 3

u

V

u

C

C

 



1 3

u

V

u

C

C

.

. 3

,

u

₃J(2.18)

:

0 0

3

0 0

0 0

3

0 0

0,

( )

0,

( )

















J

J

l

r

u

V

U

a

l

r

l

r

u

U

V

b

l

r

(2.19)

(2.19)(а) .2, a (2.19)(b) – .3.

(2.10)-(2.12), ё (2.13)-(2.19), ,

H

₁

H

₂

0

R

,

H

₃ . ,

H

0

R

:

0 1 2

(

)

H R



H



H

R

₀

 

, (2.20)

0 3 2 1

(

)

J

H R



H



H



H

R

₀

 

l

₀

r

₀,



2 2 2 2



3

( ,

1 3

)

:

1 1 1

,

3 1 3 3

,

1 1

J J Q J N Q

H



u u



R

u



u



u

u



u



u



u

u



u



U

.

1 3

( , 0,

)

u



u

u

,

( ,

u u

_{1 3}

)



H

(2.9),

X

2

H

J

u

S

u

M

u

N

u

P

u

u

L

u

K

Q

u

0

u

1

3

u

1

H

3

H

0 0

3 1

0

l

r

V

u

u

R C

C







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u

V

u

C

C

 



1 3

u

V

u

C

C





3

V

u

C

Y

T

( ,

u u

1 3

)



(2.6). ё ,

1 3 1

( ,

u u

)



H

,

0



T





T



 

, a

( ,

u u

1 3

)



H

2,

0

T

T

 





 

.

H

(2.9) ё ,

u



( , 0,

u

1

u

3

)

–

, ,

1 3

( ,

)

T



u u

(2.6) – .

(2.10)-(2.12) ,

H

_i

  

,

i

1, 2, 3

,

CU



V

. (2.21)

,

CU

V

,

r

₀

l

₀

,

[1,2].

3. х х щ х . ё

2 – (2.9) ,

(2.6):

1 3 1 3

min 0 0 0

1 3

( , ) ( , )

1 3

min

( ,

)

min

u u H u u H

R

l

r

T

T u u

u

Cu

V



 



 







_

_

 







. (3.1)

(3.1)

1 3 1 3

( ,

u u

)

u

Cu

max



  



,

( ,

u u

_{1 3}

)



H

. (3.2)

(3.2)

grad ( ,



u u

_{1 3}

)

 

( 1, )

C

,

C



0

, (3.2), ,

(3.1)

u



M

(

R u

₀

)

L



M K

u u

H

:



_

2 2 2

_

0 1 3 1 0 1 3 0 3 1 3

(

)

( ,

) :

(

)

0,

(

)

,

M L M M

u

R u



u u

u

R



u



u

R



u



U u



u



U

(3.3)



0

(

)

M N

u

R u

,

u u



N P ,

u u



P L:

   

0 0

(

)

(

)

M L M N N P P L

u

R u



u

R u



u u



u u

, (3.4)



_

2 2 2

_

0 1 3 1 0 1 1 3 0 3 3 1 3

(

)

( ,

) :

(

)

;

(

)

;

M N M N M N

u

R u



u u

u

R



u



u

u

R



u



u

u



u



U

(3.5)



_

2 2 2

_

1 3 1 1 1 3 3 3 1 3

( ,

) : ,

,

N P N P N P

u u



u u

u



u



u

u



u



u

u



u



U

(3.6)



_

2 2 2

_

1 3 1 1 3 3 1 3

( ,

) : 0,

,

P L P P

u u



u u

u



u



u



u



U

u



u



U

(3.7)

1 3 1 3

(

,

),

0,

L L L L L

u



u u

u



u



U

, , (3.1)

1 3 1 3

( ,

u u

)

u

Cu

1 3 1 3

( ,

u u

)

u

Cu

max



  



,

( ,

u u

_{1 3}

)



u



M

(

R u

₀

)

L . (3.8)



0

(

)

M L

u

R u

R

0, (2.20), :





0

(

)

M L N L

u

R u



u u

R

0

 

;



(

0

)



M L J L

u

R u



u u

R

₀

 

l

₀

r

₀ ,

, :

  

0

(

)

J L M L N L

u u



u

R u



u u

,

l

₀

 

r

₀

R

₀

 

,

(3.9)



_

2 2 2

_

1 3 1 1 3 3 1 3

( ,

) : 0,

,

J L J J

u u



u u

u



u



u



u



U

u



u



U

. (3.10)

, , (2.19)(а),

 

J L S L

u u



u u

, (3.11)(a)

, (2.19)(b), , ,



J L



S L

u u



u u

, (3.11)(b)



_

2 2 2

_

1 3 1 3 1 3

( ,

) : 0,

0

,

S L

u u



u u

 

U

u





u



U

u



u



U

. (3.12)

1 3 1 3

( ,

u u

)

u

Cu



  

2 2 2

1 3

u



u



U

1 3

(

,

)

u





u u

  , (3.13)

1 ₂

1

U

u

C



_{ }



, 3 2

1

CU

u

C



_



,

C



0

,

, ё ,

u u



S L(3.12).

, ё (3.8)-(3.11),

  

0 0

(

)

(

)

S L S M M L

u u



u u

R



u

R u

, (3.14)



_

2 2 2

_

0 1 3 1 1 3 3 1 3

(

)

( ,

) :

, 0

,

S M M M

u u

R



u u

 

U

u



u



u



u

u



u



U

,

a

u



M

(

R u

₀

)

L (3.3), (3.4),

0

, , ,

R U V C

(3.13)

u



M

(

R u

₀

)

L (3.3),



0

(

)

M L

u





u

R u

, (3.15)

  

0 0

(

)

\

(

)

S M S L M L

u





u u

R



u u

u

R u

. (3.16) (3.15) (3.13) (3.1):

u

min



u

,

(3.16) M

(

₀

)

u

R

u u



S M

(

R

₀

)

:

min

0

(

),

M

u



u

R

(3.17) . .

2 2 2 2

0 0 0 0 0 0

min

1 1 0 2 2

0 0 0

[(

) /

]

{[(

) /

]

}

(

)

[(

) /

]

М

l

r

R V

C

l

r

R

C U

V

u

u

R

l

r

R

C





















2 2 2 2

0 0 0 0 0 0

min

3 3 0 2 2

0 0 0

[(

) /

] {[(

) /

]

}

(

)

[(

) /

]

М

CV

l

r

R

l

r

R

C U

V

u

u

R

l

r

R

C



















ё ,

(3.15) (3.16).

ё (2.13) – (2.20)

(3.4)-(3.7), (3.14),

C



1

C



1

:

1.

u





u u



P L, (3.18)

2

2

1

1

C

U

V

C







,

C



1

(3.19)

0

(

0 0

,

)

R



l



r



,

2.

u





u u



N P , (3.20)

2 2

2 2

1

1

1

C

C

U

V

U

C

C



 





,

C



1

(3.21)

2

2

0

1

C

U

V

C

 



,

0

 

C

1

(3.22)

0

(

0 0

,

)

R



l



r



,

3.

u





u u



S M

(

R u

₀

)

N, (3.23)

2

2

1

C

CU

V

U

C

 

R

₀



(

l

₀



r

₀

,



)

.

(3.23), (3.24).

) (2.19)(а), (3.11)(a). ,

  

0 0

(

)

(

)

S M N S J J M N

u u

R u



u u



u u

R u

, (3.23) ,



S J

u





u u

(1)

u





u u



J M

(

R u

₀

)

N (2). (3.25)

(3.25) (1), (3.1)

u

M

(

R

₀

)

0

(

0 0

,

)

R



l



r



. (3.25) (2),

0

(

0 0

,

),

R



l



r



0

(

)

M

u

R

(2.18)

u

 (3.13):

u

M

(

R

₀

)



u

.



S L

u u

(3.11),

(2.17) (3.13)

u

M

(

R

₀

)

u



,

,

1

(

0

)

1

М

u

R



u



(3.26)

2 2 2 2

0 0 0 0 0 0

2 2 ₂

0 0 0

[(

) /

]

{[(

) /

]

}

[(

) /

]

₁

l

r

R V

C

l

r

R

C U

V

U

l

r

R

C

_C













 





_

0

R

.

0

(

0 0

,

)

R



r



l



2 2 4 2 2 2 2 2

0 0 0 0 0 0

[

V

(1



C

)



C U

]

R



2

UV

1



C

(

l



r R

)



(

l



r

)

U



0

. (3.27)

(3.24) (3.27)

, ,

0

(

)

M

u

R

u



0 0

0 _{2 2}

(

)

1

U l

r

R

V

C

C U



_







,

r

0

l

0

R

0



 

 

. (3.28)

,



0

(

)

J M

u





u u

R

,

R

₀



R

₀

 



M

(

₀

)

N

u





u

R u

,

l

₀

 

r

₀

R

₀



R

₀.

(3.29)

B) (2.19)(b), (3.11)(b). )

(3.25)(2)



0

(

)

S M

u





u u

R

,

R

₀



R

₀

 



M

(

₀

)

N

0

R

 (3.28).

(3.18) – (3.30), ,

1–3, ,

min min min

1 3

(

,

)

T



T



u

u

(3.1)

min min

1

,

3

u

u

:

2 min

1,3 1,3 0 0 0

,

0,

₂

,

1

C

u

u

l

r

R

V

U

C













_

 

 



_

_











(3.31)

1,3 0 0 0 0

2 0 0 2 0 0 min 1,3

1,3 0 0 0

2

0 0

2

0 0

,

,

0,

,

1

(

)

,

,

0,

,

1

M

u

l

r

R

R

l

r

C

V

U CU

U

l

r

C

u

u

R

R

R

l

r

C

V

U CU

U

l

r

C

  



 







 







_

_











_













 



_

_{ }





_

_{ }

_

_





_

_



_

_





_

_

_

_

_



(3.32)

1,3 0 0 0 0 3 3

min ₂

1,3 _{0 0}

2 0 0

(

)

,

,

,

,

1

M J

u

R

l

r

R

u

и

u

_C

_l

_r

V

U CU

U

l

r

C





 

 











 







_

_



_



_



_













(3.33)

1,3 0 0 0 0 3 3

2 0 0 2 0 0 min 1,3

1,3 0 0 0 3 3

2 0 0 2 0 0

,

,

,

,

1

(

)

,

,

,

,

1

J M J

u

l

r

R

R

и

u

l

r

C

V

U CU

U

l

r

C

u

u

R

R

R

и

u

l

r

C

V

U CU

U

l

r

C

    



 









 







_

_

_











_













 



_

_{ }

_





_

_{ }

_

_





_

_



_



_





_

_

_

_

_



(3.34)

1

(

0

),

3

(

0

)

M M

u

R

u

R

,

u

₃,

R

₀ ₃J

u

(2.17), (3.13), (3.28), (2.18) .

ё (3.31)-(3.34)

.

2.

2

2

0 ,

1

C

V

U

C















,

(3.31),

X

, (3.10),

min

1 3

(

,

)

T



T



u u

  (2.6)

R

₀



(

r

₀



l

₀

,



)

ё

D

0

G

0.

3.

Y

2

0 0

2

0 0

0 ,

0,

1

l

r

C

V

U

U

l

r

C



 







_

_



_

_













,

(3.32),

0

R

, (2.28), ,

0

(

0 0

,

0

]

R



r



l

R



(3.13):

u

₁min



u

₁,

u

₃min



u

₃.

R

₀



(

R

₀

, )



(3.17). ё ,

R

₀

min

1 1

(

0

)

u



u R

,

u

₃min



U

2



( (

u R

_{1 0}

))

2

, , .

4.

Y

2

0 0

2

0 0

,

,

1

l

r

C

V

U CU

U

l

r

C



 







_

_



_



_













, (3.33), (3.34),

J

u

(2.18)

u

(3.13)

u u



S N . ,

и

₃J



u

₃,

min

0

(

)

M

u



u

R

R

₀



(

r

₀



l

₀

,



)

– (3.33), 3 3

J

и



u

, (3.34) , (3.32).

ч .

, .

(3.31)-(3.34) (3.1)

ё . ,

1. . .

// . . . 2007. .60. № 1. .3-9.

2. . .

// . . . 2013. .66. № 2. .68-78.

3. . . // . 1980. .1.

.3-12.

4. . ., . . .

// . 1995. . 4.

№ 1. .827-862.

5. . ., . .

// . . .

1999. № 1. .58-66.

6. . ., . .

// . . .

1999. № 2. .31-39.

7. . . O

// . .

. 2002. № 1. .62-69.

8. . . // .

. . 2006. № 6. .160-168.

9. . ., . .

// . . . 2009. № 5. .151-164.

10. . ., . . . .:

1987. 76 .

11. . .

// . 1990. .54. . 1. .3-11.

12. . ., . . a . .: ,

1978. 270 .

х:

ч – . .- . .,

,

.: (+374 94) 44 95 60; –mail: vanavet@yahoo.com

ч – ,

.: (+374 98) 900846; -mail: nop144d@gmail.com