Applied Mathematics, 2014, 5, 642-652
Published Online March 2014 in SciRes. http://www.scirp.org/journal/am
http://dx.doi.org/10.4236/am.2014.54060
The Generalized Search for a Randomly
Moving Target
Abdelmoneim Anwar Mohamed Teamah
Mathematics Department, Faculty of Science, Tanta University, Tanta, Egypt
Email: teamah4@hotmail.com
Received 12 September 2013; revised 12 October 2013; accepted 19 October 2013
Copyright © 2014 by author and Scientific Research Publishing Inc.
This work is licensed under the Creative Commons Attribution International License (CC BY).
http://creativecommons.org/licenses/by/4.0/
Abstract
A target is assumed to move randomly on one of two disjoint lines L1 and L2 according to a stochastic process
{S ( t ) , t ∈ ℜ } . We have two searchers start looking for the lost target from some
+
points on the two lines separately. Each of the searchers moves continuously along his line in both
directions of his starting point. When the target is valuable as a person lost on one of disjoint
roads, or is serious as a car filled with explosives which moves randomly in one of disjoint roads,
in these cases the search effort must be unrestricted and then we can use more than one searcher.
In this paper we show the existence of a search plan such that the expected value of the first
meeting time between the target and one of the two searchers is minimum.
Keywords
Stochastic Process; Expected Value; Linear Search; Optimal Search Plan
1. Introduction
The search for lost targets that are either stationary or randomly moving has recently applications, such as:
searching for lost persons on roads, the search for a petroleum or gas underground, and so on (see, Abd-Elmoneim [1], Ohsumi [2], El-Rayes and Abd-Elmoneim [3], and Washburn [4]).
When the target to be found is stationary or moves randomly on the real line, this problem is of interest because it may arise in many real world situations (see El-Rayes et al. [5] and Balkhi [6]). Search problems with
stationary target on line are well studied (see El-Rayes and Abd-Elmoneim [7], El-Rayes et al. [8], Balkhi [6]
[9], Rousseeuw [10], Abd-Elmoneim and Abu-Gabl [11], and Stone [12]). In the case of randomly moving target on the line and the searcher starts search from the origin, a deal of work has been done for deriving conditions for optimal search path which minimizes the effort of finding the target (see El-Rayes et al. [5], Fristedt
How to cite this paper: Teamah, A.A.M. (2014) The Generalized Search for a Randomly Moving Target. Applied Mathematics, 5, 642-652. http://dx.doi.org/10.4236/am.2014.54060
A. A. M. Teamah
and Heath [13]). If the lost target is a valuable target as a person lost on one of disjoint roads, or is serious as a
car filled with explosives which moves randomly in one of disjoint roads, then the effort of the search (the cost
of search) must be unrestricted, in these cases using more than one searcher (see Abd El-Moneim et al. [14]).
The search problem for a randomly moving target on one of two disjoint lines will be considered, in previous
studies (see Abd El-Moneim and Abu-Gabl [15]), using a searcher for each line where each searcher starts
looking for the lost target from the origin of his line, but Abd El-Moneim et al. [14] used searchers starting
searching for the target from the origin that is the intersection point of these lines. Each of the searchers moves
continuously along his line in both directions of the starting point, in both cases the target motion is a Brownian
motion. In this article, a target is assumed to move randomly on one of two disjoint lines L1 and L2 according
to a stochastic process S ( t ) , t ∈ℜ+ , where ℜ+ is the set of real numbers. This stochastic process satisfies
the following conditions:
(i) Let c ≥ µ , where µ is the drift of the process and c is a constant. Then for any t > 0 and for some
1
ε > 0 , P { S ( t )  ct } ≤ ε t ,
2
(ii) Let µ ≠ 0, z1 ≥ z2 and t ≥ max ( z1 µ , z2 µ ) , then P { z2 ≤ S ( t ) ≤ z1 } is non-increasing with t,
(iii) Let T be a stopping time for S ( t ) , then
{
}
E S (T ) ≤ σ E (T ) + µ E (T ) ,
where E stands for the expectation value and σ 2 is the variance and
(iv) Let
S (n)
=
where
{Z i }
n
∑ Zi , n ≥ 1 ,
i =1
is a sequence of independent identically distributed random variables (i. i. d. r. v.) and
q ( j, j =
+ 1)
∞
∑ P  − ( j + 1) < S ( n ) < − j  ,
n =0
then q ( j , j + 1) satisfies the renewal theorem.
Two searcher start looking for the target from some point φ0 for the first line L1 and φ0 for the second line
L2 . We assume that the speeds of the searchers are ν1 and ν2. Assume that T is the set of real numbers and T+ is
the non-negative part of T. Foe any t ∈ T + , let S ( t ) be a random variable with S ( 0 ) = 0 , and assume that Zo
is the initial position of the target to be a random variable and independent of S ( t ) , t > 0. A search plan
( Φ, Φ ) with speed ν1, ν2 is defined such that φ : T+→T and φ : T+→T, respectively, such that
φ ( t1 ) − φ ( t2 ) ≤ ν 1 t1 − t2
φ ( t1 ) − φ ( t2 ) ≤ ν 2 t1 − t2 , ∀t1 , t2 ∈ T + , φ ∈ Φ, φ ∈ Φ
the first meeting time is a random variable valued in I+ which is defined
inf {t : φ ( t ) =
X o + S ( t ) or φ ( t ) =
Yo + S ( t )}
τ φˆ =
where Zo = X0 if the target moves on the first line and Zo = Y0 if the target moves on the second line. Let the
ˆ = φ , φ , where Φ̂ is the set of all search plans. The probsearch plan of two searchers be represented by Φ
lem is to find search =
plan φˆ (φ , φ ) , φ ∈ Φ, φ ∈ Φ , such that Eτφˆ < ∞ in this case we call Φ̂ is a finite
ˆ , where E terms to expectation value, then we call φˆ* is optimal.
search plan and if Eτ φˆ* < Eτ φˆ ∀φˆ ∈ Φ
{
{ }
}
2. The Search Plans
Let ζ1, ζ2 be positive integers and ν1and ν2 are rational numbers such that:
1) ν 1 > µ and ν 2 > µ ,
ζ1 −1
ζ −1
2) ζ1, ζ2 > 1, such
that C1 v=
=
, C ν2 2
,
1
ζ1 + 1 2
ζ 2 +1
Now we shall define sequences {Gi }i ≥0 , {ri }i ≥0 , { H i }i ≥0 for the searcher on the first line L1,
643
A. A. M. Teamah
{G } ,{r } ,{H }
i i ≥0
i i ≥0
for the searcher on the second line L2 and search plans with speed 1 as follows.
i i ≥0
Gi = ζ 1i − 1, ri = ( −1)
i +1
C1 Gi + 1 + ( −1)  , H i = ri + φ0 , i ≥ 0


i +1
i +1
i

Gi = ζ 2 − 1, ri = ( −1) C2 Gi + 1 + ( −1)  , H i = ri + φ0 , i ≥ 0


i +1
Let O be a finite set of numbers, such that
{1,3,5,, m} for
=
O
j ∈ O, i ∈ I +
we have
 < H 2 < φ0 < H1 < H 3 <  < H j − 2 < H j < 0 < H j + 2 < 
for the first searcher, and
 < H 2 < φ0 < H1 < H 3 <  < H j − 2 < H j < 0 < H j + 2 < 
for the second searcher
for any t ∈ R+, if G2i −1 ≤ t < G2i then φ ( t ) = φ0 − H 2i −1 − [t − G2i −1 ]
j +1
, then φ ( t ) = φ0 − H 2i −1 − t − G2i −1  ,
and if G2i −1 ≤ t < G2i , 1 ≤ i ≤
2
if Gi ≤ t < Gi +1 , then φ ( t )= H i + ( −1) [t − Gi ]
i
and if Gi ≤ t < Gi +1 , i ≥ j + 1, then φ ( t )= H i + ( −1) t − Gi  ,and
if G2i ≤ t < G2i +1 , then φ ( t )= H 2i − φo + [t − G2i ]
j −1
, then φ ( t ) = H 2i − φo + t − G2i 
and G2i ≤ t < G2i +1 , 1 ≤ i ≤
2
ϕ1 s ( t ) − k1 ( t ) ,
We use the following notations where k1(t), k1 ( t ) , k2(t) and k2 ( t ) are positive functions, =
ϕ2 s ( t ) + k1 ( t ) on the second line.
=
ϕ1 s ( t ) + k2 ( t ) on the first line and =
ϕ2 s ( t ) − k1 ( t ) , =
Lemma 2.1. if 0 < a, b < 1 then ab < a + b
ˆ is a search plan, and let γ1, γ2 are measurable induced by the initial position of
Theorem 2.1. If (φ , φ ) ∈ Φ
the target on the first and second line respectively, then E (τ φ ) is finite if:
i

j 2

∞
2i
 ζ 2i P ϕ ( G ) < − x +

( 1 2i
) ∑j ζ 1 P (ϕ1 ( G2i ) ≤ − x ) γ 1  ( dx ),
1
∫ ∑
1
i
=
−∞
i= +1

2



∞ ∞
2 i +1

d
.
ζ
P
ϕ
G
x
γ
x
>
−
( 1 ( 2i +1 ) ) 1 ( )
1
∫∑

φ0 i =1
 If φ0 , φ0 > 0,
φ0 j 2
∞

ζ 22i P ϕ2 ( G2i ) < − y + ∑ ζ 22i P ϕ2 ( G2i ) ≤ − y γ 2 ( dy ), 
∫∑
j

−∞ i =1
i= +1
2

∞ ∞

2 i +1
ζ 2 P ϕ2 ( G2i +1 ) > − y γ 2 ( dy ).

∫∑
φ0 i =1

φ0
(
)
(
(
)
)
( j −1) 2

 ∞

2 i +1
ζ
P
ϕ
G
x
>
−
+
(
)
(
)

∑
∑ ζ 12i P (ϕ1 ( G2i +1 ) ≤ − x ) γ 1 ( dx ) 
1
2
2
1
i
+
∫ i =
1
i=


φ0  ( j +1) 2

φ0 ∞

ζ 12i P (ϕ1 ( G2i ) < − x ) γ 1 ( dx ).
∫∑

1
i
=
−∞
 If φ0 , φ0 < 0,
j −1) 2
∞
(
∞



2 i +1
2i
∑ ζ 2 P ϕ2 ( G2i +1 ) > − y + ∑ ζ 2 P ϕ2 ( G2i +1 ) ≤ − y  γ 2 ( dy ) 
∫ i =
1
2
1
j
i
+
=

(
)
φ0 


φ0 ∞


ζ 22i P ϕ2 ( G2i ) < − y γ 2 ( dy ).
∫∑

−∞ i =1
∞
(
(
)
(
)
644
)
A. A. M. Teamah





∞ ∞

2 i +1
ζ 1 P (ϕ1 ( G2i +1 ) > − x ) γ 1 ( dx ),
∫∑

i
=
1
φ0
 If φ0 > 0, φ0 < 0,

∞
( j −1) 2
 ∞

∑ ζ 22i +1 P ϕ2 ( G2i +1 ) > − y ∑ ζ 22i +1 P ϕ2 ( G2i +1 ) > − y  γ 2 ( dy ),
∫ i =
i=
1
φ0  ( j +1) 2


φ0 ∞

2i
ζ 2 P ϕ2 ( G2i ) < − y γ 2 ( dy )

∫∑
−∞ i = 0

j 2

∞
2i
 ζ 2i P ϕ ( G ) < − x +
 γ ( dx ),

ζ
P
ϕ
G
<
−
x
(
)
) ∑j 1 ( 1 2i
) 1
∫  ∑ 1 ( 1 2i
−∞ i =1
i= +1


2
φ0
(
)
(
(
)
)
And
∞

φ0

( j −1)
∞
2








 If φ0 < 0, φ0 > 0







∑ ζ 12i +1 P (ϕ1 ( G2i +1 ) > − x ) + ∑ ζ 12i +1 P (ϕ1 ( G2i +1 ) < − x ) γ 1 ( dx ),
∫ i =
1
i=

( j +1) 2
φ0 ∞
2i
∫ ∑ ζ 1 P (ϕ1 ( G2i ) < − x ) γ 1 ( dx )
−∞ i =1
j 2
∞
 ζ 2i P ϕ G < − y +
∑j ζ 22i P ϕ2 ( G2i ) < − y
2
2 ( 2i )
∫ ∑
−∞ i =1
i= +1

2
φ0
∞ ∞
(
ζ2
∫∑
i =1
φ0
2 i +1
)
(
(
)

 γ ( dy ),
 2

)
P ϕ2 ( G2i +1 ) > − y γ 2 ( dy ),
are finite
Proof: The continuity of S(t), φ ( t ) and φ ( t ) imply that if X 0 > φ0 then X0 + S(t) is greater than φ ( t ) on
the first line until the first meeting, also if X 0 < φ0 then X0 + S(t) is smaller than φ ( t ) on the first line until the
first meeting the same for the second line by replacing X0 by Y0 and φ0 by φ0 . Hence for any i ≥ 0.
(
)
(
)
(
P τ φˆ > t= P τ φ > t and τ φ > t= P (τ φ > t ) ⋅ P τ φ > t
P (τ φ > G2i +1 ) ≤
∫−∞ P ( X 0 + s ( G2i ) < H 2i X=0 x ) γ 1 ( dx ) +
φ0
)
−∞
∫ P ( X 0 + s ( G2i +1 ) > H 2i +1
φ0
X=
x ) γ 1 ( dx )
0
we get
P (τ φ > G2i +1 ) ≤ ∫ P (ϕ1 ( G2i ) < − x ) γ 1 ( dx ) + ∫
φ0
−∞
φ0
−∞
P (ϕ1 ( G2i +1 ) > − x ) γ 1 ( dx )
Also
(
)
(
)
(
)
(
)
0
P τ φ > G2i +1 ≤ ∫−∞ P ϕ2 ( G2i ) < − y γ 2 ( dy ) + ∫φ P ϕ2 ( G2i +1 ) > − y γ 2 ( dy )
φ
−∞
0
∞
φ0
P (τ φ > G2i ) ≤ ∫−∞ P φ1 ( G2i ) < − x γ 1 ( dx ) + ∫φ P (φ1 ( G2i −1 ) > − x ) γ 2 ( dx )
0
and
(
)
(
)
(
)
0
P τ φ > G2i ≤ ∫−∞ P ϕ2 ( G2i ) < − y γ 2 ( dy ) + ∫φ P ϕ 2 ( G2i −1 ) > − y γ 1 ( dy )
φ
∞
0
Hence
( )
∞
(
)
E τ φˆ ≤ ∫ P τ φˆ > t dt
0
645
A. A. M. Teamah
from Lemma 2.1 then we get:
Gi +1
∞
∞  Gi +1
E τ φˆ ≤ ∫  P (τ φ > t ) + P τ φ > t  dt ≤ ∑  ∫ P (τ φ > Gi ) + ∫ P τ φ > Gi


i =0 
0
Gi
 Gi
( )
(
)
(

) dt

Gi +1
∞  Gi +1
∞

≤ ∑  ∫ P (τ φ > t ) + ∫ P τ φ > t  dt ≤ ∑ ( Gi +1 − Gi ) P (τ φ > Gi ) + ( Gi +1 − Gi ) P τ φ > Gi 


=i 0=
i 0
 Gi

Gi
(
∞
(
(
)
∞
)
(
) (
)
(
)
= λ1 ∑ ζ 1i +1 − ζ 1i P (τ φ > Gi ) + λ2 ∑ ζ 2i +1 − ζ 2i P τ φ > Gi
=i 0=i 0
∞
∞
= λ1 (ζ 1 − 1) ∑ ζ 1i P (τ φ > Gi ) + λ2 (ζ 2 − 1) ∑ ζ 2i P τ φ > Gi
=i 0=i 0
)
= λ1 (ζ 1 − 1)  P (τ φ > 0 ) + ζ 1 P (τ φ > G1 ) + ζ 12 P (τ φ > G2 ) + ζ 13 P (τ φ > G3 ) + 
(
)
(
)
+ λ2 (ζ 2 − 1)  P τ φ > 0 + ζ 2 P τ φ > G1 + ζ 22 P (τ φ > G2 ) + ζ 23 P (τ φ > G3 ) + 


If φ0 , φ0 < 0 , then

 ∞
E τ φˆ ≤ λ1 (ζ 1 − 1)  P (τ φ > 0 ) + ζ 1 P (τ φ > G1 ) + ζ 12  ∫ P φ1 ( G1 ) > − x γ 1 ( dx )

φ0
( )
(
+
)
φ0
∞


3


P
ϕ
G
x
γ
d
x
ζ
P
ϕ
G
x
γ
d
x
P φ1 (G2 ) < − x γ 1 ( dx ) 
<
−
+
>
−
+
(
)
(
)
(
)
(
)
(
)
)1
 1 ∫ ( 1 3
1
∫ 1 2
∫
−∞
−∞

φ0

φ0
(
)
φ0
 ∞
 ∞

+ ζ 14  ∫ P (ϕ1 ( G3 ) > − x ) γ 1 ( dx ) + ∫ P (ϕ1 ( G4 ) < − x ) γ 1 ( dx )  +  + ζ 1j  ∫ P ϕ1 ( G j ) > − x γ 1 ( dx )
−∞
φ0

φ0
(
+
φ0

−∞

∫ P (ϕ1 ( G j −1 ) < − x ) γ 1 ( dx ) + ζ 1
j +1
)
φ0
 ∞

 ∫ P ϕ1 ( G j +1 ) > − x γ 1 ( dx ) + ∫ P ϕ1 ( G j ) < − x γ 1 ( dx ) 
−∞
φ0

(
)
(
)
φ0
∞


+ ζ 1j + 2  ∫ P ϕ1 ( G j + 2 ) > − x γ 1 ( dx ) + ∫ P ϕ1 ( G j +1 ) < − x γ 1 ( dx ) 
−∞
φ0

(
)
(
)
φ0

 ∞

+ ζ 1j + 3  ∫ P ϕ1 ( G j + 3 ) > − x γ 1 ( dx ) + ∫ P ϕ1 ( G j + 2 ) < − x γ 1 ( dx )  + 

−∞

φ0
(
)
(
)

 ∞
+ λ2 (ζ 2 − 1)  P τ φ > 0 + ζ 2 τ φ > G1 + ζ 22  ∫ P ϕ2 ( G1 ) > − y γ 2 ( dy )

φ0
(
)
(
(
)
)
φ0
φ0
∞



+ ∫ P ϕ2 ( G2 ) < − y γ 2 ( dy )  + ζ 23  ∫ P ϕ2 ( G3 ) > − y γ 2 ( dy ) + ∫ P ϕ2 ( G2 ) < − y γ 2 ( dy )  + 
−∞
−∞

φ0

(
)
(
(
)
)
φ0
 ∞
 ∞

+ ζ 2j  ∫ P ϕ2 ( G j ) > − y γ 2 ( dy ) + ∫ P ϕ2 ( G j −1 ) < − y γ 2 ( dy )  + ζ 2j +1  ∫ P ϕ2 ( G j +1 ) > − y γ 2 ( dy )
−∞
φ0
φ0

(
)
(
)
(
)
φ0
φ0
 ∞


+ ∫ P ϕ2 ( G j ) < − y γ 2 ( dy )  + ζ 2j + 2  ∫ P ϕ2 ( G j + 2 ) > − y γ 2 ( dy ) + ∫ P ϕ2 ( G j +1 ) < − y γ 2 ( dy ) 
−∞
−∞

φ0

(
)
(
)
(
φ0
∞



+ζ 2j + 3  ∫ P ϕ2 ( G j + 3 ) > − y γ 2 ( dy ) + ∫ P ϕ2 ( G j + 2 ) < − y γ 2 ( dy )  + 

−∞
φ0

(
)
(
Then
646
)
)
A. A. M. Teamah
φ0

E τ φˆ ≤ λ1 (ζ 1 − 1)  g1 + (ζ 1 + 1) ζ 12 ∫ P (ϕ1 ( G2 ) < − x ) γ 1 ( dx )

−∞
( )
+ (ζ 1 + 1) ζ 14
φ0
∫
−∞
φ0
(
)
P (ϕ1 ( G4 ) < − x ) γ 1 ( dx ) +  + (ζ 1 + 1) ζ 1j +1 ∫ P ϕ1 ( G j +1 ) < − x γ 1 ( dx ) + 
−∞
∞
∞
φ0
φ0
+ (ζ 1 + 1) ζ 13 ∫ P (ϕ1 ( G3 ) > − x ) γ 1 ( dx ) + (ζ 1 + 1) ζ 15 ∫ P (ϕ1 ( G5 ) > − x ) γ 1 ( dx ) + 
∞
∞

+ (ζ 1 + 1) ζ 1j ∫ P ϕ1 ( G j ) > − x γ 1 ( dx ) +  + (ζ 1 + 1) ζ 1j + 2 ∫ P ϕ1 ( G j + 2 ) > − x γ 1 ( dx ) + 

φ0
φ0
φ0
φ0

+ λ2 (ζ 2 − 1)  g 2 + (ζ 2 + 1) ζ 22 ∫ P ϕ2 ( G2 ) < − y γ 2 ( dy ) + (ζ 2 + 1) ζ 24 ∫ P ϕ2 ( G4 ) < − y γ 2 ( dy ) + 

−∞
−∞
(
)
(
(
+ (ζ 2 + 1) ζ 2j + 2
)
(
φ0
)
∞
∫ P (ϕ2 ( G j + 2 ) < − y ) γ 2 ( dy ) +  + (ζ 2 + 1) ζ 2 ∫ P (ϕ2 ( G3 ) > − y ) γ 2 ( dy )
3
φ0
−∞
∞
)
(
∞
)
(
)
+ (ζ 2 + 1) ζ 25 ∫ P ϕ2 ( G5 ) > − y γ 2 ( dy ) +  + (ζ 2 + 1) ζ 2j ∫ P ϕ2 ( G j ) > − y γ 2 ( dy ) + 
φ0
φ0
∞

+ (ζ 2 + 1) ζ 2j + 2 ∫ P ϕ2 ( G j + 2 ) > − y γ 2 ( dy ) + 

φ0
(
)
∞
∞

 φ0

E τ φˆ ≤ λ1 (ζ 1 − 1)  g1 (ζ 1 + 1)  ∫ m1 ( x ) γ 1 ( dx ) + ∫ q1 ( x ) γ 1 ( dx ) + ∫ q1 ( x ) γ 1 ( dx ) 
φ0
φ0
 −∞


( )
φ
∞
∞

 0

+ λ2 (ζ 2 − 1)  g1 (ζ 2 + 1)  ∫ m1 ( y ) γ 2 ( dy ) + ∫ q1 ( y ) γ 2 ( dy ) + ∫ q1 ( y ) γ 2 ( dy ) 

φ0
φ0
 −∞

where
∞
=
g1 P (τ φ > 0 ) + ζ 1 P (τ φ > G1 ) + ζ 12 ∫ P (ϕ1 ( G1 ) > − x ) γ 1 ( dx )
φ0
(
(
)
)
∞
(
)
=
g1 P τ φ > 0 + ζ 2 P τ φ > G1 + ζ 22 ∫ P ϕ2 ( G1 ) > − y γ 2 ( dy )
φ0
=
m1 ( x ) ζ 12 P (ϕ1 ( G2i ) < − x ) ,
(
)
m1 ( y ) ζ 22 P ϕ2 ( G2i ) < − y ,
=
( j −1)
=
q1 ( x )
∑
2
ζ 12i +1 P ϕ1 ( G2 j +1 ) > − x 
2
ζ 12i +1 P ϕ2 ( G2i +1 ) > − y 
j =1
( j −1)
=
q1 ( y )
∑
i =1
∞
=
q1 ( x )
=i
∑
( j +1)
2
ζ 22i +1 P ϕ1 ( G2i +1 ) > − x 
and
=
q1 ( y )
∞
=i
∑
( j +1)
2
ζ 22 j +1 P ϕ2 ( G2i +1 ) > − y 
the other cases can be proved by similar way
647
,
A. A. M. Teamah
Lemma 2.2. Let an ≥ 0 for n ≥ 0, and an+1 ≤ an, {dn} n ≥ 0 be a strictly increasing sequence of integers with d0
= 0 then for any k ≤ 0,
∞
∑ [ d n +1 − d n ] ad
n k
=
n +1
≤
∞
∞
∑ an ≤ ∑ [ d n +1 − d n ] ad
n d=
n k
=
k
n
(see EL-Rayes et al. [5])
Theorem 2.2. The chosen search plan satisfies
q2 ( x ) ≤ L ( x ) , q2 ( y ) ≤ L ( x )
q2 ( x ) ≤ L′ ( x ) , q2 ( y ) ≤ L′ ( y ) , φ0 , φ0 > 0
m1 ( x ) ≤ L′′ ( x ) , m1 ( y ) ≤ L′′ ( y )
q1 ( x ) ≤ L′′′ ( x ) , q1 ( x ) ≤ L′′′ ( y ) , φ0 , φ0 < 0
where L ( x ) , L′ ( x ) , L′′ ( x ) , L′′′ ( x ) , L ( y ) , L′ ( y ) , L′′ ( y ) , L′′′ ( y ) are linear functions.
Proof. We shall prove the theorem for q2 ( x ) and q2 ( y ) , since the other cases can be proved by a similar
way.
∞
=
q2 ( x )
∑ ζ 12i +1 P ϕ ( G2i +1 ) > − x ,
q2 ( y )
=
∑ ζ 22i +1 P ϕ ( G2i +1 ) > − y ,
i =1
∞
i =1
φ0 > 0 ,
φ0 > 0
(i) if x ≥ φ0
∞
q=
q2 (φ0 ) + ∑ ζ 12i +1 P  − x < ϕ ( G2i +1 ) ≤ −φ0 
2 ( x)
i =1
and y ≥ φ0
∞
q=
q2 (φ0 ) + ∑ ζ 22i +1 P  − y < ϕ ( G2i +1 ) ≤ −φ0 
2 ( y)
i =1
(ii) if 0 ≤ x ≤ φ0
∞
q2 (=
x ) q2 ( 0 ) + ∑ ζ 12i +1 P  − x < ϕ ( G2i +1 ) ≤ 0 
i =1
and 0 ≤ y ≤ φ0
∞
q2 (=
y ) q2 ( 0 ) + ∑ ζ 22i +1 P  − y < ϕ ( G2i +1 ) ≤ 0 
i =1
(iii) if x ≤ 0
∞
q2=
( x ) q2 ( 0 ) − ∑ ζ 12i +1 P 0 < ϕ ( G2i +1 ) ≤ − x 
i =1
and y ≤ 0
∞
q2=
( y ) q2 ( 0 ) − ∑ ξ 22i +1 P 0 < ϕ ( G2i +1 ) ≤ − y 
i =1
we have x ≥ φ0 and y ≥ φ0 in (ii)
q2 ( x ) ≤ q2 ( 0 ) and q2 ( y ) ≤ q2 ( 0 )
∞
q=
q2 (φ0 ) + ∑ ζ 12i +1 P  − x < ϕ ( G2i +1 ) ≤ −φ0 
2 ( x)
i =1
648
A. A. M. Teamah
∞
q=
q2 (φ0 ) + ∑ ζ 22i +1 P  − y < ϕ ( G2i +1 ) ≤ −φ0 
2 ( y)
i =1
but,
∞
q2 (=
x ) q2 ( 0 ) + ∑ ζ 12i +1 P  − x < ϕ ( G2i +1 ) < 0 
i =1
∞
q2 (=
y ) q2 ( 0 ) + ∑ ζ 22i +1 P  − y < ϕ ( G2i +1 ) < 0 
i =1
from [1]
=
q2 ( x )
∞
∞
∑ ζ 12i +1 P ϕ ( G2i +1 ) > 0 ≤ ∑ ζ 12i +1ε G
2 i +1
≤
=i 1 =i 1
=
q2 ( y )
∞
∞
∑ ζ 22i +1 P ϕ ( G2i +1 ) > 0 ≤ ∑ ζ 22i +1ε G
2 i +1
≤
=i 1 =i 1
ζ 13
ζ 12 − 1
ζ 23
ζ 22 − 1
, 0 < ε <1
we define the following
1)=
d n G=
ζ 12 n +1 − 1
2 n +1
(
(ζ
=
d n G=
2 n +1
2) k=
( n ) ϕ=
(n)
k=
( n ) ϕ=
(n)
2 n +1
2
n
)
)
−1
∑Wi , where {Wi }i ≥0
i =1
n
∑Wi
i =1
is sequence of (i. i. d. r. v.)
where {Wi }i ≥ 0 is sequence of (i. i. d. r. v.)
 x 
3) a ( n ) =
P  − x < k ( n ) ≤ 0  =
∑ P  − ( j + 1) < k ( n ) ≤ − j 
j =0
 y 
a (n) =
P  − y < k ( n ) ≤ 0  =
∑ P  − ( j + 1) < k ( n ) ≤ − j 
j =0
4) n1 is an integer such that =
d n2 b1 x + b2
d n1 b1 x + b2 ; n2 is an integer such that =
∞
j + 1)
5) U ( j ,=
∑ t2 P  − ( j + 1) < k ( n ) ≤ − j 
n=0
∞
U ( j ,=
j + 1)
6) α1 ζ 12
=
=
α2 ζ
2
2
∑ t2 P  − ( j + 1) < k ( n ) ≤ − j 
n=0
(ζ
(ζ
)
− 1)
2
1
2
2
−1
If n > d n and d n then a(n), a( n ) are non-increasing, see [15], and we can apply Lemma 2.2 in suitable
steps.
q2 ( x )=
− q2 ( 0 )
∞
∞
n1
2 n +1
∑ ζ 12i +1 P  − x < ϕ ( G2i +=
1 ) ≤ −0 
 ∑ ζ 1 a ( dn ) + ∑
=i 1
n= 1
n1
≤ ∑ ζ 12 n +1 + α1
n= 1
n1
≤ ∑ ζ 12 n +1 + α1
n= n1 +1
ζ 12 n +1a ( d n )
∞
∑ ( d n − d n −1 ) a ( d n )
n= n1 +1
∞
∑ a (n)
n 1=
n d n1
=
n1
x 2
≤ ∑ ζ 12 n +1 + α1 ∑ U ( j , j + 1)
n 1 =j 0
=
and
649
A. A. M. Teamah
q2 ( y ) −=
q2 ( 0 )
∞
∞
 ∑ ζ 22 n +1a ( d n ) + ∑
∑ ζ 22i +1 P  − y < ϕ ( G2i +=
1 ) ≤ 0
n2
=i 1
=
n 1
=
n n2 +1
∞
n2
≤ ∑ ζ 22 n +1 + α 2
=
n 1
∑
=
n n2 +1
=
n 1
ζ 22 n +1a ( d n )
∞
∑ ( d n − d n −1 ) a ( n )
n2
≤ ∑ ζ 22 n +1 + α 2
ζ 22 n +1a ( d n )
=
n n2 +1
∞
n2
∑ a (n)
≤ ∑ ζ 22 n +1 + α 2
=
n 1=
n d n2
y 2
n2
≤ ∑ ζ 22 n +1 + α 2 ∑ U ( j , j + 1)
=
n 1 =j 0
Since U ( j , j + 1) , U ( j , j + 1) satisfies the conditions of renwal theorem (see [15]) hence U ( j , j + 1) ,
U ( j , j + 1) is bounded for all j by a constant, so
q2 ( x ) ≤ q2 (φ0 ) + M 1 + M 2 x =
L( x )
q2 ( y ) ≤ q2 L (φ0 ) + M 1 + M 2 y =
L( y )
ˆ then E|Z0| is finite.
Theorem 2.3. If there exist a finite search plan (φ , φ ) ∈ Φ
Proof. If Eτ φˆ < ∞ , then P (=
τ φ < ∞ ) 1 or p (=
τφ < ∞ ) 1
( ) ( )
then P (τ φ < ∞ ) + (τ φ < ∞ ) =1 , that is=
Z 0 φ τ φˆ − S τ φˆ
with probability 1, so
( ) ( )
( )
S (τ ) ≤ τ then E S (τ ) ≤ Eτ
Z=
φ τ φˆ + S τ φˆ ≤ τ φˆ + S τ φˆ
o
( )
Hence E Z 0 ≤ Eτ φˆ + E S τ φˆ , but
φˆ
φˆ
φˆ
φˆ
and E Z 0 < ∞.
Remark A direct consequence of theorems 1, 2 and 3 in Section 2 is the existence of a finite search plan
φˆ ∈ Φˆ ( t ) if and only if E Z 0 ≤ ∞ , and then the set of search plans, which defined in Section 2, satisfies the
conditions of Theorem 1 if the expectation value of initial position of the lost target is finite.
3. Existence of an Optimal Path
ˆ ( t ) be a sequence of search plans, we say that φˆn converges to φˆ as n tends to ∞ if
Definition. Let φˆn ∈ Φ
for any t ∈ I + → , φˆn ( t ) converges to φˆ ( t ) on every compact subset.
+
Theorem 3.1. Let for any t ∈ ℜ , S ( t ) be a process with continuous sample paths. The mapping

(φ ,φ ) → E (τ Φ ) ∈ ℜ+ is lower semi-continuous on Φ ( t ) .
Proof. Let ξ be a sample point on L1 corresponding to the sample path ξ ( t ) of X o + S ( t ) and ξ be a
sample point on L2 corresponding to the sample path ξ ( t ) of Yo + S ( t ) . Let {φn }n ≥1 be a sequence of
converges to φ ∈ ΦV2 ( t ) . Given t ∈ ℜ+ , we define
search paths which converges to φ ∈ ΦV1 ( t ) and {φn }
n ≥1
for any n ≥ 1
{ξ : min ξ ( x ) − φ ( x ) > 0} ,
=
B ( t ) {ξ : min ξ ( x ) − φ ( x ) > 0} ,
=
Bn ( t )
0≤ x ≤t
n
0≤ x ≤t
{ξ : min ξ ( y ) − φ ( y ) > 0}
and=
B ( t ) {ξ : min ξ ( y ) − φ ( y ) > 0}
=
Bn ( t )
0≤ y ≤t
0≤ y ≤t
650
n
A. A. M. Teamah
Let ξ ∈ B ( t ) , ξ ∈ B ( t ) and since {φn }n ≥1 converges uniformly on [ 0,t ] to φ , then there exists an integer n (ξ ) such that for all n ≥ n (ξ ) and for any 0 ≤ x ≤ t
φn ( x ) −=
φ ( x ) < ε 0.5 min ξ ( x ) − φ ( x )
0≤ x ≤t
and ξ ( x ) − φ ( x ) ≤ ξ ( x ) − φn ( x ) + φ ( x ) − φn ( x ) ,
then ξ ( x ) − φn ( x ) ≥ ξ ( x ) − φ ( x ) − φ ( x ) − φn ( x ) ≥ 2ε − ε =
ε >0.
Hence ξ ∈ Bn ( t ) for all n > n (ξ ) and so B ( t ) ⊂ lim inf Bn ( t ) ,
n →∞
inf Bn ( t ) ) dt ≤ ∫ lim inf P ( Bn ( t ) ) d t
∫ P ( B ( t ) ) dt ≤ ∫ P ( nlim
n →∞
→∞
∞
∞
∞
0
0
0
∞
∞
by the same way we can get
∞
inf Bn ( t )  dt ≤ ∫ lim inf P ( Bn ( t ) ) dt
∫ P ( B ( t ) ) dt ≤ ∫ P  nlim
→∞
n →∞

0
0
0
since the sample paths are continuous then
∞
we get
∫ p (τ φ
{
{
)
}
{
Bn ( t ) = τ φn  t , B ( t ) = τ φ  t , Bn ( t ) = τ φn  t
∞
∞
∞
}
and B = {τ φ  t}
(
)
∫ p (τ > t ) dt ≤ ∫ lim inf p (τ
∫  p (τ > t ) + p (τ > t ) dt ≤ ∫ lim inf  p (τ > t ) + p (τ
> t ) dt ≤ ∫ lim inf p τ φn > t dt and
0
∞0
n →∞
φ
0∞
φ
φ
0
0
0
n →∞
φn
φn
n →∞
φn
)
> t dt , we obtain
)
> t  dt

then
∞
∞
0
0
inf p (τ Φ
∫ p (τ > t ) dt ≤ ∫ nlim
→∞

where Φ n =
(φn , φn )
(
)
hence E (τ Φ ) ≤ E lim inf τ Φ . By Fatou lemma, we get
n
n →∞
(
)
n
)
> t dt ,
( )
E (τ Φ ) ≤ E lim inf τ Φ n ≤ lim inf E τ Φ n .
n →∞
n →∞
Since Φ is sequentially compact (see [2]), then by the same way Φ is sequentially compact. It is known
that a lower semi-continuous function over a sequentially compact space attains its minimum.
4. Conclusion
We consider, here, the search for a lost target on one of two disjoint lines, where the target moves randomly according to continuous stochastic process which satisfies some conditions. Theorems conclude that there exists a
finite search plan if and only if the expectation value of the initial position of the target is finite. Existence of optimal search plan is proved.
Acknowledgements
The author would like to thank the reviewers and Editorial Board of AM Journal for their helpful suggestions
which would improve the article.
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