On a problem of escape along a prescribed curve

PMM U.S.S.R.,vol.46,pp.553-555 Copyright Pergamon Press Ltd.1983.Printed

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ON A PROBLEM OF ESCAPE ALONG A PRESCRIBED CURVE* A. AZAMOV in which a pursuer can move over the whole plane A differential game is considered, while the escaper can move along a prescribed curve with a with unit velocity. An escape strategy is constructed, ensuring bounded velocity greater than unity. a positive constant lower bound for the distance between the players. A regular curve f of smoothness class Cz is pre1. Statement of the problem. The 0, o> I. scribed on a plane, along which an escaper moves with a velocity not exceeding pursuer, whose velocity does not exceed unity in absolute value, moves over the whole plane A further refinement of the problem statement (informand tries to overtake the escaper. ativeness andclassof admissible strategies of the escaper, the concept of escape possibility, The paper's purpose etc.) can be traced from the proof of the fundamental theorem in Sect.4. is to construct an escape strategy permitting evasion from capture, as well as to derive an It is close in spirit to /1,2/. estimate for the distance between the pursuer and escaper. the An escape problem in the case when there are an arbitrary number of pursuers, while stright line, was solved in /l/. escaper moves in a neighborhood of an arbitrarily prescribed for both players was proved in /2/ in the case when the The existence of optimal strategies If in the game at hand the curve P is replaced by a graph, then once curve P is a circle. again a nontrivial game arises.

2. Preliminary

constructions.

We

introduce

the following

assumptions.

Assumption A.

The curvature of curve Pis bounded (in modul&) by a number i/p,p> 0. 1, so that If p>i, then it is obvious that the curvature is bounded also by the number we can set p= 1. Therefore, without loss of generality we can take p\
Assumption B. Curve r either is closed or is of infinite length in both directions from each of its points. Obviously, if curve I' has a finite length, even if in one direction only, then for specific dispositions of the players at the start of the game the pursuer overtakes the escaper in finite time. See Sect.5 for the motivation of Assumption A. This Assumption enables us to localize the problem. Let Qa be an arbitrary point on curve P and H be a rectangle centered at point Qo, two sides of which areparallelto the tangent to curve r at point Q0 and are of length kp, while the other pair of sides are half the size. Hereandsubsequently I k is a positive number depending only on p and e and specially selected, while kc1/1/5. Lemma 1. In the neighborhood in an appropriate coordinate system f(O)=1'(O)= 0. The inequalities

H of each point 00Er the curve as the graph of a function Y = f(z).

I\’ (I) 1C 1I I.@’ - z2)-“’ < x 12 1 <

hold

for function

S (I)

=

1 [I +

f“ @)I”

k (1_ k2)-'1' dz <

( I ,. (1 -

P can be represented, where f EC', I z I \i kp, (2.1 1

k?)+

f(z).

Proof. We introduce a Cartesian coordinate system as follows: point Q~ is the origin, the axis 01 coincides with the tangent to r at point Qa, and the positive semiaxis OY is directed toward the center of curvature if the curvature of curve P at point Q. is nonzero or is otherwise arbitrary (Figure). By virtue of the regularity of curve P it can be represented in every case by the graph of a function y =f(z) on some interval (--E,E). For values of z taken from this interval, using Assumption A and integrating, we find *Prikl.Matem.Mekhan.,46,No.4,pp.694-696,1982 553

554

--r/p .s

f’ (r)[l

+

(z)jf”

f’2

,c

r/p

Hence follows the first inequality in (2.1) for Ir\
y= t

Therefore, the representation tion y =f(.z) can be continued k <

kz)-“‘z

of curve F as the graphof onto the segment j~J
funcsince

l/l/%

3, Escaper's

strategy. We introduce a number 6>0 then as follows: if P,Q,< kzp, 6 = P,Q,. (*). We consider the circle S :z2+yz= F. If the inequality Pop,> k*p obtains for the initial positions P, and Q,,of the pursuer and escaper, respectively, then we set the escaper's velocity equal to zero until PQ= k8p occurs. Here and subsequentthe distance between at the current time f and PQis 1Y p and Q are the players' positions points P and Q. If, however, P,Q, < k2p, then by definition P. E S. Thus, we can take it that PO ES. To determine the escape direction on the circle S we divide the normal to curve F For definiteness let P,E~-. Then the at point Q. into two semicircles Sk:2 =*((s* -.p)? I, i.e., escaper is instructed to move away from point Q,, towards the side of increase of along the part Y = f(~),.z>O, of curve F (up to the point kp,f(kp)). 6=

J'oQo, otherwise

4. Estimate when

of the distance

P, = (0,4)

y,),O and

when

Proof. If (4 Id =

I f (I)

I Y, -

6 1=

8 -

I I Y, -

6 Id

-

Lemma 3. Let an instant

T

Lemma 2. Let

Then

(62 =o.

z&“’

g

I+0 +

be defined

f (d

by the

PQ>6*forOGtGT,

Proof.

With

due regard

to Lemma

1<(1+

<

that

-IO,

z.

Yo <

f (4 6. The

We

y,>O.

have

0

condition:

last

inequality

Q = (kp,f (kp))when

PQ
is equival-

t = 2’.

T

Then (4.1)

- PP, ,,P,Q - PP,

1 ensues PP,
Since

we reckon

2 we have

PQ>PoQ But from Lemma

We set P,= (0,6) P, = (10,Yo). for any point Q =(&f(z)) E F, O<:rG@*

P,Q>P,Q

symmetry, henceforth in (2.1)) and

In view of the total (see the last inequality

Therefore, f (d (Yo - 6) = ent to the one required.

PQ.

Y0 < 0.

k)“(l

-k*).

=y

s (2)

$

4

I3

(1 -

l+k < \-------I 0

k+

In addition

J'+Q2 P.Q.--QQ.. Q.= (2,'3 Consequently,

to prove

the first

inequality

(9 + ?I~)"' - (1+ Allowing for the conditions inequality ~+6~. Therefore, which, pQ>,6'

in

111 d kp< 1/1/c and (4.2) follows from z > (1+ k) a-b

its turn, is a consequence in for rc[O, T/ is fulfilled for

-

f (z) >

to derive (4.2)

6*

6dk2p<'/s , it can be shown the relation +

that(~'+6~)'/'>

z*p-’ k <

of the condition

k<

the proof Analogously to the above, fication of the inequality

(4.1) it is enough

k) o-11

(a -

i)/(o+l)-I.

Thus,

min

of the second

inequality

in

(4.1) is reduced

(9 + @)'b 2, T + p-k* + 6 Taking (see

into

account

the second

that

inequality

1:= kp, T = in

(2.1)),

appears

in the original

text

a-‘s(kp)

we replace

(k2p2+ 6*)"' > *) This

the estimate

ko-’

- Ed.

(1 -

<

(4.3)

kpa-1

(4.3) by the stronger La)-‘l’o

+

to the veri-

k*o +

6

inequality

555 We square we have

The

this inequality

inequality

and replace

6 by the larger

quantity

k?p. Then

after

cancellations

t > a-* (1- k2)-l + 3k'+ 4ko-'(1- k*)' obtained

is a corollary

of i>o-=+4P+4kn-1

which, in its own turn, follows from the condition proof of Lemma 3 it suffices to take k =

kg (a - i) (20)~1.

Thus,

to

complete

the

(4.4)

min I

in Sect.3, then in If the escaper adopts the strategy constructed 5. Conclusion. (4.2). This allows the further continuathe instant of time t= T ensure the second estimate Since each time the escaper goes on a path of length s(kp) boundtion of the escape process. edfrombelowby the constant kp (see the second inequality in (2.1)), by virtue of Assumption B escape is possible for all t>O. By the same token we have proved the following theorem (see the definition of 6..and the remark after Assumption A). Thenescape Theorem. Let Assumptions A and B be fulfilled in the game being examined. The escape process can be carried out is possible from any initial positions P,,Q~,.P,#Qw such that the following estimate for the distance PQ between escaper and pursuer is observed: PQ > (P,Q,J2,if P,Q, < k*min (1,P) PQ > hamin (1,p*),if PO&> k*min (1,P) where

k is determined by formula (4.4). From the proof of the second inequality in (4.1) of Lemma 3 we can note that in case we can ensure the estimate P,Q, < k*min {I.PI PQ > k'rnin(1,p2) beginning with the instant T= s (kp)/o If the curvature of curve T is not bounded, then the escaper cannot always ensure that the distance PQ can be estimate from below by a positive constant. For example, ifas r we take a hyperbolic spiral or the graph of the function Y = zsin(l/z),z>O, then by starting from many initial positions the pursuercanget arbitrarily close to the escaper. Ontheother hand, the theorem's proof remains in force for regular curves having selfintersections. An analogous theorem can be proved for space curves. REFERENCES 1. CHERNOUS'KO 2. FLYNN J-O.,

F.L., A problem of evasion Lion and Man: the boundary

from many pursuers. PMM Vo1.40, No.1, 1976. constraint. SIAM J. Control, Vol.11, No.3, 1973.

Translated

by N.H.C.