Pursuit Games (Mathematics in Science & Engineering, 120)

Pursuit Games

Academic Press Rapid Manuscript Reproduction

This is Volume 120 in MATHEMATICS IN SCIENCE AND ENGINEERING A Series of Monographs and Textbooks Edited by RICHARD BELLMAN, University of Southern California The complete listing of books in this series is available from the Publisher upon request.

Pursuit Games An Introduction to the Theory and Applications of Differential Games of Pursuit and Evasion

Otomar H6jek Department of Mathematics and Statistics Case Western Reserve University Cleveland, Ohio

Academic Press, Inc. New York San Francisco London 1975 A Subsidiary of Harcwrt Brace Jovanovich, Publishers

COPYRIGHT 0 1975,BY ACADEMIC PRESS, INC. ALL RIGHTS RESERVED. NO PART OF THIS PUBLICATION MAY BE REPRODUCED OR TRANSMITTED IN ANY FORM OR BY ANY MEANS, ELECTRONIC OR MECHANICAL, INCLUDING PHOTOCOPY, RECORDING, OR ANY INFORMATION STORAGE AND RETRIEVAL SYSTEM, WITHOUT PERMISSION IN WRITING FROM THE PUBLISHER.

ACADEMIC PRESS, INC. 111 Fifth Avenue, New

York, New York 10003

United Kingdom Edition published by ACADEMIC PRESS, INC. (LONDON) LTD. 24/28 Oval Road. London N W l

Library of Congress Cataloging in Publication Data Hijek, Otomar. Pursuit games. (Mathematics in science and engineering ; Bibliography: p. Includes index. 1. Differential games. I. Title.

QA272.H34 519.3 ISBN 0-12-317260-8

75-33342

PRINTED IN THE UNITED STATES O F AMERICA

11.

Series,

Contents

vii

Introduction Conuentions

xi

1

I EXAMPLES

1.1 1.2 1.3 1.4 1.5 1.6 1.7

Navigation Problem Simple Pursuit in the Plane One-DimensionalRocket Chase Kelley's Game: Pursuit on a Sphere Homicidal Chauffeur Two Cars Unpleasant Examples

1 3 9 10 20 23 26 33

II BASIC CONCEPTS

2.1 Differential Equations: Miscellany 2.2 Controls, Strategies, Winning Positions 2.3 Principle of Suboptimality Ill STROBOSCOPIC AND ISOCHRONOUSCAPTURE

3.1 Forcing to Origin 3.2 Unorthodox Linearisation 3.3 Affine Targets 3.4 Necessary Conditions 3.5 General Targets 3.6 Time Delays 3.7 Holding 3.8 Convex Sets, Pontrjagin Difference 3.9 Measurable Selection 3.10 Richter's Theorem 3.1 1 Reachable Sets V

33 38 47 55 58 67 78 85 89 95 100 105 117 122 125

CONTENTS

135

IV ISOCHRONOUS CAPTURE

4.1 Winning Sets 4.2 Necessary Conditions

135 140

V CAPTURE 5.1 Necessary Conditions 5.2 Sufficient Conditions 5.3 Large Targets 5.4 Invariant Targets

149 149 155 162 176 183

V I ALGEBRAIC THEORY 6.1 6.2 6.3 6.4

Game Space, Control Order Min-Max Controllability Equivalent Descriptions Invariants and Semi-Invariants

209

VII NONLINEAR GAMES 7.1 7.2 7.3 7.4

183 190 195 204

Essential Points Essential Points on Large Targets Necessary Conditions for Small Targets Isochronous Capture

209 213 223 235

Vlll STRATEGIES 8.1 Compactness Lemma 8.2 Optimal Strategies 8.3 Design of Strategies

239 239 246 252

Index

263

vi

Introduction

The purpose of this book is to present systematic methods for winning in differential games of pursuit and evasion, and to illustrate the scope and application of the developed procedures. These games appear in the abbreviated schema stochastic.

< '
0 ) i f t h e wind component i n t h e d i r e c t i o n of motion i s smaller t h e r e . This a l s o i n d i c a t e s t h e mild i m p l i c i t assumptions, and i l l u s t r a t e s t h e reasonable 'information p a t t e r n ' employed: a t t i m e t and p o s i t i o n z, t h e p i l o t bases h i s decision on t h e knowledge of t h e wind v e l o c i t y v e c t o r a t t h e same time t (and at time

i n p o s i t i o n s close t o

2).

We w i l l change t h e problem somewhat. eiCp i n (1)w i l l be replaced by

p

F i r s t , t h e term

subject t o

(PI

< 1. Since

it i s i n t u i t i v e l y obvious t h a t p i l o t should use maximal a v a i l able c o n t r o l t o minimise a r r i v a l time, t h i s modification i s i n s i g n i f i c a n t , while t h e advantages of having a convex set a s c o n t r o l c o n s t r a i n t s e t ( t h e u n i t d i s c i n place of t h e u n i t c i r c l e ) a r e immense. A major change w i l l be t h e replacement of

w(z,t)

by

w ( t ) , independent of t h e s t a t e v a r i a b l e ("gleichformigen Winde2

EXAMPLES

feld").

Condition (2) t h e n s t a t e s t h a t optimal s t e e r i n g main-

t a i n s a constant d i r e c t i o n ; and t h i s i s a complete mystery:

a

wind gust u n r e s t r i c t e d i n magnitude can, of course, blow t h e a i r p l a n e o f f any s t r a i g h t path.

For a p p r o p r i a t e l y bounded wind

v e l o c i t i e s , t h e conclusion does make sense (Exercise 1 i n 3.1). Another paradox may not be apparent a t f i r s t glance. Optimal s t e e r i n g

of

cp(t)

a t time

t

i s determined on t h e b a s i s

w(z,t); i f t h e p i l o t could p r e d i c t t h e future wind behav-

iour, w h y can no advantage be gained from such knowledge? answer i s provided i n 5.4:

An

t h i s i s indeed t h e case f o r a some-

what s p e c i a l c l a s s of gnmes, t o which t h e navigation problem belongs. For Zermelo's problem and s o l u t i o n see E . Zermelo : iiber das Navigationsproblem b e i ruhender oder v e r k i l e r l i c h e r Windverteilung, Zeitschr. f . angew. Math. u. Mech. 11 (1931) 114-124.

The case of constant wind i s an i l l u s t r a t i v e example i n Section

1.17 of G. Leitmann, An Introduction t o Optimal Control, McGrawH i l l , New York, e t c . , 1966. 1.2

Simple Pursuit i n t h e Plane Two players move i n t h e Euclidean plane

motion:

R2 with simple

each has a bound on h i s speed, b u t t h e r e a r e no

f u r t h e r r e s t r i c t i o n s (e .g., allowed).

abrupt d i r e c t i o n a l changes a r e

One, t h e pursuer, wishes t o capture t h e other,

quarry, i.e., a t t a i n perfect coincidence of t h e i r t e r m i n a l positions. The answer i s obvious: i f u > fl holds f o r t h e pursuer's speed -bound. a and t h e quarry's p, then termination i s assured i n f i n i t e t i m e , whatever t h e i n i t i a l p o s i t i o n s and a c t i o n of quarry; on t h e o t h e r hand, i n t h e case a s P

3

PURSUIT GAMES

quarry can avoid capture forever from any positions not i n contact initially. Let us discuss b r i e f l y some aspects of simple motion,

f i r s t f o r a single player ( f o r the flm of it, t o make possible the analysis below, and t o prepare an analogy f o r subsequent sections). I f the player's position a t time t E R1 is denoted by speed

x(t) E R

l & ( t )I.

2

, then t h e

velocity vector is

;(t),

Thus the dynamical constraint i s

and the 5

a;

actually, another formulation i s preferable, v i z 1 2 (1) = u; measurable u: R + R (U(t)l L a. t What i s r e a l l y meant here i s t h a t x ( t ) = x(0) + u(s)ds J

u(.)

a s indicated.

place the origin a t

x(O), so t h a t

some Punction

0

for

Just t o simplif'y matters x(0) = 0.

Where can the player get t o a t time

t ? The constraint

on u ( - ) yields (x(t)l s Conversely, any point control u, e.g.,

y

u(s) =

t 0

with

(u(s)lds s at. IyI

(;&

L

ut

can be 'attained' by

for 0 s s

IYI

f o r lyl < s s t. Thus t h e a t t a i n a b i l i t y s e t Ao(t) = {y: (yI < at] (Unfortunately, current terminology reserves 'reachable s e t ' f o r a d i f f e r e n t use).

See f i g . 1.

4

EXAMPLES

Point

F i g . 1 Simple motion i n t h e plane. x moves anywhere within Ax(3) a t

time t = 3; i f i t s p o s i t i o n y a t t = 1 o r z a t t = 2 is known, t h e p o s s i b i l i t i e s reduce t o A ( 2 ) o r A Z ( l ) . Y L e t us r e t u r n t o t h e game, i n t h e case

a

> p.

Probably

t h e f i r s t capture stratepy t h a t comes t o mind i s f o r t h e pursuer t o aim, constantly, a t t h e quarry, thereby following an appropriate curve of pursuit (Huygens t r a c t r i x , i f quarry's motion is uniform).

It would be a formidable and boring t a s k

t o obtain t h i s curve e x p l i c i t y ; f o r t u n a t e l y t h i s may be 1 avoided. If t h e pursuer's motion i s x: R 4 R2 and t h e 1 2 t h e equations of motion a r e quarry's y: R + R

,

(2)

.

x = u , y = v

f o r s u i t a b l e measurable c o n t r o l s

5

u,v: R

1

-+

R2

with a l l

PURSUIT GAMES

values

l u ( t ) l s a , / v ( t ) l s p.

e r s ' distance by

1 * Ix

r; = Adt

(3)

r

- yI .

Ix

=

- YI 2

A t any time denote t h e play-

Then

= (x-y)'(k-$)

-

= (x-y)'u

(x-y)'v.

The procedure suggested f o r t h e pursuer i s t o take

-2

rE = a rr

by Cauchy's inequality. have

(x-y)'v s -ar + r p = -(a-p)r

Therefore, as long as

r

>

0, we

s -(a-P),

r(t)s.r(o) This shaws t h a t capture with

-

(a-p)t = (xo-y0I

( r = 0)

-

(a-p)t.

m u s t occur, a t some time

8

I X0-Yo I a-p

Ekercises 1. For a mapping x: R1

lu(t)( s a

t h e two conditions

Rn

almost everywhere; x ( t ) = x(0)

l u ( t ) { s a a.e.

+

= u,

Jtu(s)ds, 0

a r e often t r e a t e d a s i f they were synonymous.

Prove c a r e f u l l y t h a t t h i s i s so i f ' x ( * ) absolutely continuous' i s added t o t h e f i r s t , but mere continuity i s i n s u f f i cient.

(Hint:

t h i s involves some simple but r a t h e r d e l i c a t e

r e a l analysis. ) Exercises 2 t o

7 concern t h e case

a

>p

of simple pur-

s u i t i n t h e plane. (Cutting Corners) The pursuer, s t a r t i n g a t xo, f i r s t s t e e r s t o quarry's i n i t i a l position yoj on reaching 2.

it, he (she?) repeats t h e process. Set up an accounting of times tk and positions 5, yk, with t h e requirement xk+l = k'

determining tk+l and an estimate of

6

lyk+l-ykI.

EXAMPLES

Prove t h a t (4) holds.

3.

The pursuer f i r s t steers t o

( F o l l m t h e Leader)

quarry's i n i t i a l position, and t h e n follows h i s path, using Show t h a t , again, (4) holds.

excess speed t o catch up.

4.

(Neutralisation)

Referring t o (l), t h e pursuer

f i r s t n e u t r a l i s e s quarry's action, and then uses l e f t - o v e r c a p a b i l i t y t o f o r c e termination: direction

y

-x

and magnitude

t i o n always occurs a t time

Jxo

u = v + w

lu-Bl.

w

with

i n the

Prove t h a t termina-

- yol/(a-p)

p r e c i s e l y (even

i f quarry cooperates).

5.

(A Moveable F e a s t )

Take a moving coordinate system,

with o r i g i n a t t h e quarry's p o s i t i o n y ( t ) : z = y(t)

satisfies

condition.

=

u

- v(t),

with

The new c o n t r o l v a r i a b l e is

new coordinate

z = 0 a s termination w = u v ( t ) , with

-

constraint

w E [u

-

v(t):

or, i f constant bounds a r e desired,

IUI

s a],

Iwl s a

- p.

Show t h a t

e i t h e r choice i s equivalent t o one of t h e capture procedures already outlined.

w

where

6.

i n t h e second case l e t

(Hint:

has magnitude

a

u = v and d i r e c t i o n yo xo.)

-p

-

+w

(Capture with A n t i c i p a t i o n ) The quarry announces h i s

future policy

v: R + R2 t o t h e pursuer; based on t h i s i n f o r -

mation, t h e pursuer proceeds with capture.

Prove t h a t (4)

holds.

7.

(Approximate Capture under Time Delays)

A t any time

t 2 0 t h e pursuer is allowed information on quarry's c o n t r o l v ( s ) o r p o s i t i o n y ( s ) only f o r s s t 6 (6 > o i s a

-

given c o n s t a n t ) .

Prove t h a t approach t o within a d$stance p6

can be guaranteed, and estimate t h e needed time.

8. pursuer:

Show t h a t t h e case

as p

i s unfavourdble t o t h e

t h e r e i s an extremely simple evasion c o n t r o l f o r t h e

quarry. 7

PURSUIT GAMES

9.

Treat simple p u r s u i t on a s t r a i g h t l i n e ; i n particu-

lar,show t h a t again capture always occurs i f otherwise.

u

> p,

and never

Prove t h a t , by a ' n e u t r a l i s a t i o n technique,'

cap-

ture t i m e can be prescribed, but not capture p o s i t i o n . 10. Consider pursuit on a s t r a i g h t l i n e , by two pursuers, of one quarry, a l l moving with simple motion.

Prove t h a t ,

from same i n i t i a l p o s i t i o n s , capture occurs even i f t h e pur-

suers a r e both slower t h a n t h e quarry, b u t t h e capture t irne cannot be prescribed; and sketch t h e corresponding set i n

2

f o r which 'bracketing' occurs. Remarks The reader i s encouraged t o experiment with other reasonable pursuit strategies.

E.g.,

t h e method of c u t t i n g corners

(Exercise 2 ) might be modified, with t h e pursuer only reaching half way a t each step, bringing t h e method c l o s e r t o t h e curve-of-pursuit

s t r a t e g y from t h e text (which then suggests

further questions).

O r , t h e pursuer might overshoot, e.g. by

a f a c t o r 1.1, i n an attempt t o p r e d i c t quarry's motion.

An-

other analogue of t h i s involves 'programed' time i n s t a n t s

to = 0 < t 1 < t2
0, i n x 5 -F. As t + 0, t h e boundaries (having a p a r a b o l i c envelope i n x 2 - E ) approach x = -E, y 2 1. 2.

On t a k i n g

u

4

1, show t h a t

-

x ( t > 2 xo + (yo

f o r any quarry c o n t r o l then t h e r e e x i s t s

some

~ ( 0 ) . Conclude t h a t , i f

>0

such t h a t

x(t) = 0

xo < 0,

occurs f o r

t E [ O , e ] j and a c t u a l l y o b t a i n t h e l e a s t v a l u e of

( i n dependence on

(Yo

8

- 1)t + 2 tz

- 1).) 3.

xo, y o ) .

(Answer:

,/((yo

-

1)2

-

2xo)

The above e x e r c i s e e x h i b i t s a capture time bound

obtained on t a k i n g a p a r t i c u l a r pursuer c o n t r o l .

14

8

0,

Show t h a t

EXAMPLES

it i s a c t u a l l y independent of t h i s : c o n t r o l v(.),

such t h a t


1. e e5 . Experiment w i t h g r a p h i c a l s o l u t i o n f o r o t h e r shapes

of t a r g e t (e.g.,

a u n i t c i r c l e a t t h e o r i g i n ) ; granting t h a t

u =

pursuer's responses a r e piecewise constant

2

1, motion

w i l l be along t h e parabolas ( 2 ) with v e r t e x ordinates -1 < v < 1, and

v

=

2

1 a s t h e i n t e r e s t i n g cases.

Obtain,

h e u r i s t i c a l l y , t h e boundary of t h e s e t of winning positions. Consider t h e two-dimensional version of t h e game 1 t r e a t e d , and, i n p a r t i c u l a r , t h e motion x : R -b R2 of a

6.

player with

= u,

l u ( t ) l s 1. If t h e o r i g i n i s placed a t

x(O), with t h e r e a l a x i s i n t h e d i r e c t i o n of

G(O), show t h a t i s a d i s c with IG(0) It; t h a t t h e r e i s a doubling-

t h e s e t of possible p o s i t i o n s a t t i m e

t 2/2

radius

and c e n t r e

back e f f e c t s t a r t i n g a t t i m e

@ =

t i o n regained a t doubled time 28.

t

2

0

IG(0) I, with i n i t i a l posi(Warning:

c o n t r o l - t h e o r e t i c concept of a t t a i n a b l e set

4

l a t t e r i s i n R , not not o c c w f o r t h e s e . )

7.

R2

-

-

t h i s i s not t h e i n our case t h e

and a c t u a l l y t h e phenomenon can-

..x + A .x + B x = u , Y

For t h e game with n-dimensional equations of motion

=

~

s e t up t h e appropriate f i r s t - o r d e r equation, i n case

B = 0 = C, with

i s a reduction, with

x = y R2n

v

2".

In the

a t termination, show t h a t t h e r e

a s s t a t e space, and equations

analogous t o (1) ( t h e s p e c i a l case A = UI, B = 0

+

n = 3, s c a l a r matrix

i s t h e so-called I s o t r o p i c Rocket Game).

(Answer : x = y 15

V,

$

= -Ay +

u.)

PURSUIT GAMES

on changing n o t a t i o n . ) Remarks The game t r e a t e d i s t h e one-dimensional v e r s i o n ( f r i c t i o n l e s s c a s e ) of t h e i s o t r o p i c rocket game, and a l s o of t h e homicidal chauffeur game; s e e R. I e a a c s :

D i f f e r e n t i a l Games, Wiley, New York, e t c . ,

1%-7 -

For g e n e r a l dimensions, t h i s i s c a l l e d t h e Boy and Crocodile game i n J. F. Mizbenko: P u r s u i t and evasion problems i n d i f f e r e n t i a l games, I z v e s t i a Akad. Nauk

SSSR 5 (1971)3-9.

1 . 4 Kelley's Game:

P u r s u i t on a Sphere

2,

Two p l a y e r s move on t h e two-sphere S2 i n each with a f i x e d bound on h i s speed; t h e game ends a t coincidence of p o s i t i o n s . The motivation is t h a t " i n a dogfight, t h e planes t e n d t o move i n a c i r c u l a r fashion" ( i n a w r i t t e n text one cannot adequately convey t h e a p p r o p r i a t e hand motions).

The simpli-

f i c a t i o n does away w i t h one s i g n i f i c i a n t a s p e c t of a c t u a l combat:

t h a t t h e r o l e s of pursuer and quarry a r e not fixed,

b u t may w e l l switch back and f o r t h . The outcome i s not t o o s u r p r i s i n g (denote t h e p u r s u e r ' s

p ) : i f a > p, t h e purs u e r can f o r c e t e r m i n a t i o n from any i n i t i a l p o s i t i o n , w i t h i n speed bound by

a, t h e quarry's by

a bounded t i m e i n t e r v a l ; i n t h e case avoid c a p t u r e a t a l l times tion

a = p

t >0

a

< B t h e quarry can

(and t h e stand-off s i t u a -

i s rather too sensitive t o d e t a i l s i n the

s p e c i f i c a t i o n of t h e p l a y e r s ' s t r a t e g i e s ) . This i s r e a d i l y seen a s follows.

I n t h e case a > p

f i r s t assume t h a t t h e p l a y e r s a r e not a t d i a m e t r i c a l l y

16

EXAMPLES

opposite p o i n t s i n i t i a l l y .

Then t h e r e i s a unique s h o r t e r a r c

y of a g r e a t c i r c l e j o i n i n g t h e i r p o s i t i o n s . By a p a r a l l e l s h i f t along y, m e a neighbourhood of quarry's p o s i t i o n t o t h e pursuer's ( t h i s ' a c t i o n a t a d i s t a n c e ' serves t o i d e n t i f y t h e quarry c o n t r o l ) . u = v

t h e f i r s t component n e u t r a l i s e s quarry's action,

f W j

t h e second, w of a

-

The pursuer t h e n uses t h e c o n t r o l

IwI = u

with magnitude

-

p

i n the direction

y, serves t o decrease t h e players' d i s t a n c e ( a t a r a t e If t h e i r i n i t i a l posi-

p, see 1.3) u n t i l capture occurs.

t i o n s a r e opposite, then any constant c o n t r o l u

>

IuI = u

with

applied over a s h o r t i n t e r v a l w i l l achieve non-

f3

opposing p o s i t i o n s .

By a l i k e reasoning, i n t h e case

a

0.

4 2 a; i n the

obvious manner. By assumption, t h e v e c t o r s

t h e r e i s a b a s i s for

a,b

of t h e form a,b,c3,

Rn

(4)

X,Y,C3,.

remain independent i f

a r e independent, so t h a t

x,y

*

...,cn.

Then

*,cn

a r e c l o s e enough t o

a,b.

Apply

t h e Gram-Schmidt orthogonalization process t o t h e sequence

(4), obtaining orthonormal v e c t o r s el,e2,. 1x1 = 1, we have

since

..,e n .

el = x, and, i n t h e second s t e p ,

- ( x # y ) x l * = 1 - (x'y)2 = 1 - 2'9. colunn v e c t o r s e2,. ..,en i n t o t h e (n,n-1) matrix Iy

Collect t h e E(x,y);

COB

(1) holds s i n c e (x,E(x,y)) is orthonormal.

coordinate of

y'E(x,y)

is

el

=

x

and

The f i r s t

y'e2 = Isim( f r o n ( 5 ) ; t h e r e -

0, s i n c e

maining coordinates a r e both

Since

ek

e2, and hence t o

y

is perpendicular t o also.

QZD

Returning t o Kelley ' s game ( a c t u a l l y , f o r dimensions n

2

2), we may chmse t h e s t a t e space d e s c r i p t i o n = E(x,y)u,

The c o n t r o l values i n

R

n-1

$

= E(y,x)v.

a r e constrained 5 y

( u ( t )I s a.

t h e i n i t i a l p o s i t i o n s a r e on t h e u n i t sphere. Iv(t)l s 0 i s the angle between x,y, and r = Isimp(, then 18

If

EXAMPLES

rc

(6 1

=

+

w

IwI

r;

u = -v

subject t o

t (1

+ y':)

= -(x';

Write

a

d 2 +s i n cp = dt

with

w

E

-

(x'y)

R

t o be chosen subsequently,

Then, using ( 2 ) ,

a-p.

rE = (-x'E(Y,x) + Y'E(X,Y))V =

o

-

lsi*1(1,0,

)

- y'E(x,y)u.

= -x'E(y,x)v

n-1

2

...,O)W s

- Y'E(X,Y)W -r (a-a)

,...,

on t a k i n g w' = (u-p)(l,O 0 ) . Thus $ s -(u-p) < 0, and s i 9 = 0 can be a t t a i n e d i n f i n i t e t i m e ( f o r obtuse angles cp

an abvious modification must be employed).

Exercises 1. I n t h e c o r o l l a r y show that t h e neighbourhood of t h e n-1 point a E S on which E( ) i s defined may be taken a s

t h e e n t i r e hemisphere with apex a t b,c3,

. . ., c n

Note t h a t here

complex notation, so t h a t X H E(x) 1 2 S + R Furthermore, i n ( 6 ) ,

.

r; = -1m

3 . Assume t h a t a continuous el

R3

E(x) = i x

in

is a well-defined mapping

G(u xw

E(x)

v) satisfies ( 3 ) .

be t h e f i r s t b a s i c u n i t vector i n

x w E(x)el

Conclude t h a t in

choose

The a n a l y s i s remains v a l i d i n t h e t r i v a l case of

2.

that

(Hint:

suitably.)

n = 2 ( p u r s u i t on t h e c i r c l e ) .

Letting

a.

verify

Rn-',

defines a continuous ( t a n g e n t ) vector f i e l d .

x b E(x)

cannot be defined over a l l of

S2

(you may use Poincarg's theorem on continuous vector

f i e l d s , t h e Hedgehog Theorem. ) Extend t h e negative result of Exercise 3 t o a l l n-1 with an dimensions n 2 3 . (Hint: i n t e r s e c t S 2 appropriate l i n e a r space, thereby reducing t o S .)

4.

5.

For

n = 3 use standard polar coordinates

19

b'9

of

PURSUIT GAMES

a point

with

x E F?

1x1 = 1, and show t h a t cosecosp

sine

-sing This s a t i s f i e s conditions (3) f o r a l l

how does t h i s

9,'p;

match up with Exercise 3?

k

6. Verify t h a t , i n t h e preceding s i t u a t i o n , t h e equation = E(X)U reduces t o

6 7.

Carry out t h e preceding two exercises f o r

(Answer: c

k

= ul,-sinrp*B = u2'

f o r polar coordinates

= cos 9

k

and

ek,

sk = s i n

8 ,8 ,9

c s c 1 2 3

-S2S38,

8.

=

1

2

3

on

n =

4.

S3, with

c c 1 2

5 , s3Q2= u2,

h3

= ul.)

The solution of Kelley's game proposed i n t h e t e x t

would s t e e r

x

t o t h e point diametrically opposite y

a t t h e i n i t i a l time, the angle

i s obtuse.

if,

Obtain a

solution f o r t h i s case.

1 . 5 Homicidal Chauffeur A consituent p a r t of s e v e r a l games is t h e motion

x: R1

+

R"

constraints:

of a point i n n-space subject t o two dynamical t h e speed i s constant, and t h e curvature i s

bounded a p r i o r i . metrised curve

t

Recall t h a t t h e curvature pb

x(t)

i s t h e magnitude of

H

of a para-

EXAMPLES

a d i r e c t computation of

i s constant i f f

Now, i,w =

2

then y i e l d s

u2

x

remains perpendicular t o

u s l / p reduces t o The perpendicularity requirement

3112; then t h e c o n s t r a i n t

5 s2/p

for

s

=

I;().

can be expressed i n terms of t h e mapping

E(m) ( s e e Corollary

i n 1.4) j we then obtain t h e c o n t r o l - t h e o r e t i c formulation

..x = E(;)u,

(1)

1 n-1 measurable u: R + R

,

2

lu(t)l

5

s 7

9

( x ( 0 )I = s. I n t h e homicidal chauffeur game t h e pursuer has such a

i n i t i a l conditions including motion (parameters : with speed bound

sl,pl),

s2

t h e quarry has simple motion,

( a s i n 1.3); t h e game ends if Ix-Yl

f o r a given capture radius

c

2

0.

S

The primary equations of

motion a r e = E(~)u,

(2)

with f i r s t - o r d e r version

i1=

x2,i2 = E(x2)u,

thus t h e n a t u r a l phase space i s s i o n 3n

- 1.

5

= v; end:

Rn

x Sn'l x Rn with dimen-

Ix, - $ 1

s e;

An obvious reduction i s obtained i n terms of

new coordinates ( 31

= V,

=

k

= y

- 3'

- v,

$

= x2

(not those of ( 2 ) ) :

= E(y)u; end: 1x1

control constraints

21

5

E,

PURSUIT GAMES

and a requirement on i n i t i a l values, phase space i s

Rn x S n'l,

The reduced

Iy(0)l = sl.

with dimension

-

2n

1.

Exercises I n 1.4 t h e

1. (Two-Dimensional Homicidal Chauffeur)

case

n

able

8

= 2

here it is n o t .

was t r i v i a l :

ie

.

y = s e 1

t o replace

i n (3):

- "1

x1 = s1c0se

- v2

x = s sine 2

e constraints: (Hint:

1

= u,

IuI < sl/pl,

Exercise 2 i n 1.5.)

2.

The dimension

i l y large.

Use a new v a r i -

2n

2 2 v1 + v s s2, end: 2

- 1=3

2

2

X1+X2
)v,

1 n-1 2 u,v: R + R constrained by IuI s s1/p 1' and w i t h termination condition involving only

X,Y.

The corresponding f i r s t - o r d e r dynamical equation i s

23

PURSUIT GAMES

e a s i l y obtained, w i n g

Y

- 2,

space has dimension 4n formal dimension 4n.

il= iJy4 = i3 = $;

the s t a t e

with t h e dynamical equation of

I n terms of new s t a t e v a r i a b l e s = Y1

t h i s reduces t o

i1 = X2

(1)

=

2

- Y3> x2

- 3JG2

with s t a t e space dimension

= Y2J

5=

E(X2)U,

=

5'4,

5

= E(?)V,

To apply t h i s we must

3n-2.

assume t h a t t h e o r i g i n a l termination condition depends on x

-y

only ( r a t h e r t h a n on

x,y

separately).

Let u s consider, s p e c i f i c a l l y , t h e case of dimension n = 2J and proceed t o lower t h e s t a t e space dimension (from

4 t o 3) i n t h e manner suggested by Exercises 1 and

2 in

1.5;

f o r t h e moment t h i s formal reason s h a l l be our s o l e justification.

The n o t a t i o n i s s i m p l i f i e d by using new ( r e a l valued)

control variables

p,q: u = s p, v = s2qJ with c o n s t r a i n t s S 2 l s1 IP(t)I ->, Is(t)l *

a

p 1

2

The one-dimensional ' c i r c u l a r ' v a r i a b l e s w i l l be w r i t t e n a s x2 = sle icp ,

3=

s2eit;

then, from t h e l a t t e r two r e l a t i o n s i n (l),

(31

P, 4

cp =

= 9.

F i n a l l y , new s t a t e v a r i a b l e s a r e introduced:

e = e-icpx and r e a l 1

8 = cp = -izp

(4 1

.

Alternately, w = e-iJrx

(5)

1

and

w = -iwq

- $. -

sl

complex

Then, from (1)t o ( 3 ) ,

+

eeie, 5

s

2

= p

8 satisfy

+ s1eie

-

s2,

8

= p

-

q.

- q.

I n e i t h e r s e t , t h e three r e a l equations a r e e a s i l y obtainedj

( 4 ) corresponds, modulo n o t a t i o n a l changes, t o t h e 'kinematic equation', p. 238 of 24

EXAMPLES

R. I s s a c s :

D i f f e r e n t i a l Games, Wiley, New York, e t c . ,

1957

-

This r e v e a l s another i n t e r p r e t a t i o n ( r e a d i l y v e r i f i e d by ret r a c i n g t h e successive s u b s t i t u t i o n s ) :

( 4 ) d e p i c t s t h e game

on t h e pursuer's radar screen, i . e . , i n a movable coordinate system which has i t s o r i g i n a t t h e pursuer's c u r r e n t position, and with r e a l a x i s i n t h e d i r e c t i o n of h i s motion; s i m i l a r l y ,

( 5 ) shows t h e game on t h e quarry's r a d a r screen. Suppose t h e t a r g e t There i s a f u r t h e r use f o r (b), (5). i s not c i r c u l a r , b u t r a t h e r ship-shaped, with a chosen point a s t h e quarry p o s i t i o n .

The bow w i l l then ccmstantly point i n

t h e quarry's d i r e c t i o n of notion (say, side thrusters a r e not allawed f o r ) j t h e termination condition w i l l then involve not only

x,y

but a l s o

i.

Analogously,

i f t h e pursuer i s an

a i r p l a n e with f i x e d guns, t h e t a r g e t set i s a fan-shaped region with a x i s i n t h e p u r s u e r ' s d i r e c t i o n of motion.

In

t h e s e cases we of (5) o r (4) w i l l profoundly simplify t h e termination condition. Remarks The game i s t r e a t e d i n Section 9 . 2 of I s s a c s ' book, f o r circular targets.

More d e t a i l , together with extensive (and

i n t r i g u i n g ) p l o t s of t h e results, appear i n T. L. Vincent, E. M. C l i f f , W. J. Grantham, W. Y. Fkng: A problem of c o l l i s o n avoidance, EES S e r i e s Report 39, Engineering Experiment S t a t i o n , The University of Arizona, 1972.

The r e s u l t s a r e summarised i n T. L. Vincent, E. M. C l i f f , W. J. Grantham, W. Y. Fkng: Some aspects of c o l l i s i o n avoidance, AIAA Journal 1 2 (1974) 3-4, T. L. Vincent: Avoidance of guided p r o j e c t i l e s , p. 267-279 i n The Theory and Application of D i f f e r e n t i a l Games (ed. J. D. Grote), Reidel, Dordrecht and Boston, 1975. 25

PURSUIT GAMES

The case of a ship-shaped (or rather, lens shaped) t a r g e t i s t r e a t e d i n t h e l a s t reference.

A preliminary treatment, with

t h e t a r g e t replaced by a d i s c displaced i n the direction opposite t o the velocity vector, appears i n T. L. Vincent, W. Y. Peng: Ship c o l l i s i o n avoidance, Navy Workshop i n D i f f e r e n t i a l Games, Annapolis,

1973

The case of perfect capture i s t r e a t e d i n E. Cockayne: Plane [ s i c ] pursuit with curvature cons t r a i n t s , SIAM J. Appl. Math. 15 (1967)

1511-1516.

The a s s e r t i o n t h e r e i s t h a t pursuer can force capture from any i n i t i a l position i f , and only i f ,

In G. T . Rublein: On pursuit with curvature constraints, SIAM J. Control 10 (1972) 37-39

t h e method i s extended t o t h e case of dimension

n = 35 sharp

i n e q u a l i t i e s i n (6) ensure perfect capture from a= i n i t i a l positions.

See a l s o t h e remarks folluwing 7.3.

1.7 Unpleasant Examples The reason f o r t h e section heading i s t h a t t h e examples described here, simple instances of f a r more general situations, cannot be t r e a t e d by t h e methods t o be developed l a t e r i n t h e book (surely t h e r e i s an obligation t h a t the reader be made aware of t h i s circumstance). The f i r s t i s t h e celebrated Lion and Man game; it i s probably b e s t t o reproduce, with t h e kind pelmission of t h e publisher, two pages from J. E. Littlewood: A Mathematician's Hiscellally, Methuen, London, 19

26

EXAMPLES

'Lion and Man' ' A l i o n and a man i n a closed arena have e q u a l maximum speeds. What t a c t i c s should t h e l i o n employ t o b e sure of h i s 1 meal?

It was s a i d t h a t t h e 'weighing-pennies' problem wasted of war-work, and t h a t t h e r e was a pro-

10,000 s c i e n t i s t - h o u r s

p o s a l t o drop it over Germany. This one,2 though 25 y e a r s old, has r e c e n t l y swept t h e country; b u t most of us were t e a s e d no more t h a n enough t o a p p r e c i a t e a happy idea b e f o r e a r r i v i n g a t t h e answer, ' L keeps on t h e r a d i u s OM'. I f L i s o f f OM t h e asymmetry helps M. So L keeps OM, M a c t s t o conform, and i r r e g u l a r i t y on h i s p a r t helps L. L e t u s t h e n s i m p l i f y t o make M run i n a c i r c l e C of r a d i u s a w i t h angular v e l o c i t y W . Then L (keeping t o t h e on

r a d i u s ) runs i n a c i r c l e touching C', a t P, say, and M i s caught i n time less t h a n n/W. This follows e a s i l y from t h e 2 2

equations of motion of L, namely = w, G2+r2W2 = a w It i s , however, i n s t r u c t i v e t o analyse t h e motion near P. For t h i s

E2

(a-r)*/K,

and t

.

< c o n s t . + Kj(a-r)-$dr.

-

The i n t e g r a l converges ( a s r + a ) with p l e n t y t o s p a r e p l e n t y , one would guess, t o cover t h e use of t h e simplifying hypothesis.

* The p r o f e s s i o n a l w i l l e a s i l y v e r i f y t h a t when

M s p i r a l s outwards t o a c i r c l e , and, w i t h obvious n o t a t i o n x = rM rL, w e have (W varying), we w r i t e

-

-2 > (W 2rL )(x/GL) = X, where X/x + m. Then t < c o n s t . + X -1dx, and i n t h i s t h e i n t e g r a l i n c r e a s e s more slowly than

s

X

x-ldx:

X

it i s a g e n e r a l l y s a f e guess i n such a case t h a t t h e

i n t e g r a l converges.

*

1 'The 2

3

curve of p u r s u i t ' ( L running always s t r a i g h t a t M) t a k e s an i n f i n i t e time, s o t h e wording has i t s p o i n t . Invented by R . Rado (unpublished). The c a s e when L s t a r t s a t 0 i s p a r t i c u l a r l y obvious, on geometrical grounds.

27

PURSUIT GAMES

A l l t h i s notwithstanding, t h e 'answer' i s wrong, and M

can escape capture, (no matter what L does).4 This has just been discovered by Professor A . S. B e s i c w i t c h ; here i s t h e f i r s t (and only) version i n p r i n t . I begih with t h e case i n which L does keep on OM; very easy t o follow, t h i s has a l l t h e e s s e n t i a l s i n it (and anyhow shows t h a t t h e 'answer' is wrong). S t a r t i n g from M ' s p o s i t i o n % a t t = 0 t h e r e i s a polygonal path Mo%M 2... with t h e pro-

perties:

(i)

MnMn+l

i s perpendicular t o OM,,

( i i ) the t o t a l

lew - h i s i n f i n i t e -, (.i i i ) t h e -path s t a g s i n s i d e a c i r c l e round 0 infijide t h e arena. I n f a c t , i f 4, = Mn,lMn w e have I

h i , and a l l i s secured i f we t a k e fin = cn-314 with a s u i t a b l e c. L e t M run along t h i s path ( L keeping, i s perpendicular t o LoMo, a s agreed, on OM). Since Mo% L z s not catch M while M Ls o," Mo%. Since L i s on 1 0%) M1M2 i s perpendicular t o Ll%, and L does not catch M while M is on M1M2. This continues f o r each successive Of

,

= 0% +

and f o r an i n f i n i t e t i m e s i n c e t h e t o t a l length i s

MnMn+l,

infinite.

*

I add a sketch, which t h e p r o f e s s i o n a l can e a s i l y complete, of t h e astonishingly concise proof f o r t h e q u i t e general case. Given Mo and Lo, M s e l e c t s a s u i t a b l e 0

( t o secure t h a t t h e boundary does not i n t e r f e r e with what follows), and ' c o n s t r u c t s ' t h e polygon MoM1M2.. . described above, b u t runs along another one, MOM;M;.

. ., associated

i s drawn with it, but depending on what L does. MOM: perpendicular t o LoMo, No i s t h e f o o t of t h e perpendicular

from

0

so t h a t

M;,

on

MOM;,

NoM;

=

MiM;

i t so t h a t an)

4

OMi2

5

and

Mi

i s taken beyond

fil(=MoY). If

L

is at

when

L1

i s drawn perpendicular t o

LIM;,

and

N M' = fi2; and so on. 1 2

Clearly

OM:

Of,

from M

No

Mi -

2

M

0

is a t

taken on

OMk-12 s

and t h e new polygon i s i n s i d e t h e same c i r c l e

I used t h e comma i n 1. 9, [previous page]

does not a c t u a l l y cheat.

28

t o mislead; it

EXAMPLES

a s t h e o l d one.

Since

again i. n f i n i t e length.

M' M' t h e new polygon has n-1 n an And a s before L f a i l s t o catch M.*

L i t t l e can be added

- t h e exposition i s eminently c l e a r

and concise (even though t h e discussion of convergence of S(a-r$dr

=

1

-2(a-r)f,

and t h e l a s t footnote, may leave t h e

reader s l i g h t l y uncomfortable). For an account of subsequent developaents, and f u r t h e r references, s e e J. Flynn:

Pursuit i n t h e c i r c l e : l i o n versus man, pp. 99-124 i n D i f f e r e n t i a l Games and Control Theory (ed. E. 0. Roxin, P. T. Liu, R. L. Sternberg), Dekker, New York, 1974.

Games s i m i l a r t o t h i s w i l l not be t r e a t e d i n t h i s book. It i s s i n g u l a r i n t h a t t h e p l a y e r s ' c a p a b i l i t i e s a r e perfectly balanced; t h e confinement of play t o t h e d i s c induces con-

s t r a i n t s on t h e s t a t e v a r i a b l e j and it i s p r e c i s e l y t h e i n t e r play of t h e s e two e f f e c t s t h a t makes t h e game i n t e r e s t i n g . The next game, Obstacle Tag, i s due t o Isaecs.

The two

players move, with simple motion, i n Euclidean n-space, n e i t h e r being allowed t o e n t e r t h e ' o b s t a c l e ' , an open subset G

of

R n j equivalently, t h e s t a t e v a r i a b l e s of both players

a r e constrained t o l i e i n t h e closed set

RqG.

This g e n e r a l i s e s s e v e r a l of t h e games already mentioned:

simple pursuit, G = fl; Kelley's game, G = I s \ S 2 j and Lion and 2 Man, G = {x E R : 1x1 > 1). We s h a l l only t r e a t a very s p e c i a l case: n = 2, G i s 2 t h e i n t e r i o r of t h e u n i t d i s c i n R , and a > p holds f o r t h e pursuer's and quarry's speed bounds. f o r , e.g.,

The s t a t e c o n s t r a i n t

t h e pursuer can be t r e a t e d by a somewhat clumsy

equation of motion:

29

PURSUIT GAMES

11

x = u,

i f 1x1

and

x'u2

>

1

0 i f 1x1 = 1.

> p) is

A t h r e e s t a g e procedure f o r capture ( i n our case a

The pursuer f i r s t c a r r i e s out simple pur-

e a s i l y described.

s u i t , u n t i l t h e f i r s t t i m e t h a t he meets t h e obstacle; then he moves a s i n simple p u r s u i t on t h e c i r c l e ( s e e Exercise 2 i n 1.4) w i t h

y/lyl

a s n o t i o n a l opponent; f i n a l l y , he proceeds

a s i n simple p u r s u i t again. Any one of t h e s e s t a g e s may be t r i v i a l , of time duration 0.

It i s r e a d i l y v e r i f i e d t h a t , from any i n i t i a l positions,

each p u r s u i t s t a g e terminates a t a f i n i t e t i m e , whatever t h e a c t i o n of t h e quarry.

Rather obviously t h e s t r a t e g y i s not

time optimal, t h e f i r s t two s t a g e s being maintained f a r t o o long. A paradoxial game, i n

:=

R I , has dynamical equation

(P-q)

2

1

-8

with p l a y e r s ' c o n t r o l s constrained by

3

0

5

p,q s 1; t h e pur-

s u e r wishes t o a t t a i n (and t h e quarry, t o avoid) x an i n i t i a l p o s i t i o n

xo

>

5

0

from

0, e . g . , xo = 1. I f pursuer chooses

p = q, then

t h e s t a t e v a r i a b l e moves towards t h e o r i g i n , with t h e game ending a t t i m e

t

=

8. Apparently t h i s i s t h e b e s t choice

f o r t h e pursuer, i n terms of minimising t h e termination time. Second, suppose t h a t t h e quarry chooses h i s c o n t r o l

q

thus : q = 1whenever p s Then (p-q)

2

1

,q

=

0 if p

r 1/4, b e s t quarry's estimate, so t h a t

1 >2

$

2

1

8

,

EXAMPLES

t 8

x ( t ) z 1+

:

t h e s t a t e variable constantly mwes away from

t h e origin. What happens i f both players simultaneously make t h e indicated choices? The paradox is not i n t h i s , since such incompatible s t r a t e g i e s can be s e t up i n almost a l l games (quarry counters pursuer's choice

p = q by

q = -p), but i n

t h a t t h e s t r a t e g i e s a r e optimal.

A

Fig. 1 Obstacle Tag. Three stages i n t h e capture of quarry by pursuer avoiding a c i r c u l a r obstacle. 31

PURSUIT GAMES

Exercise,

Prwe t h a t t h e obvious analogue of Exercise 3 i n 1 . 2 provides a s o l u t i o n t o o b s t a c l e t a g w i t h minimal time t o capture. Remarks The p r e s e n t author has t h e impression t h a t t h e paradoxic a l game i s due t o L. D. Berkovitz, but has been unable t o t r a c e t h e reference.

The o b s t a c l e t a g game appears on p. 152

of R. I s s a c s :

D i f f e r e n t i a l Games, Wiely, New York, e t c . ,

1967 *

PROBLEM Generalise 1.2, 1.4 and t h e Exercise above, and solve t h e simple p u r s u i t problem on n-dimensional Riemannian manifolds with boundary. (The appropriate form of t h e equation of motion i s

= u,

u'G(x)u defining t h e d i s t a n c e . )

g

2

a.

32

, where

G(*)

is t h e matrix

CHAPTER I1 BASIC CONCEPrS This i s a very dry chapter:

it introduces, and coments

upon, t h e concepts subsequently used:

player controls, s t r a t -

Other than t h i s , t h e r e i s some

egies, and winning p o s i t i o n s .

minor pedagogical content t o a number of t h e exercises; and mathematicial, i n t h e two propositions from 2.3 needed t o s t a r t o f f Chapters IV, V, and VII.

-- t h e s e a r e

The standard

assumptions introduced i n 2 . 1 a r e not r e a l l y used u n t i l nonl i n e a r games a r e t r e a t e d i n Chapter VII.

Thus a cursory f i r s t

reading might be confined t o t h e main text of 2.2, returning when and a s need i s f e l t . 2.1

D i f f e r e n t i a l Equations:

Miscellany

With minor exceptions, we s h a l l be t r e a t i n g a game whose

s t a t e space i s Euclidean, -

say Rn

f o r some n; with dynamical

equation of t h e form

players' control constraints

t h e data being pursuer's c o n s t r a i n t set s t r a i n t set tion

Qi and a t a r g e t s e t

R

P, and quarry's con-

( o r termination condi-

x E a). Usually t h e above w i l l be abbreviated, a s i n

end:

x

E 33

Rj

s t a t e space:

RL'.

PURSUIT GAMES

I n (2) t h e data a r e t,q

are

dumnly

so

a, and n; t h e symbols x,p,

variables ( i . e . , bound variables, j u s t as t h e and

'3dS)j

in

f , P, Q,

is is

a convention.

The data c o n s t i t u t e

a p a r t i a l description of t h e game; i t s s p e c i f i c a t i o n w i l l be completed implicitly, by studying only winning s t r a t e g i e s , The i n i t i a l point i s not p a r t of t h e data:

etc.

t h e same

game is played from various i n i t i a l positions. U n t i l t h e i n t e r p l a y between t h e two players' controls p,q

becomes e s s e n t i a l , one could w e l l t r e a t t h e p a i r as a

s i n g l e e n t i t y u = (p,q), a control u with values i n a given subset of some Euclidean space. The following w i l l be referred t o as standard aeeump-

tions : -

ASSUMPTIONS

The players' constraint s e t s , P Ra ; t h e

a r e nonvoid and compact subsets of a closed subset of

Rn.

The mapping

and

target s e t

f : R n+2a

+ R"

Q, R

is

i s con-

First, a

t i n u m a , and s a t i s f i e s two f u r t h e r requirements.

l o c a l Lipschitz condition, uniformly i n t h e control variables: every point c

> 0,

xo E Rn

has a neighbourhood U =

{X:

IX-XoI

4

such t h a t If(X,P,d

f o r some X

E

R1

and a l l

-

f(y,p,q)I

x E U, y E U, p

XlX-Yl

E

P, and q

E Q.

Second, t h e growth condition limSup sup IXl+ P,q

- = m i n 1x12/3). ~

The t h r u s t of our approach may be i l l u s t r a t e d on t h e 'decoupling' of t h e analogous problem i n l i n e a r control theory.

For t h e control syatem

optimal control

t

I+ u ( t ) ,

f

= Ax

-u

one f i r s t finds an

and only then, a s a separate ques-

tion, does one attempt t o synthesise t h i s by a feedback device

4 = Ax

F: x n u

such t h a t optimal t r a j e c t o r i e s s a t i s f y

- F(x) (i.e.,

F ( x ( t ) ) = u ( t ) optimal).

The object

studied i s then not ( 9 ) but

G(t and d i s c o n t i n u i t i e s i n

t

= f(x(t),p(t),q(t) ) j

p(t)

no longer pose any concept-

u a l problem. The over-all e f f e c t i s t h a t t h e question of feedback s t r a t e g i e s is avoided.

I f t h e game is such t h a t one of

our

r e c i p r o c i t y theorems applies (see 3.1, 3.3, 3.7, 5.4) then t h e feedback problem i s s h i f t e d i n t o control theory, where, probably, it properly belongs. Another question which we s h a l l avoid i s t h a t suggested by t h e paradox i n 1.7:

t h e e f f e c t of both players using

s t r a t e g i e s simultaneously.

Quite reasonable resolutions of

t h i s have been proposed (e.g., 45

a r t i f i c a l time lags

6

are

PURSUIT GAMES

introduced a t appropriate places i n t h e information flow diagram, and subsequently one takes 6 + O+). The a t t i t u d e adopted i n t h i s book is that, however t h i s is resolved, the quarry chooses a control t

H

q ( t ) , which the pursuer count-

ers, systematically, via a strategy. That t h e so posed prablem i s interesting is shown by the application t o control under unpredictable perturbations, and t h e 'unorthodox l i n e r i s a t i o n ' of 3.2; there the second player i s purely notional nature i n t h e first case, and s t u p i d i t y i n t h e second. This d e f i n i t e l y prefers t h e pursuer t o t h e quarry. O f course, t h e quarry's winning positions are precisely those which a r e I & winning f o r the pursuer, so t h a t a determinat i o n of t h e l a t t e r fixes the former. However, the specificet i o n of a successf'ul quarry strategy involves a construction

-

f o r a l l times t 2 0: often f a r more d i f f i c u l t than t h a t of a pursuer strategy, over a bounded time i n t e r v a l . Entering t h e plea of greater simplicity, we s h a l l t r e a t , almost exclusively, the pursuer's problem; our position i s weakened by legitimate instances of i n t e r e s t i n g evader's problems, as i n c o l l i s i o n avoidance between a i r c r a f t treated i n T . L. Vincent, E. M. Cliff, W. J. Grantham, W. Y. Peng: A problem of c o l l i s i o n avoidance, Engineering Experiment Station Report No. 39, The Univers i t y of Arizona, 19'72. A s i t u a t i o n i n which the roles of t h e two players are

symmetric might be indicated by t h e following. I n addition t o t h e dynamicel equation and control constraint s e t s , l e t there be given three s e t s i n s t a t e space: pursuer's t a r g e t R, quarry's ' l i f e - l i n e ' , and s t a t e constraint r . The pursuer desires t o reach R a t f i n i t e time while remaining

c

i n r \ c a t a l l previous times; the quarry wishes t o reach c,remaining i n r\R i n t h e meanwhile. A t present t h i s

46

BASIC CONCEPTS

One complication i s t h a t , i n

problem seem8 t o o d i f f i c u l t . i n t e r e s t i n g cases, a l l of

r / c , r\n

c,r

a,

a r e closed, whereupon

a r e not ( t r i v i a l i t i e s excepted); and, b a r r i n g some i n c i s i v e new approach, even t h e s i m p l i f i e d problem of a or even c o n t r o l system with (pointwise) s t a t e congame s t r a i n t s A i s unrewarding unless A is, a t l e a s t , closed.

-

2.3

-

P r i n c i p l e of Suboptimality I n t h e l a s t s e c t i o n t h e b a s i c terminology and notation

was e s t a b l i s h e d .

It i s w i t h reluctance t h a t we introduce

f u r t h e r notation here, namely fi(t,X,P,d,

T(x,p,q),

As usual, one must s t r i k e

T(X,U),

Tb).

some balance between cumbersome (and,

often, i n a c c u r a t e ) d e s c r i p t i v e phrases, and a r i s i n g t i d e of

symbols o r abbreviations t h a t t h e reader deciphers with i n creasing impatience. The game i s indicated by ~ ( t E) PI q ( t ) E Qj x E R; s t a t e space: Rn.

= f(x,p,d;

(1)

end:

x E Rn

Under t h e standard assumptions, an i n i t i a l p o s i t i o n and player c o n t r o l s

p ( - ) , q ( * ) determine uniquely a (gen-

+

e r a l i s e d ) s o l u t i o n t o t h e d i f f e r e n t i a l equation on R ; l e t US denote its value a t n(O,x,p,q)

t

0 by

2

fi(t,x,p,q).

E.g.,

always

= x i and, i f t h e dynamical equation i s l i n e a r , t h e

variation-of-constants formula provides t h e e x p l i c i t s i o n (6) of 2 . 1 for

expres-

YC(*). n and player c o n t r o l s p ( * ) , Given ( l ) , a p o s i t i o n x E R q(.), define t h e f i r s t termination t .

,

T(x,p,q) = i n f i t

2 0:

n(t,xdbq)

E 01.

The set involved may be empty, and then, of coursej i t s infimum i s

+m.

Since

R

i s closed and t h e s o l u t i o n 47

PURSUIT GAMES

t w n(t,x,p,q)

continuous, t h e infimum ( i f f i n i t e ) i s a c t u a l l y

a t t a i n e d : thus

E

fi(e,X,P,d fi(t,x,p,q)

4R

for

e o

= T(x,p,q),

s t s e.

and (pursuer) s t r a t e g y

x E Rn

Given (l), a p o s i t i o n

one has t h e termination times controls

R for

T(x,a[q],q)

0,

f o r various quarry

q ( * ); t h e 'worst' of t h e s e i s T(x#) =

F i n a l l y , w e define t h e minimum

SUP

9('

time

T(x,C"qI,q).

1

T(x) = i n f T(x,a), U

infimum taken over a l l s t r a t e g i e s

0.

Thus, i f the intermed-

i a t e concepts a r e eliminated,

T(x) = i n f sup i n f { t : n(t,x,a[q u 4') As mentioned above, t h e innermost infimum i s a minimum; so i s t h e outermost, under reasonable conditions (see 8.1)j no comment on t h e middle supremum.

It i s almost immediate t h a t

T(x)

is f i n i t e iff

x

is

a winning p o s i t i o n (see Exercise 1); thus

x)., T(x) i s a real-valued function on t h e s e t of a l l winning p o s i t i o n s .

In

analogy with c o n t r o l theory, T ( * ) w i l l be c a l l e d t h e minimal

time function; it makes a convenient reference concept, but seldom i s it a v a i l a b l e computationally. PROPOSITION 1

q

a quarry control.

s T (x,o[ql,q),

x be a position, u

For any f i n i t e time for

a s t r a t e g y , end

t

t h e s t a t e response endpoint

belongs t o W(0,e-t) (2)

Let

with

8 = T(x,a), so t h a t

T(y) s T(x,o)

48

- t.

0c t


,t) = u(t) q(t) = f(x(t))

- Ax(t),

where

+

q(t)

u(*) i s a s described i n

t h e F i r s t Reciprocity Theorem. (Proof) dent of X(*)j

from (131,

0

n e u t r a l i s e s , so t h a t

controls.

x(.)

Choose a constant i n

t h e n t a k e t h e admissible quarry c o n t r o l 75

Q

9,

i s indepent o obtain

PURSUIT GAMES

q(t)

f(x(t))

=

-

Ax(t), t o conclude

k

= AX =

- a ( * ) + q = AX - p + ( f ( x ) - Ax) -p

f(x)

almost everywhere. Thus indeed p s t e e r s t h e t r a j e c t o r y from X(O) = x0 t o x ( e ) = 0. The conditions on 0 (and Q) can be eliminated e n t i r e l y

w

by the following formulation. x ( * ) t o Q i n (ll), l e t t h e corresponding image:

Retaining t h e s t a t e response

0 be i t s t r a j e c t o r y a r c , and V

0 = {X(t): 0

5

t

el,

5

- AX:

V = {f(x)

However, t h e o r i g i n a l v e r s i o n i s preferable.

x E 03.

T h u s , for t h e

1

c o n t r o l l e d pendulum and i n i t i a l point ( ~ ~ it0 i)s obvious t h a t 1 e ( t ) decreases from F t o 0 monotonously (invariance), whereupon [sin e

- aeJ s

( t h e estimate) f o r t h e values of

lsin

1

-

1

u

used there.

a

From t h e F i r s t Reciprocity Theorem, Q ought t o be r a t h e r small t o make be small:

P

-Q Y

large; hence, i n (l), f ( x )

- Ax

should

linearisation.

Exercises I n t h e f i r s t t h r e e of t h e s e we examine t h e consequences of an 'ordinary' l i n e a r i s a t i o n of ( 1 ) . Assume t h a t one i n s t a l l s a device which implements t h e feedback c o n t r o l appropriate t o

..e + e =

(14 1

7;

-I< T

5

1

1. Obtain t h e switching locus f o r (14).

wer:

semicircles with radius 1 . ) 2.

( P a r t i a l ans-

1

For t h e i n i t i a l p o i n t s ( ~ , 0 )and (1,O) f i n d t h e

switch on t h e locus from Exercise 1. (Hint: 76

For o r i e n t a t i o n ,

STROBOSCOPIC AND ISOCHRONOUS CAPTURE

t h e lowest corners i n f i g . 4 have distance, from t h e c e n t r e (l,O), approximately 1.00039 and 1.01135.)

3.

Prove t h a t each t r a j e c t o r y near t h e o r i g i n ( o t h e r than on t h e curve (4)) has an end-point, on t h e locus from Exercise 1, beyond which it cannot be continued. Exercises 4 t o 7 t r e a t t h e controlled van der Pol equation

;; +

(15)

2c(x2

-

1);

+x

= u, -1 < u ( t ) < 1.

Note t h a t here t h e uncontroll'ed motion, corresponding t o u

=

0, has an unstable focus a t t h e o r i g i n .

4. There a r e two standard procedures f o r bringing (15) t h e phase plane method, beginning

t o a f i r s t - o r d e r system: with

= y, and t h a t i n t h e Lignard plane, using t h e energy

i n t e g r a l , x + 20

($ -

x ) = y.

Comment on t h e advantages of

t h e s e i n respect t o unorthodox l i n e a r i s a t i o n . The switching locus of (15) i s not obtained by ele-

5.

mentary i n t e g r a t i o n .

Obtain q u a l i t a t i v e information about i t s

d i s p o s i t i o n near t h e o r i g i n . (Answer: it enters the o r i g i n from t h e second and f o u r t h quadrants, u l t i m a t e l y tangent t o t h e y-axis.) 6. Apply unorthodox l i n e a r i s a t i o n t o (15) and show t h a t an e n t i r e neighbourhocd of the o r i g i n can be s t e e r e d t o t h e origin.

(Hint:

f o r smell t 2

x IYI

I;

7.

>

t h e associated l i n e a r c o n t r o l system i s

..x - 2 e i + x

Jw(t)l L 1

= W,

0, i t s reachable sets a r e e n t i r e l y within

6.) Obtain s p e c i f i c estimates f o r Exercise 4. e-Atbdt

with 7 =

- 6;

&?Estimate

from t h e o r i g i n of

2

x y =

=

e"'

(

(Hints:

c s i n v t + qcosvt sinqt

)

R(t)

and a l s o t h e l e a s t d i s t a n c e

2

Find an upper bound on t

6. 77

in

PURSUIT GAMES

terms of

~ , 6 , and then vary

over (O,l).)

6

Remarks Unorthodox l i n e a r i s a t i o n was described i n 0. Hgjek:

Pursuit games: a survey, pp. 281-291 i n The Theory and Application of D i f f e r e n t i a l Games (ed., J. D Grote), Reidel, Dordrecht and Boston, 1975.

.

and i t s proof s t r o n g l y depend on t h e

The

t a r g e t being t h e o r i g i n (so t h a t

u

completely n e u t r a l i s e s

q). Nevertheless it would be highly d e s i r a b l e t o extend i t s scope t o termination conditions of t h e form M x = 0, a s t r e a t ed by t h e Second Reciprocity Theorem; and possibly, t o gene r a l capture.

k

= Ax

-a

For feedback c o n t r o l s t h e r e i s no problem:

+ q one simply t a k e s

q = f(x)

in

- Ax.

3.3 Affine Targets This s e c t i o n t r e a t s t h e game

&

(1) in

Rn,

= Ax

- p + q;

E P,

p(t)

q ( t ) E Q; end: x

E

R

with t h e t a r g e t set an a f f i n e manifold R = [x:

(2)

described by an (m,n) matrix

M

M x = Mb]

and a point

b

E

Rn.

There w i l l appear an obvious analogy with t h e Reciprocity Theorem of 3.1 (which is, i n f a c t , t h e s p e c i a l case b = 0).

M = I,

To unburden t h e formulation of t h e result below, t h e

needed apparatus i s introduced f i r s t . be t h e n u l l space of M, so t h a t system associated with (l), (3 1

jr

= -Ay

Let

R = b + N.

N = [x:

Define a c o n t r o l

- v; v ( t ) E vt

with c o n s t r a i n t sets Vt = ( P

78

+ e-AtN)

Mx = O}

Q.

STROBOSCOPIC AND ISOCHRONOUS CAPTURE

SECOND RECIPROCITY THEOF@M Assume t h a t t h e c o n s t r a i n t set

i s compact.

P

Then an i n i t i a l p o s i t i o n

forced t o t h e a f f i n e manifold onously a t t i m e

can be

x 6 Rn

$ sl t r o b o s c o p i c a l l y and isochr-

w i t h i n t h e game (1) i f , and only i f ,

8

x € e-A8(R(8) + a),

(4 1 where

i s t h e reachable s e t a t

R(8)

of t h e a s s o c i a t e d

8

c o n t r o l system (3). Furthermore, u(q,t) e v(e

(5) (for a l l

q

strategy

u

E

Q, t

E

- t)

+ q modulo e

-A( 8-t

)m

[ 0 , 8 ] ) can be used t o determine a winning

from an admissible c o n t r o l f o r (3) ( s e e ( 4 ) ) and

vice versa. (Proof)

This i s an expansion of t h a t of t h e F i r s t Reciprocity

Theorem.

F i r s t , assume t h a t a stroboscopic s t r a t e g y

t e r m i n a t i o n a t time

8,

e - J;-At(o(q(t),t) r e f e r e n c e p o i n t so E

-

MeA e ( x Choose a f i x e d u ( t ) = u(%,t)

- %.

u

forces

q ( t ) ) d t ) = Mb.

Q, and d e f i n e u by

Take a r b i t r a r i l y a p o i n t

q

E Q and

time

s E [0,8]j consider t h e admissible quarry c o n t r o l u for s = 0 with v a l u e q i n [O,s] and v a l u e 90 i n ( s , e ] : we o b t a i n A8

(61

Me

and f o r g e n e r a l

MeA8 (x

-

(x

- Joe - A t u ( t ) d t )

s, af'ter rearranging,

e J:-Atu(t)dt

+ J Se'At(u(t)

= Mb.

- o ( q , t ) ) d t ) = Mb.

+ q

0

Then S

-

-At MeA8J e (u(t) + q u(q,t))dt = 0 0 by subtraction; and, a s 8 i s f r e e t o vary wer [ 0 , 8 ] ,

ReA(e-t)(u(t) + q 79

- u(q,t))

=

o

(almost a l l t ) .

PURSUIT GAMES

U s e t h e n u l l space t o r e i n t e r p r e t t h i s a s

(71

E

u(t) + q

a ( q , t ) + e-A(e-t)N.

Now, u

has values i n P; using t h e Lemma from 3.1 and t h e Pontrjagin difference, we f i n d t h a t v ( t ) = u ( e - t ) s a t i s f i e s v(t)

E

(P

+ e‘AtN) il. Q

and i s i n t e g r a b l e t o g e t h e r with

Thus

u.

v

= Vt

a.e.

is an admissible

c o n t r o l f o r (3), and (6) y i e l d s MeA8x

- MJ:eAsv(s)ds

= Mb,

(4) holds; f i n a l l y , (7) provides (5). Conversely, assume t h a t ( 4 ) holds, and v is a c o n t r o l appropriate t o t h e p o i n t i n R(8). S e t t i n g u ( t ) = v ( 8 - t ) one obtains (6); and v ( t ) E Vt y i e l d s

so t h a t

u(t) To construct t h e s t r a t e g y u

+

q

E P + e-A(e’t)N.

apply Filippov’s Lemma (3.9);

t h e r e result measurability preserving mappings 1 u : Q X R -+ P, w : Q ~ R ’ + N such t h a t u ( t > + q = a ( q , t ) + e-A(e-t)V(q,t).

i s a winning s t r a t e g y . -A(@-t) we have u q = e V J so

It now only remains t o v e r i f y t h a t For any quarry c o n t r o l

q(0 )

that

u

-

U

A0

= Me

(x

-

w

=Mb+o.

The theorem, though simple enough i n p r i n c i p l e , has q u i t e complicated d e t a i l s .

It seems appropriate t o i l l u s t r a t e i t s

workings on an elementary example. 80

STROBOSCOPIC AND ISOCHRONOUS CAPTURE

W e t r e a t t h e game

Example

= y

in

- p,

$

= q j end: x = y

0

Rc, with -0th control values i n [ - 1 J 1 ] . This resetn-3s

t h e reduced description of t h e one-dimensional rocket chase i n

1.3 except f o r t h e t a r g e t s e t . A sketch of t h e players' vectograms f o r several points i n

R2

suggests t h a t points above t h e t a r g e t

forced upward:

x = y

can be

possibly away from t h e t a r g e t i n i t i a l l y , but

ultimately toward it; and symmetrically below t h e t a r g e t . Again t h i s i s reminiscent of rocket chase; but t h e resemblance ends there, see Exercise 3. We have A

=( 0" i),

M = (l,-l)J b = 0; eAt =

(: 1).

The constraint s e t of t h e associated control system i s determined via Exercise 1: MeAt( so t h a t

Vt

;)=

+ (t-l)y,

x

consists of a l l points whose coordinates

vk

satisfy v for a l l

1

+ (t-l)v2 + ( t - l ) q

q, 191 s 1, i.e., Jvl + (t-l)v21

E.g*

= p

5;

1

E

[-1,11

- It-11.

, v1 = v2 f o r t = 0, (vl-v2/21 < 1/2 f o r t = 1/2, (vll s 1 f o r t = 1, Ivl+v2/21

s 1/2 f o r t

= 3/2,

vl= -v2 f o r t = 2, and

Vt =

for

t > 2.

Subsequent computations a r e simpli-

f i e d i f we apply Exercise 2 (with

81

90 =

0 ) , and use only

PURSUIT GAMES

P

These c o n s t r a i n t sets a r e segments on t h e x-axis,

n Vt.

symmetric about t h e origin; denoting t h e r i g h t endpoints by p ( t ) , we have p(t) = t for

os t

f o r 1 4 t s 2.

s 1, p ( t ) = 2-t

The associated c o n t r o l system may t h e n be w r i t t e n a s

G

= -y

The reachable set

e Thus

R(8)

endpoint

-

i=

J w ( t ) l s 1.

0;

c o n s i s t s of a l l points

R(B)

eAtp(t)

p(t)w,

8

(wf))

dt =

(Jo

i s a symmetric segment on t h e y-axis, with r i g h t y (t )

,

for 0 4 e s 1 for 1 s e s 2 ,

v ( t ) = {:::1-2)2/2 (obviously t h e extremal c o n t r o l s w

a r e constants

2

1). The

sets of winning p o s i t i o n s from (4) are, using (8) and Exercise 1, determined by

IX

(9)

v(e).

+ (e-i)yJ 4

These a r e p a r a l l e l s t r i p s , based on t h e segments x = y for

e

= 0, J x

- y/2(

4

1

for

e

R(8):

= 1/2

1x1 5 $ f o r 8 = 1, Ix + y/21 s 7/8 f o r e J x + yI s 1 f o r e = 2.

= 3/2

Exercises Vt

1. Show t h a t another d e s c r i p t i o n of t h e c o n s t r a i n t set v E Vt i f f At Me v E (MeAtP) f (MeAtQ);

i n t h e theorem i s t h a t

and t h a t (4) i s equivalent t o

MeAex E MR(8) + Mb.

82

STROBOSCOPIC AND ISOCHRONOUS CAPTURE

2.

p if

The c o n s t r a i n t sets

need not be compact even i f

Vt

i s such. Verify t h a t both p a r t s of t h e proof c a r r y through Vt i s replaced by t h e compact set

n

(P-%) here

90

i s any fixed point i n

((P+e'AtN) 5 Q);

Q.

around (7), con-

(Hint:

s i d e r t h e s e t of values of

3.

u.) I n t h e example determine t h e set of a l l winning

points, and decide whether it i s open or closed.

(Hint:

t h e enveloping curve t o t h e p a r a l l e l s t r i p s i n (9). answer:

find

Partial

t h e boundary of t h e set of

t h i s comprises f o u r a r c s :

winning p o s i t i o n s c o n s i s t s of two curves, each being two half rays connected by a parabolic a r c . )

4.

Assume t h a t t h e sets

pact, convex, and s y m e t r i c . which can be forced t o

P,Q

i n (1)a r e nonvoid, com-

Prove t h a t t h e set of p o s i t i o n s

stroboscopically a t t i m e

R

closed, convex, and symmetric about

e'Aeb.

8

is

(Hint : you w i l l

need Exercise 2 . )

5.

Obtain a version of t h e r e c i p r o c i t y theorem more

c l o s e l y r e l a t e d t o t h a t of 3.1 by t r e a t i n g t h e c o n t r o l system

5

= Ax

- u;

u(t) E

u p ,

U,(t) = (pce'A(e't)N)

Q;

note t h a t t h i s depends on t h e termination t i m e t h e r parameter.

(Hint:

r a t h e r than

P a r t i a l answer:

v.

a s a fur-

i n t h e proof of t h e theorem, use

t i o n on a winning p o s i t i o n i s

6.

8

u

t h e c o n t r o l - t h e o r e t i c condi-A8

x E Re(t) + e R.) Make appropriate simplications i n t h e preceding f o r

t h e case t h a t t h e n u l l space

N

i s i n v a r i a n t under A.

7 . I n t h e game (1)assume t h a t t h e pursuer i s allowed t o choose a l i n e a r feedback component i n h i s controls, replacing and

p by u(t)

Fx + u

E U c span

with, however, both values P.

Show t h a t t h e matrix

Fx E span P F

can be

PURSUIT GAMES

chosen so a s t o make

N = {x:

new c o e f f i c i e n t matrix

I n 3.2 we t r e a t e d

8.

F, i f f

i n v a r i a n t , under t h e

ANc N

+ span P.

a game induced by t h e one-dimen-

..x + x = u

s i o n a l equation

k

-

A

M x = O}

- v.

Show t h a t n e i t h e r

x = 0

nor

y i e l d s an i n v a r i a n t t a r g e t ; apply t h e condition from

= 0

t h e preceding exercise; i f possible, design an appropriate feedback s t r a t e g y component.

(Answer:

F

0 0 =(-1 a ) .)

Generalise t h e r e c i p r o c i t y theorem t o allonomous

9.

games, i n t h e form t r e a t e d i n 3.1, Exercise 8, and with termination condition

M x = Mb.

( P a r t i a l answer:

i n t h e approach

approach of Exercise 6, t h e c o n s t r a i n t set f o r t h e associated c o n t r o l system i s

where

X(.)

i s t h e fundamental matrix s o l u t i o n . )

Remarks That t h e present formulation of t h e Second Reciprocity Theorem i s p r e f e r a b l e t o t h e more involved one of Exercise 5,

i s due t o E. Rechtschaffen (personal communication). There a r e s e v e r a l d i s s i m i l a r i t i e s with t h e F i r s t Reciproc i t y Theorem.

F i r s t , t h e associated c o n t r o l system i s not

autonomous, s i n c e t h e c o n s t r a i n t s e t s It w i l l be autonomous i f

A

Vt

may depend on time.

i s a s c a l a r matrix (e.g.,

simple motion); or, more generally, i f t h e n u l l space i n v a r i a n t under for a l l

A.

Indeed, then

A N c N, so t h a t

A = 0, N

e-AtN

is =

t, and

=V=(P+N)&Q t ( s e e Exercises 6-8, and a l s o t h e e x e r c i s e i n 7.3 f o r f u r t h e r V

developments). Second, it i s p o s s i b l e t o have

Vt

nonvoid f o r some t

and empty f o r others, a s a c t u a l l y happened I n t h e example.

84

N

STROBOSCOPIC AND ISOCHRONOUS CAPTURE

This w i l l be examined i n t h e next s e c t i o n . A less spectacular phenomenon i s t h a t even t h e dimension of t h e epan of may Vt

change discontinuously:

i n t h e i l l u s t r a t i o n , t h e dimensions

a r e , i n t u r n , 0, 1, 0, -1. Finally, t h e associated c o n t r o l system w i l l steer one point t o another, b u t , notwithstanding t h e r e c i p r o c i t y , a winning s t r a t e g y does not f o r c e t h e i n i t i a l p o s i t i o n t o a

it does f o r c e t o t h e t a r g e t set, b u t not t o

terminal p o i n t :

a s i n g l e point i n t h e t a r g e t . sence of t h e term w control.

in

This may be t r a c e d t o t h e pre-

(5), which may vary with t h e quarry

More f o r c i b l y , however, t h e Necessary Condition from

3 . 1 f o r f o r c i n g t o a s i n g l e t o n w i l l u s u a l l y f a i l i n t h e games treated

3.4

.

Necessary Conditions

W e refer again t o t h e game

4

(1)

-P

+

q; P ( t > E PI q ( t ) E Q

Rn, and t a r g e t set

with s t a t e space R = {x:

= Ax

ll

an a f f i n e manifold

Mx = Mb].

I n (1) l e t

PROWITION

P be compact.

A necessary and

s u f f i c i e n t condition f o r presence of i n i t i a l p o s i t i o n s which can be forced t o t h e t a r g e t s t r o b o s c o p i a l l y a t time

8

is

that (2)

(MeAtP)

(MeAtQ)

4 fi whenever

0

s t s

0;

and a necessary condition i s t h a t

( 31 I n case

At

Me

At (Q-Q) c Me (P-P) whenever 0 s t s 8.

P i s compact and convex, and both

P,Q a r e symme-

t r i c , (2) may be replaced by

(4 1

At At Me Q c Me P whenever 0 s t s 8.

(Proof) According t o t h e Second Reciprocity Theorem of 3.3,

PURSUIT GAMES

such positions e x i s t i f f

R ( B ) =/

fl,

R(8) =

i.e., -At

4 $.

e Vtdt 0 By Filippov's Lemma, t h i s i s equivalent t o all

4 fl

Vt

f o r almost

t E [ O , e ] , and thence, f o r a l l t E [0,01. ( 2 ) is now a reformulation of t h i s (see Exercise 1 i n

3 . 3 ) , and t h e remaining assertions are then e a s i l y obtained (see 3 .l, Necessary Condition and Corollary 1). QE ,D COROLLARY 1 I n (l), i f t h e r e e x i s t

positions which can

e,

be forced t o t a r g e t stroboscopically a t one time t h e r e a l s o e x i s t such positions a t a l l times

then

t E [O,e].

The proposition involves t h e flrndamental matrix solution eAt, so t h a t t h e conditions are usually not d i r e c t l y v e r i f i able; however, they are a source of working c r i t e r i a . COROLLARY 2

M(Q-Q)

c M(P-P) i s necessary, and ( i f Q

M Q c I n t MP

i s compact)

s u f f i c i e n t , f o r presence of poei-

t i o n s which can be forced t o t h e t a r g e t stroboscopically a t s t r i c t l y positive times. (Proof) The f i r s t inclusion is t h e case

t

MQ, there i s a

t h e second holds, then, from compactness of ball

t B

H

B

eAt

about t h e o r i g i n such t h a t

i s continuous with value

+ MeA t Q cMeAtP

1 ball

B1.

Then

B + M Q c MP. I

for small t > 0

at

Since

0, we w i l l have

and a possibly smaller

(MeAtP) 2 (MeAtQ) z,B1 =/

so t h a t ( 2 ) holds.

of ( 3 ) . I f

= 0

fl

for small t > 0,

&ED

A more precise r e s u l t i s presented below.

The player

control orders are defined i n 6.1; under our assumptions, pursuer has control order

k if

consisting of t h e zero vector, j = 0,. ..,k-2

but not f o r

NECESSARY CONDITION

MA'(P-P)

not t h e

j = k

- 1.

= 0

( t h e set

empty s e t ) holds f o r

I n (1)assume t h a t

P i s compact.

A necessary condition f o r presence of points which can be

forced t o termination stroboscopically a t a s t r i c t l y positive 86

STROBOSCOPIC AND ISOCHRONOUS CAPTURE

t i m e is that

(5)

quarry c o n t r o l order

and, i f

k

2

pursuer c o n t r o l order,

denotes t h e l a t t e r , t h a t

(6 1

MA~-’(Q-Q)c MA

k-1 (P-P).

eAt i n t o a power series. If k denotes t h e pursuer c o n t r o l order, then, f o r t i n some (Proof)

I n ( 3 ) expand

i n t e r v a l (0,e)

e>

with

0.

MeAt(Q-Q) c MeAt(P-P) = O ( t k-1 ); hence, by induction on

j = 0

y i e l d s ( 5 ) , and a l s o (6) f o r

,...,k-2, MAj(Q-Q) = 0. j = k - 1. &ED

This

3 Let P,Q be compact, convex, s y m t r i c . If ( 5 ) holds and (6) is replaced by COROLLARY

(7)

M A ~ - ’ Q ~ I n t MA

k- 1 P,

e > 0, t h e set of p o s i t i o n s

then, f o r s u f f i c i e n t l y small

which can be forced t o termination, stroboscopically a t time

e, has nonvoid i n t e r i o r . (Proof)

Now we have MAJP = 0 = MAJ&

f o r j = 0,...,k-2

by ( 5 ) and t h e assumptions on P,Q; with (7), t h i s ensures At At t h a t Me Qc Int(Me P) f o r small t > 0, again by power s e r i e s expansion.

It follows t h a t t h e s e t s

i n t e r i o r (compactness used h e r e ) . sets

Vt

have nonvoid

Hence so do t h e reachable

R(B), and, by t h e r e c i p r o c i t y theorem, t h e s e t s of

winning p o s i t i o n s a l s o . Example

Q,ED

Consider t h e game with n-dimensional equations

1+

A

i-

1

u(t)

E

A x = 2

U,

7 + B1;

U, v ( t ) E V; end:

Both t h e pursuer’s c o n s t r a i n t set 87

U

i-

B$

=

V;

x = y.

and t h e quarry‘s

V

are

PURSUIT GAMES

non-void,

compact, convex, and symmetric.

The case of A2 = 0 = B2 i s t h e l i n e a r analogue of t h e game of two c a r s ( j u s t a s rocket chase i s t h e l i n e a r version of t h e homicidal chauffeur game); a l s o s e e 8.3. The formula4n i s simple: t h e t i o n i n terms of a f i r s t - o r d e r system, i n R

,

result has

M = (I 0

-I

0).

i s always s a t i s f i e d , b u t no choice of t h e data w i l l y i e l d t h e s u f f i c i e n t condition The necessary condition of Corollary 2 (since both c o n t r o l orders a r e 2:

-i2-i17( ;)”’

thus, (5) holds).

containment r e l a t i o n (4) i s

(8)

(I,O)exp(

(I,O)exp(

o2

The

;7.

Conditions (6) and (7) a r e V c U and V c I n t U respectiven ly ( i n t e r i o r i n R ), independent of t h e c o e f f i c i e n t matrices. F’urther developments appear i n t h e e x e r c i s e s . Exercises

1. For t h e n-dimensional rocket chase, v e r i f y t h a t one of t h e necessary conditions f a i l s . 2.

I n t h e example, write

f o r t h e matrix appearing i n (12), and analogously f o r with t h e matrices

Bk.

Re-interpret

88

( 8 ) a6

D(t),

STROBOSCOPIC AND ISOCHRONOUS CAPTURE

V

c D’l(t)C(t)U

(Answer: and obtain t h e f i r s t three terms of D’k. 1 = I + D-’(t)C(t) (B1-Al)t + 1 2 + I 2 (Bf+2Al-3AlB, + 2(B2-A2))t 2 +

. ..

-

3.

I n t h e preceding s i t u a t i o n , assume t h a t

V = {x: x‘Hx

s u]. Prove t h a t (8) holds f o r small t > 0 ( i . e . , t h e condit i o n from t h e lemma i s s a t i s f i e d ) i f V c U and e i t h e r

i s negative d e f i n i t e , o r (A2

A1=

B1

- B2)‘H

and

+ H(A2

- B2)

i s negative d e f i n i t e .

4.

Specify t h e preceding conditions i n t h e case of t h e

example, with both

5.

b a l l s about t h e o r i g i n .

U,V

Generalise t h e example t o t h e following s i t u a t i o n .

The game is induced by d i f f e r e n t i a l equations

u(t)

E

U, v ( t ) E Vj end:

x = y

( a l l involving r-dimensional vectors, e t c . ).

Apply Corollary

3 , t r e a t i n g s e p a r a t e l y t h e cases n > m, n = m, n < m, and making s u i t a b l e assumptions where appropriate.

3.5

General Targets

I n t h i s s e c t i o n we examine f u r t h e r consequences of adopting stroboscopic and isochronous capture. t h e used p o r t i o n of t h e t a r g e t set

R

One is t h a t

must be f l a t (Proposi-

t i o n 1); t h e implication i s t h a t , i f t h e s u r f a c e of

R

is

w e l l rounded, a s u c c e s s f u l s t r a t e g y must f o r c e t h e s t a t e response w e l l i n t o

R.

For a game of simple capture with s e v e r a l p u r s u e r s , t h e

89

PURSUIT GAMES

t a r g e t set obviously decomposes i n t o l i n e a r subspaces.

More

generally, one might consider games whose t a r g e t set is a countable union of simpler sets.

A second result i s t h a t , i n

t h e context of stroboscop$c end isochronous capture, t h i s type of problem i s i r r e l e v a n t . The game i s

with

- P + 9i

= Ax

(1)

P(t)

E

PJ q ( t )

E Q

a s s t a t e space.

Rn

PROPOSITION 1 If t h e player c o n s t r a i n t s e t s a r e compact,

then, f o r any i n i t i a l p o s i t i o n and stroboscopic s t r a t e g y , t h e

set of a l l s t a t e responses a t each f i x e d end-time ( t o t h e various quarry choices) has convex closure. (Proof)

The s e t i n question c o n s i s t s of a l l p o i n t s At

e with

(x

- J0e-As ( f J ( s ( s > , s-) s ( s > ) d s >

varying and a l l o t h e r e n t r i e s fixed.

q( )

Apparently

it i s t h e image, under a p a r a l l e l s h i f t and nonsingular l i n e a r mapping ( t h e e f f e c t of x and eAt), of t h e set of a l l points Jie-As(o(q(s),s)

- q(s))ds.

Each such point belongs t o

t

Jo

FsJ where

FS

=

-AS

{e

(fJ(q,x)

-

E

9):

which has convex closure by Eemma 2 i n 3.10.

Q}J

Thus, t o com-

p l e t e t h e proof, it i s s u f f i c i e n t t o v e r i f y t h e converse inclusion.

t

w(s)ds

SO

By d e f i n i t i o n , each p o i n t of

with measurable

s

Fs

0

i s of t h e form

W ( S )E Fsj from F i l i p p w ’ s

Lemma, t h e r e e x i s t s a measurable w(s) =

t

6

-AS

q(s)

E

Q such t h a t

(a(q(s)Js)-q(s))

( i n t e g r a b i l i t y follows from compactness of t h e c o n s t r a i n t

STROBOSCOPIC AND ISOCHRONOUS CAPTURE

sets).

QED

I n (1)assume t h a t

COROLLARY

Q

- Qd P -

P.

If t h e t a r g e t set

then no p o s i t i o n outside

R

a r e compact, and t h a t

P,Q

contains no segments,

R

can be forced t o

stroboscopi-

R

c a l l y and isochronously. (Proof)

By assumption, t h e only convex subsets of

single points.

are

R

From t h e Necessary Condition of 3.1, no

i n i t i a l p o s i t i o n can be forced t o a point t a r g e t i n p o s i t i v e game.

QJD

PROPOSITION 2 I n game (1)assume t h a t t h e c o n s t r a i n t sets a r e compact, and t h a t t h e closed t a r g e t set R =

where each

%

is an i n t e r s e c t i o n of sets

4

k=l

which a r e

Z

a n a l y t i c i n t h e sense t h a t Z = {x: @ ( x ) = O},

(2)

with

Rn+ R

@:

@ ( x ) a n a l y t i c i n t h e coordinates of

i n i t i a l p o s i t i o n which i s forced t o

R

x

E

,

1

Rn.

Then each

stroboscopically and

isochronously i s a c t u a l l y forced t o an a f f i n e manifold e n t i r e -

4 of

l y contained i n one of t h e portions

(Proof)

R.

This w i l l follow d i r e c t l y from Proposition 1 once we

v e r i f y t h a t , whenever a convex s e t

C

i s contained i n R

4.

= US-&,

i s i n some Note t h a t t h e a f f i n e span of C c Rn is t h e a f f i n e span of a t most n + 1 p o i n t s of C. We s h a l l prove, by induction on m, t h a t t h e a f f i n e span of any m points of C i s e n t i r e l y i n some 4. The a s s e r t i o n i s true t r i v i a l l y f o r m = 0 o r 1; assume it holds f o r any m points, end consider m + 1

t h e a f f i n e span of

points

xo,

...,xm

-

C

in

C.

For a f i x e d a

E [0,11, t h e m

ax + (1 a)xo a r e i n C j by assumption, t h e r e j 1 e x i s t s an index k such t h a t , f o r any tl, tm i n R with = 1, we have points

(3 1

x t3

43 ct j (axj +

...,

(1'U)X0)

91

=

x0 + a C t j ( X j

- xo).

PURSUIT GAMES

Keeping t h e

t

many values

a but only countably many i n d i c e s

v a r i a b l e , note t h a t there a r e uncountably

j

holds f o r some fixed a l l t E R1 w i t h 3 determine t h e sets Setting a E [0,11.

C t . = 1 ) . Since t h e functions

a

J

l i m i t argument).

Q

which

eo = 1

- a,

e j = atJ we f i n d t h a t , f o r ejxj E $ f o r a l l 8 with 3 8

QE ,D

0

= 1 is

t r e a t e d by an- obvious

I n ( 1 ) assume t h a t P,Q a r e compact and If t h e t a r g e t s e t R i s a s i n Proposition 2

COROLLARY

Q

Thus ( 3 )

a r e a n a l y t i c , ( 3 ) must hold f o r a l l

t h e appropriate index k, = 1 ( t h e exceptional case

C e3

k.

and i n f i n i t e l y many a ' s (and,again,

k

- Qc/ P - P.

but contains no s t r a i g h t l i n e s (e.g., R i s compact), t h e n no p o i n t outside R can be forced t o R stroboscopically and isochronously

.

The remainder of t h i s s e c t i o n is devoted t o an elementary

The s u b j e c t

examination of a f f i n e subsets of a n a l y t i c t a r g e t s .

properly belong6 t o a n a l y t i c geometry but, because of Proposit i o n 2, it i s of considerable i n t e r e s t f o r game theory. The very s p e c i a l case of quadratic hypersurfaces i s t r e a t e d i n the exercises. Consider a set

Z c Rn

of the form (2), and any f i x e d

a E Z. Expand @ i n t o a power series centered a t a; t h i s converges absolutely, so one may group terms t o obtain

point

m

@(a + x ) where

ipk(x)

=

C @,(XI

k=l

i s a k-ary form i n t h e coordinates of

(we a r e suppressing n o t a t i o n a l dependence on a ) .

x E R"

Now, one

may always write

(4) where

yk(xl,.

variables

xi

..,%)

E

Rn.

is a symmetric k - l i n e a r form i n t h e k I n p o i n t of f a c t (omitting t h e indices

k; f o r d e t a i l s Bee t h e exercises), 92

STROBOSCOPIC AND ISOCHRONOUS CAPTURE

(5)

Y(X1

+

1 ,...,"k) ".+\ ) - C@(X1+ . . . + xi+...+\) k! i c *( ...+ xi+ . . . + j z . +J. . .+%I - ...+(-1)k- 1C + ( x , ) A

MX1+

=

A

Xl+

a

i<j

(circumflexes i n d i c a t i n g omitted terms).

LEMMA If a f f i n e manifold

L

( 61

yk(xl,.

(Proof) have

i s a l i n e a r subspace of Rn, then t h e a + L i s contained i n Z i f f

First l e t

a

..,\)

= 0 f o r a l l k and xi E L.

+ L c Z; for

any

x E L and t E R1 we

a + t x E a + L c Z, so t h a t 0 = +(a + t x ) =

But then all t h e c o e f f i c i e n t s

Cm , t k@.,(XI.

+k(x) = 0 whenever

x

L:

thus (6) follows from ( 5 ) .

Conversely, i f (6) holds, then @(a+

from (4), so t h a t

for a l l x k( x ) = 0 x ) = 0, i.e., a + L c Z. QE ,D

As an i l l u s t r a t i o n , l e t

g'x

+ @o,

and choose

symmetric.

a

E

Rn

E L

have degree 2, + ( x ) = X'HX +

@

with

,+(a) = 0; t a k e

H

Then @ ( a + x) = (2a'H

+

g')x + x'Hx,

so t h a t Y1(x) = (2a'H + g ' h , Y2(%9x2) = x1'm2* Then a l i n e a r subspace

L

has

a

+ L c @-l(O)

iff

(2a'H + g')x = 0, y'Hx = 0 for a l l

x,y

E

L.

p o s s i b l e only f o r

If, e.g.,

H is positive definite, t h i s is

L = 0.

We formalise p a r t of t h i s i n t h e following d e f i n i t i o n :

Rc Rn, an i s o t r o p i c subspace of R i s any a f f i n e manifold a + L c R which cannot be increased (i.e., such t h a t t h e d i e n s i o n of t h e l i n e a r subspace L i s maximal).

given a subset

93

PURSUIT GAMES

I n p a r t i c u l a r , every set

i s t h e union of i t s i s o t r o p i c

R

subspaces; t h e s e need not be d i s j o i n t and my w e l l be s i n g l e tons

.

Exercises 1. For n-dimensional rocket chase i n reduced form,

i=

- v, i

y

= u j end:

v(t) E V

with quarry‘s c o n s t r a i n t

=/

0, and

1x1

=

e

0, prove t h a t

2

e

n o n t r i v i a l isochronous and strobo8copic capture i s impossible. (Hint:

f i n d t h e i s o t r o p i c subspaces of t h e t a r g e t set; apply,

t o aach of these, t h e Necessary Condition from 3.4.) 2.

m

Prove t h a t t h e i s o t r o p i c subspaces of

%

U k=1 p r e c i s e l y t h e unions of i s o t r o p i c subspaces of t h e

i s o t r o p i c subspace of t r o p i c subspsces of

.Qln R2

4.

are

9j

each

i s t h e i n t e r s e c t i o n of i s o -

Exercises 3 t o 6 e s t a b l i s h formula ( 5 ) , deternining a k-linear x).

y

f r o n i t s associated k-ary form

I n any case, d e f i n e y

3.

Prove

complex t.

k

j=o

if

@ ( x ) = y(x,

...,

by ( 5 ) . k

( t - j ) = k!

(-I)’(:)

(Hint:

@,

Af(t) = f ( t + l )

for

-

f(t)

k = 0,1,

-

... and

i s t h e forward

d i f f e r e n c e op%rator, and A i s w r i t t e n a s E I, conclude a k binomial formula f o r , and then apply t o f ( t ) = t k , f o r k which obviously A f = k!)

4. 5.

*we

y(x,

..., x )

@ ( x ) . (Hint: previous exercise.)

~ ( x ) i s a polynornial of degree s k i n n i s t h e forward d i f f e r e n c e x f R and

Show t h a t i f

t h e coordinates of

=

4.

,

uperator with fixed s t e p y f Rn, then sk

- 1.

Conclude t h a t

has degree sl i n t h e coordinates of 94

x.

,y(x)

has degree

STROBOSCOPIC AND ISOCHRONOUS CAPTURE

6.

Conplete t h e v e r i f i c a t i o n t h a t symmetric k - l i n e a r form.

. .,%)

y(xl,.

7. Given t h e quadratic form @ = S12 + 2 + 0 5; i n R6, show t h a t - t h e rows of

0

0

span an i s o t r o p i c subspece of

0 @

t i o n of maximality - 1

0

0

+

2 13

is a

- 142 - 552

1

= 0.

(Do not omit v e r i f i c a -

&

- q+lck

r 2 s 2 Extend t o t h e form @ = Sj i n Rn (0 c r .< s s n ) . Conjecture a fornula f o r t h e dimension of

8.

a l l i s o t r o p i c subspaces, involving if

@

Q! only.

( P a r t i a l answer:

has f'ull rank, t h e dimension i s (n-]signature1)/2).

Remarks The following converse t o Exercise 8 can be proved for a i n n-space:

nonsingular quadratic f o r a Q!

any b a s i s of an

i s o t r o p i c subspace of @ can be augmented t o a b a s i s f o r r e l a t i v e t o which @ has t h e matrix

Rn,

( i f 0 i s singular, t h e r e i s an a d d i t i o n a l bordering by zero matrices). The idea and d e t a i l s of Exercises 3 t o 6 a r e due t o

Prof'. L. Takics ( p r i v a t e communication).

3.6

T i m e Delays

I n a l i n e a r game consider t h e added complication t h a t t h e r e a r e delays i n t h e transmission of information: delay 6

2

a time

0 i n t h e p u r s u e r ' s observation of quarry's action,

and a t i m e l a g X

2

0

i n t h e system r e a c t i o n t o p w s u e r ' s

c o n t r o l s ( i t i s convenient t o t r e a t t h e s e diverse e f f e c t s 95

PURSUIT GAMES

The f i r s t i s adequately t r e a t e d by t h e con-

simultaneously).

for

cept of t h e time-lag pursuer s t r a t e g i e s , exercises i n 2.2;

t h e stroboscopic s t r a t e g i e s used here, t h i s s i m p l i f i e s f u r t h e r , The second appears a s a s suggested by p ( t ) = a ( q ( t 6),t).

-

a modification of t h e dynamical equation:

k

(1)

= AX

- p(t-X)

+ q(t);

t h e remaining data a r e formally unchanged: constraints

p(t)

players' control

E P, q ( t ) E Q, and an appropriate terminaNote t h a t we a r e not allowing any t i m e delays

t i o n condition.

within t h e system i t s e l f (such a s k ( t )

=

.

Ax(t) + Bx(t-8)+. .);

t h i s would lead t o problems of a q u i t e d i f f e r e n t nature. The formulation suggests t h a t , i f t h e game i s played over an i n t e r v a l [O,t], t h e pursuer i s q u i t e i n e f f e c t i v e Over t h e l a s t i n t e r v a l [t

- h,t],

and cannot counter quarry's moves

over a f u r t h e r i n t e r v a l of l e n g t h 6 . tion-of-constants

Referring t o t h e v a r i a -

formula f o r t h e s t a t e response

x(t),

t h e pursuer has no influence on t h e component

eAt{

e -AS q ( s ) d s =

t-X-6 As

eAs q(t-s)ds. 0

q ( * ) v a r i e s over t h e quarry controls, t h e point8 (3)

make up t h e set

(4) independent of i n i t i a l p o s i t i o n and terminal time. Now, i f pursuer aims a t a t a r g e t s e t

+ A; or,

R, he can ensure

is strict* A, ly required, pursuer may aim a t t h e a u x i l i a r y t a r g e t R

a t l e a s t t h e outcone

R

i f t h e outcome

R

s i n c e then he can ensure t h e outcome ( R % A ) + Ac R. As w i l l be seen immediately, t h i s q u i t e p l a u s i b l e

96

-

STROBOSCOPIC AND ISOCHRONOUS CAPTURE

reasoning does provide a moderately s u c c e s s f u l r e s u l t .

But it

does not resolve t h e problem completely ( t h e proposition prov i d e s a s u f f i c i e n t condition f o r winning t h e game, but not a necessary o n e ) j and obviously it i s i r r e l e v a n t i f t h e r e i s a constant winning pursuer s t r a t e g y .

W e r e f e r t o (l), data including X compact and contain 0. position t o target

0, 6 r 0, and ( 4 ) .

R

Then pursuer can f o r c e an i n i t i a l

+

at time

= AX

-

t r o l system

(5) x

2

Assume t h a t t h e player c o n s t r a i n t s e t s a r e

PROPOSITION

can be steered t o

u(t)

U,

2

E

i f , within t h e con-

0

U = P

- ,A(X+8) Q, Y

t by an admissible c o n t r o l ~ ( 0 ) .

at

R

t

Furthermore, t h e r e i s a stroboscopic winning s t r a t e g y u with delay

6,

(6 1 for

u(q,s+6) q E Q, 0 s s

(Proof) Since

u

+

h

+

=

6 s t.

steers

x

to

system, eA t (x Abbreviate

A(X+6) u(s+h+6) + e 9 R

at

t within t h e c o n t r o l

t - J,e"'u(s)d.)

E

h + 6 = 9 ; f o r any quarry c o n t r o l

t o define a pursuer response P(S) = u ( q ( 8 (interpreting

q(s) = 0

R.

q(*)

use (6)

p, r

6 ) , ~ )= U ( S

s < 0).

for

+

A)

+ eA0q(s

- 6)

Then t h e corresponding

s t a t e response ( 2 ) has

t At x ( t ) = e (x-s e-Asp(s-X)ds 0

At

= e

If

(x P

+

st-e

- J' 0e-As u ( s ) d s ) + and

Q

e-A8q(s)ds

0

As

e

0

q(t-s)ds

+

eAtl

E

R + A.

t-e

e

-As

q(s)ds)

&ED

a r e convex a i d symmetric about 0, then a

97

PURSUIT GAMES

necessary and s u f f i c i e n t condition f o r t h e c o n s t r a i n t set

U

of t h e c o n t r o l system ( 5 ) t o be non-void is t h a t

Neither of t h e time delays a c t u a l l y involves t h e s t a t e v a r i a b l e d i r e c t l y ; t h i s makes it p o s s i b l e t o avoid t h e d i f f i c u l t i e s inherent i n d i f f e r e n t i a l equations with time l a g . Note, however, t h a t a quarry c o n t r o l on [O,t] determines t h e

- XI.

pursuer response on [ - X , t

If quarry i s i n e f f e c t i v e , Q = 0, t h e n

a

= 0

a l s o : para-

doxically, a pursuer c o n t r o l f o r c e s an e r r o r - l e s s outcome a

time

X

i n t o t h e future.

Example

Consider t h e game, i n

R2, with dynamical

equation w

x = y + q, y = -x + p, I q ( t ) l s 1 (without t h e quarry c o n s t r a i n t t h i s would be r e l a t e d t o % + x = p + h ) . and player c o n s t r a i n t s I p ( t ) l s 2, It follaws from Corollary 1 i n

3.1 t h a t t h e o r i g i n of R2

is

not a possible t a r g e t f o r stroboscopic and isochronous cap-

ture; it w i l l t u r n out t h a t non-stroboscopic isochronous capt u r e , and even g e n e r a l capture, a r e a l s o ruled out. Assume t h a t , nevertheless, it i s a t l e a s t d e s i r e d t o f o r c e t h e s t a t e v a r i a b l e c l o s e t o t h e origin.

Retracing our

s t e p s , t h e reason t h a t stroboscopic and isochronous capture f a i l s i s t h a t t h e necessary condition Q cP is not s a t i s f i e d : P

i s a segment on t h e x-axis, and

Q

one on t h e y-axis. Now,

t h e y could be aligned by a r o t a t i o n by a r i g h t angle; and one n a t u r a l l y thinks of condition (7). Introduce an a r t i f i c i a l t i m e l a g

6

i n t h e quarry

+ pur-

suer l e g of t h e information p a t t e r n (keeping X = 0 ) i n such a way t h a t

AS

e Q c P.

A simple computation y i e l d s 6 = n/2 (or,

a l l odd multiples); and then

98

STROBOSCOPIC AND ISOCHRONOUS CAPTURE

+

with

x = u

a s t h e a s s o c i a t e d c o n t r o l system ( 5 ) .

It i s

e a s i l y e s t a b l i s h e d t h a t a l l i n i t i a l p o s i t i o n can be s t e e r e d t o 0 w i t h i n t h i s c o n t r o l system.

Thus, r e t u r n i n g t o our game,

a l l p o s i t i o n s can b e forced t o t h e s e t 0 + ( w i t h time l a g x/2) and isochronously.

a

stroboscopically

Another simple compu-

t a t i o n d e s c r i b e s t h e e r r o r term A, a s t h e reachable s e t

R(n/2) of t h e a s s o c i a t e d c o n t r o l system:

a lens-shaped s e t

bounded by a r c s of two c i r c l e s , with r a d i u s 2 and c e n t r e s a t

-+ (1+ i ) .

1. ( S t i r r e d t a n k )

A hopper feeds m a t e r i a l t o a long

conveyor, which d e l i v e r s t h e m a t e r i a l i n t o a l a r g e tank; solvent i s c o n t i n u a l l y added, and an e q u a l volume of s o l u t i o n l e d t h e c o n c e n t r a t i o n i s measured, as a guide f o r a d j u s t i n g

Offj

t h e hopper feed r a t e .

Taking account of t h e t i m e l a g , and

small system e r r o r s , o b t a i n t h e dynamical equation of t h e system; use t h e v a r i a b l e s suggested i n t h e f i g . 1. I n terms (Partial

of t h e chosen parameters, determine t h e e r r o r term. 1 answer: = kc + a(t to) + c(t).)

-

Consider e x t e r n a l c o n t r o l

2.

u

of a harmonic o s c i l l a -

t o r s u b j e c t t o small b u t u n p r e d i c t a b l e p e r t u r b a t i o n s

..x + x

a

>

=

+

u

2 0.

v, w i t h c o n s t r a i n t s I u ( t ) l

g

.a, I v ( t ) ( s j3

Given t h a t t h e o p e r a t o r has time-delay

i n r e a c t i n g t o t h e p e r t u r b a t i o n s , how s m a l l can Assume

er:

6 i

IT.

(Hint:

6

E

and (O,n]

x ( t ) be made?

u s e t h e a r t i f i c i a l time delay n; answ-

J x ( t ) I s 28.)

3.

Prove an analogue of t h e proposition, w i t h a time-

dependent c o n s t r a i n t s e t i n t h e c o n t r o l system: 05;

v,

t r A c 6 ,

ut=

P

E

~ )Q

for

t > A + 6.

Ut = P

for

PURSUIT GAMES

Remarks The material is based on 0. Ha'jek:

Pursuit games with delays, Funkcialaj Ekvacioj (to appear).

3.7 Holding Suppose that one of the players in a game wishes to maintain some favourable configuration over an interval of time; pausing to catch one's breath is a pleasing but irrelevant interpretation. We modify slightly (but see Exercise l), and replace 'maintaining state configuration' by the notion of holding within some (possibly auxiliary) target set. Explicitly, a point x E R is held within R by a strategy if the state response endpoints satisfy x(t) E R for 100

STROBOSCOPIC AND ISOCHRONOUS CAPTURE

t

all

0

2

and a l l quarry controls.

Capture with holding w i l l c o n s i s t , f i r s t , of f o r c i n g t h e s t a t e v a r i a b l e t o t h e t a r g e t w i t h i n a bounded t i n e i n t e r v a l

[O,eI,

and of subsequent holding within t h e t a r g e t a t a l l t i n e s Note t h a t nothing changes i f we r e q u i r e iso-

after capture.

chronous capture a t t h e

8 , and holding f o r a l l

t

2

If

8.

stroboscopic s t r a t e g i e s a r e t o be used, t h e n t h e results of

3.5 suggest t h a t t h e a f f i n e manifolds a r e appropriate a s t a r gets. Thus, consider t h e game i n

- p + q;

k = AX

(1)

where

M

and

b

Rn,

p ( t ) E P, q ( t ) E

Qj

a r e given, and t h e t a r g e t

M x = Mb,

end:

R

is t h e i n d i c a t -

Anticipating t h e r e s u l t below, we i n t r o -

ed a f f i n e manifold.

duce t h e associated c o n t r o l system

9

(2)

where

Thus =

- u;

u(t)

E

U;

end:

Mx = Mb,

a r e a s above, and

U = (P

(3)

N

A, M, b

= Ay

+

L ) 2 Q, L = [x:

.I.

~ = jo f x o r j = 0~1,. ,

~

L i s t h e l a r g e s t A-invariant subspace of t h e n u l l space [x: M x = O} of M ( c f . 6.3, Exercises 16 and l 7 ) i t h e

obvious property A L c L 1 for a l l t E R .

has t h e consequence t h a t

eAtL = L

If pursuer's c o n s t r a i n t set

THIRD N C I P R O C I T Y THEOREM

i s compact, t h e n t h e holding problems f o r t h e game ( 1 ) and t h e c o n t r o l system ( 2 ) a r e equivalent. In detail, a pmition i n (1)a t t i n e

8

2

0

x

E

Rn

can be forced t o

R

with-

and held t h e r e subsequently, by a

stroboscopic s t r a t e g y u, i f , and only i f , x to

R

within (2) a t tine

8

can be s t e e r e d

and held t h e r e subsequently, by

an admissible c o n t r o l u. 8

Moreover, t h e r e l a t i o n 101

PURSUIT GAMES

(41

o(q,t)

PF

u ( t ) + q modulo L

may be used t o determine e i t h e r of (Proof)

u,u from t h e o t h e r . The techniques from t h e two r e c i p r o c i t y theorems w i l l

be put t o a y e t f u r t h e r use. F i r s t , l e t u be a s u c c e s s f u l s t r a t e g y ; s e l e c t 90 E QY 90; choose a r b i t r a r i l y h 2 8 , and set u ( t ) = o ( Q , t )

-

q E Q, s E [ O , h ] ; and consider t h e piecewise constant quarry The s t a t e c o n t r o l with values i n [O,s], q i n (s,+m). response has

x ( h ) E R, h l;-Atu(t)dt

-

S

e-At(u(t) + q - a ( q , t ) ) d t ) = Mb. 0 On s u b s t r a c t i n g t h e version of ( 5 ) with s = 0, we obtain

(5)

MeAx(,

+

-

Ah -At (u(t) + q u ( q , t ) ) d t = 0. Me Joe Here one may keep s f i x e d while varying h ( t h e r e s t r i c -

tions are

h

5: 8 ,

h z s only).

(6) holds f o r a l l h E R1

By a n a l y t i c i t y of

(and

AX hw e

,

s 2 0); t h e r e f o r e t h e i n -

t e g r a l term i n (6) s a t i s f i e s S

J0

E n {e-*'N:

x E

RL) = L,

and thus t h e integrand u(t) + q

- u(q,t)

At

E e

L = L

a.e.

q E Q. An a p p l i c a t i o n of t h e lemma from 3 . 1 y i e l d s (4); s i n c e q was a r b i t r a r y and u has values i n P,

for a l l

(P

u(t) f i n a l l y , ( 5 ) with point

x(h)

E

R

Second, l e t

+

L)

s = 0 shows t h a t for a l l h u

2

iL Q u(.)

=

U;

steers

x

t o the

8.

be an appropriate c o n t r o l within ( 2 ) .

Use (4) and F i l i p p u v ' s L e m from 3.9 t o define a stroboscopic strategy u

with values i n

a u x i l i a r y mapping w

P

(by d e f i n i t i o n of U ) , and an

with values i n

102

L.

Then t h e assumption

STROBOSCOPIC A N D ISOCHRONOUS CAPTURE

Me

Ax

x - Joe-Atu(t)dt)

= Mb f o r X 2

e

i s a successful strategy:

u

easily yields that

(x

h

e-Atw(*)dt = Mb + 0. Assume t h a t t h e d a t a i n (1) s a t i s f y t h e following

LEMMA

t h r e e requirements : The t a r g e t s e t and rank

R = {x:

Mx

with

= 01

of type (m,n)

M

mj

P,Q a r e compact and b o t h contained i n an m-dimensional f o r an (n,m) matrix B; subspace {By: y E d e t M(X1 A)-$ does not v a n i s h i d e n t i c a l l y ( e . g . , d e t ME

-

4 0).

within

R4

I n this situation, i f a point by a s t r a t e g y

R

u, t h e n

u

x E R

can be held

i s n e c e s s a r i l y strobo-

scopic and of t h e form

(7)

~ [ q l ( t =) q ( t ) + u ( t ) x

a.e.

with

U(*) independent of x and q. (Proof) Let a be a s t r a t e g y holding t h e p o i n t

R.

The s t a t e responses

for a l l

t

2

xo

within

x ( * ) w i l l , t h e r e f o r e , have Mx(t) = 0

0, so t h q t t h e i r Laplace transforms xh(s) s a t i s f y

(8)

&(s)

= 0.

From t h e d i f f e r e n t i a l equation,

(9) (note t h a t

Sqs)

q,u[ q ]

- xo = &(s) - $ [ q l ( s )

a r e measurable and bounded,

have Laplace transforms).

We solve ( 9 ) f o r

i n (8) (omitting most e n t r i e s M(SI

M(~I

- A>-%

+ $(s) 80

x^

t h e y do

and replace

s):

- A )-1(-8 = M(~I

+ ^s + x0)

- A>-%

+

M(~I

According t o t h e second assumption one may write 103

= 0,

8

- A)”.~.

=

&,

PURSUIT GAMES

From t h e l a s t assumption, t h e m-square matrix

^u

1,

=

- A)-%

s) has an inverse a t

( e n t r i e s a r e r a t i o n a l functions of l e a s t f o r l a r g e 1s

M(s1

so t h a t

- A ) - $ ) - h ( s I - A ) -1xo

$ + (M(s1

which, pre-multiplied by

B, y i e l d s

8

=

6

+ O(S)Xo

with t h e i n c i d e n t a l information

G(e)

-

= B ( M ( ~ I A)"B)-'(~I

- ~1-l.

Inverse Laplace transforms now provide (7); U(t)x

i s a func-

t i o n ( r a t h e r than an operator) s i n c e both o [ q ] J q a r e such.

BED Exercises 1. I n (1)shuw t h a t holding a point

over an i n t e r v a l [0,9] entire.

over LO,+-)

with

(Hint:

9

> 0 is

x E R

within

R

equivalent t o holding

t h e point i s a n a l y t i c i t y i n t h e

.

argument following (6 ) ) 2. P = {bu:

(An'$

,...,

4

P

-

verify t h a t m ' ( s 1 A)'$ 4 0 f o r some row m' 0, and solve (10) f o r t h a t row.)

(Hints: M

i s one-dimensional, n such t h a t rank -1 g u s l}, f o r a v e c t o r b E R Ab,b) = n. Prove t h a t u is, again, l i n e a r i n x.

I n t h e lemma, assume t h a t

of

Remarks I n connection with t h e c o n t r o l - t h e o r e t i c side, see references under 'core of t a r g e t ' i n E. B. Lee, L. Markus: Foundations of Optimal Control Theory, Wiley, New York, etc., 1967;

104

STROBOSCOPIC AND ISOCHRONOUS CAPTURE

and a l s o 0. Hijek:

Cores of t a r g e t s i n l i n e a r c o n t r o l system, Math. Systems Theory 8 (1974) 203-206.

It would be d e s i r a b l e t o Simplify t h e l a s t assumption of t h e

lemma, t h a t det M(h1 observable case:

- A)-$ 4 0,

a t t h e very l e a s t f o r t h e

an observed c o n t r o l system i n

;C

-

= AX

U,

5

= bj

u(t)

E

Rn, U,

i s s a i d t o be observable i f t h e p a r t i t i o n e d (nm,n) matrix

n; t h e condition i s independent of U ) .

has maximal rank ( i . e . ,

From ( 3 ) it follows t h a t o b s e r v a b i l i t y i s equivalent t o L = 0. Compactness of t h e c o n s t r a i n t sets may be ommitted i f , i n t h e proof, one uses Mikusi6ski's operators, A r e s u l t analogous t o t h e theorem holds f o r s t r a t e g i e s which have t h e added property t h a t u [ q ] is piecewise con-

tinuous whenever

q

i s such

E. Rechtschaffen:

Equivalences between d i f f e r e n t i a l games and optimal c o n t r o l s ( p r e p r i n t ) , P o n t i f f c i a Univ. Cat6lica do Rio de Janeiro

(1975).

3.8 Convex S e t s , Pontrjagin Difference A set

in

S

Rn

i s convex i f it contains t h e e n t i r e

segment between any two of i t s points, owc

+ (1-a.)~ E

S

whenever x,y

E S,

0 s u s 1.

,

n a Given a t r i p l e S,a,c c o n s i s t i n g of a set Sc R n n and a v e c t o r c E R we s h a l l say t h a t c i s point a E R

,

,

an e x t e r i o r normal t o S

at

a 105

if

PURSUIT GAMES

c'x

(1)

5

c'a

for a l l x

x

The geometry of t h i s i s t h a t t h e v e c t o r obtuse angle t o

c

E

-a

S.

b e a r s an

( t h e cosine being nonpositive); and a l s o

t h a t t h e 'supporting' hyperplane {x: c'x = c'a} through w i t h normal

c

determines two half-spaces, contains t h e set

{x: c'x s c'a}

subspace, (1) implies t h a t x

E

c'tx

5

of which

I n case

S.

c'a

S, so t h a t t h e only p o s s i b i l i t y is

a

is a linear

S

t E R1

f o r all c'x = 0:

and

t h e concept

then coincides with t h a t of t h e orthogonal complement A

s

= ~y

LEMMA 1

nonvoid s e t

o

y'x =

E R":

If a point

S, and

t h e vector

c = b

-

a

E S].

for a l l x

E

a

b S

not

is

i n a convex, closed,

i s t h e p o i n t c l o s e s t t o b, then i s an e x t e r i o r normal t o

0

S

a t a.

Moreover, t h e corresponding supporting hyperplane separates S

and

Convexity is not needed f o r t h e existence of a point

(Proof)

a E S any

i n a d d i t i o n t o (l), c'a < c'b.

b:

closest t o

b (but it does imply uniqueness).

Take

x E S; then, f o r a E [0,11, t h e point a x + ( 1 a ) a E S,

so it i s no c l o s e r t o

b

than

a:

lax + (1-z)a

- b(

- bl.

la

2

Rearrange and t a k e squares,

d i v i d e by

2a

Thus indeed from b

w

4

>

0

c = b = S

and t a k e

-a

(a

it

+

0,

- b)'(x - a ) 2

0.

c

s a t i s f i e s (1). That

3 a, and then

lcI2 > 0 y i e l d s

COROLLARY 1 Every closed, convex set i n

i n t e r s e c t i o n of half-spaces (Proof)

40 c'b

n R

follows

>

c'a.

is the

{x: c'x s u}.

Excluding t h e t r i v i a l cases of 106

pt'

or

Rn,

Lemma 1

STROBOSCOPIC A N D ISOCHRONOUS CAPTURE

s h m s t h a t t h e r e e x i s t half-spaces

H containing

S, and a l s o

that

Sc i s a c t u a l l y an e q u a l i t y . By s e p a r a b i l i t y of t i o n countable.

n

{H:

half-space H 2 S]

Q,ED

one may always t a k e t h e i n t e r s e c -

Rn,

The polyhedra a r e t h e f i n i t e i n t e r s e c t i o n s of

half-spaces. If a set-valued mapping

COROLLARY 2

t

TS,

s

7 J 0 So s edeas c c, 2

etc.

By Corollary 1, it s u f f i c e s t o prove t h a t t h e mean

(Proof)

H

c'f(e) s a

a.e.,

then c+

t

A t every boundary point, S

t

0

0

0

S.,

(Proof)

4 F;

a w i t h bk

nearest t o a

S, a t some point of

Consider any p o i n t

k

+

bk

a

E as.

i s an

S.

Then t h e r e e x i s t s points

according t o Lemma 1, t h e points

have e x t e r i o r normals

= bk

a, s i n c e lak-al s la k-b k I + lbk-aI s la-b k

The u n i t v e c t o r s

S.

has a n o n t r i v i a l e x t e r i o r normal.

is a l s o compact, t h e n every v e c t o r i n Rn

e x t e r i o r normal t o

also

%S S

Let t h e r e be given a nonvoid, convex s e t

LEMMA 2 S

if

J;(e)ae

analogously f o r t h e second moments

+

whenever

= {x: c'x s a]

This i s an elementary e x e r c i s e i n i n t e g r a t i o n :

H = J C.

bk

has a l l

s ea e c c f o r t > 0,

values a r e i n t h e half-space

If

H St

C, then

values i n a closed, convex s e t

t

t

dk = ck/ Ick I

I

- ak =/

ak 0.

+ Ibk-aI

E S

Now,

+

0.

a r e a l s o e x t e r i o r normals t o

107

PURSUIT GAMES

ak'

dk'x s

dpk, f o r a l l x E

S,

nce hey a r e on t h e compact u n i t sphere [x: 1x1 = 11, they have a subsequence converging t o some vector d =/ 0. On taking limits i n ( 2 ) f o r fixed x E S we find t h a t indeed d'x

5

d'a

for a l l x E S

For the second assertion, note that the continuous r e a l -

valued f'unction x w c'x, when r e s t r i c t e d t o t h e compsct set S, a t t a i n s i t s m a x i m a t some point

w

PROPOGITION 1 For subsets of A

+

a E S: Rn,

+ X implies

X =, B

thus (1)holds.

A

3

B

i s cloaed and convex, and X compact, convex, nonvoid. (Proof) Proceed by contradiction: the conclusion f a i l s , and some point b E B i s not i n A. By Lemma 1, the point a E A closest t o b yields c = b a as non-zero exterior normal t o A a t a . By Lemma 2, t h i s c i s an exterior n o m l t o X a t some point x E X. Since A + X 2 B + X, we have b + x = a f x1 f o r some a1 E A and x1 E Xj thus 1 if

A

-

c'b contradicting

c

For any s e t

4

5

,

c'a + 0,

&ED

0. S

-

c'al + c'xl c'x 2 0 s c'(a-b) = -Icl =

t h e r e e x i s t s a l e a s t convex set

i n Rn

containing S. It can be obtained e i t h e r a8 the interesection of a l l convex s e t s C containing S, o r as t h e s e t of a l l convex co5binations of points i n S,

(3)

{

r

k=1

akYk: Yk E

ak

2

r

3%=

'8

= 1,2,"']

The usual term is convex span ( o r h u l l ) of Sj our notation w i l l be cvx S .

108

STROSOSCOPIC AND ISOCHRONOUS CAPTURE

If

i s c a l l e d a polytope; it can be

i s f i n i t e , cvx S

S

shown t h a t t h e polytopes a r e p r e c i s e l y t h e compact polyhedra.

r s n

Obviously one cannot r e s t r i c t t h e case

n = 2).

always t a k e

i n (3) (consider, e.g.,

By Carathebdory's Theorem one may, however,

r = n

+

1j

a consequence i s t h a t t h e convex span

of a compact s e t i s again compact (while, e.g.,

t h e convex

2

y = l/(l+x ) i n

span of t h e bell-shaped curve

i s not a

R2

closed s e t ) . A point

a

of a convex s e t

S

i s extreme if it i s not

an i n t e r i o r point of any t r u e segment i n with

x,y

in

S

and

0

S:

a = ax + (1-a)y

< a < 1 implies t h a t

x = a = y.

By

t h e Krefn-Milfman Theorem, every compact, convex s e t is t h e convex span of i t s extreme points.

%2

i s not required i n ( 3 ) , t h e r e s u l t is t h e S (and i f = 1 is also omitted, one obtains t h e l i n e a r span of, or l i n e a r space gene r a t e d by, s ) . Each convex set has non-void i n t e r i o r within If

0

c&,

a f f i n e manifold generated by

t h e generated a f f i n e manifold. The reader i s r e f e r r e d t o many e x c e l l e n t books on convex

sets, e.g. H. G. Eggleston: Convexity, Cambridge University Press, New York, London, 1958; B. Grunbaum:

etc.,

Convex Polytopes, Interscience, London,

1967;

T. Bonnensen, W. Fenschel: Theorie d e r Konvexen K6rper, Springer, Berlin, 1934.

The second t o p i c of t h i s s e c t i o n i s t h e Pontrjagin d i f f erence operation.

Consider t h e problem of c u t t i n g out a

shaped p r o f i l e by a m i l l i n g machine with c u t t e r head diameter 26; it i s desired t o determine t h e p o s i t i o n s of t h e c u t t e r

head centre, admitting t h a t c e r t a i n moderately sharp inner corners w i l l not be shaped p e r f e c t l y .

R2

t o be milled off, and

D

If

denotes [d

109

S

E R2:

is t h e s e t i n Id/

5

61, then

PURSUIT GAMES

t h e c u t t e r head p o s i t i o n s

c

must have

c + d E S

for a l l

d E D: 2

C = { c c R :

c+DcS}

A c l o s e l y r e l a t e d problem i s t h a t of finding t h e curve a t a

distance

6

from a given curve (above, t h e boundary of

Generalising only s l i g h t l y , given two subsets

S).

in

P,Q

Rn, t h e i r Pontrjagin d i f f e r e n c e i s

(4 )

P C Q = {x: x + Q c P } ,

t h e l a r g e s t set

X

with

X

+

Q c P.

Note t h a t here t h e con-

tainment may be proper, a s i n f i g . 1. By inspection one ob-

tains the a l t e r n a t i v e description

(5) Thus

PfQ

n

(P-9). q€Q i n h e r i t s various p r o p e r t i e s from P%Q=

P:

if

is

P

closed, o r compact, convex, an a f f i n e manifold, t h e n so i s P i t Q.

i s symmetric i f both

From ( 3 ) , P 5 Q

P

and

Q

are;

thus we have U M M A 3

nonvoid; t h e (Proof:

Assume t h a t P

Q

P,Q

a r e convex,

symmetric, and

i s nonvoid i f , and o m i f , Q C P.

a convex symmetric s e t i s nonvoid i f f it contains 0 . )

The following two r e s u l t s j u s t i f y t h e d i f f e r e n c e nota-

t ion. PROPOSITION 2 (on Cancellation)

For subsets of

Rn

we

have

(6 1 if

( P + X) P

i s closed, convex and X

(Q

+ X)

= Pf Q

i s compact, convex, and non-

void.

(Proof) Referring t o (4), conditions

equivalence i s a s s e r t e d f o r t h e two

x + Q +X c P

110

+

X,

x

+ Q c P.

STROBOSCOPIC AND ISOCHRONOUS CAPTURE

One implication is inmediate; t h e other follows from Proposit i o n 1. &ED

3

If U i s closed, convex, and Q is compact, convex, and nonvoid, then COROLLARY

(U + Q) 2 Q = U.

(7)

Thus, if Q i s a d i r e c t summand of P, i n the sense t h a t P = U + Q f o r some closed, convex s e t U, then necessarily U = PrQ.

(Proof) (7) is a s p e c i a l case of (6), with 0 as one of the sets.

The second assertion follows a s P 2 Q = (U + Q) 2 Q = U.

&eD A c l a s s i c a l concept i n convex s e t theory i s t h a t of asymptotic directions: a vector c E Rn is an asymptotic

direction of a convex s e t S

+

if t h e ray

X + R C =

{x+ec:

era]

l i e s e n t i r e l y i n S. Note t h a t necessarily x E S; i f S i s closed (or open), then t h e condition is independent of the end point x, and S i s compact i f f 0 is i t s only asymptotic direction (or S = g). PROPOSITION 3 If P is closed and convex, then P P + is a closed wedge (W, with W = w’= cvx W = R W), and coincides with t h e s e t of asymptotic directions of P. If P i s closed, convex, and symmetric, then P P i s the l a r g e s t l i n e a r subspace L with P + Lc Pj moreover,

*

P = (PrP) + Q with Q compact, convex, and symmetric. (Proof) P f P is closed and convex since x,y a r e i n P P then so is x + y:

thus

P

*P

(x

P is such; i f

+ y ) + P = x + (y + P) c x + P c P;

i s a closed wedge. 111

Each element of P

P is an

PURSUIT GAMES

asymptotic d i r e c t i o n :

+

P + R (PEP) = P + (P*P)=

P.

i s an asymptotic d i r e c t i o n , then P + R c c P, so t h a t c E P f P.

Conversely, i f

c

+

P is, i n addition, symmetric, then so i s

If

P it P i s a l i n e a r subspace, L, t h e l a r g e s t with

thus P

+

*

P - P;

L c P.

Q = P

Set

, so t h a t

L

nL

i s closed, convex,

Q

and symmetric; we propose t o show t h a t

P= L

+

Q

and t h a t

i s compact.

Q

Obviously L + Q c ( P k P ) + P c P . Conversely, every

I

can be w r i t t e n a s

x E Rn

with

y -+ z

-

E L ; here, i f x E P, then z = y x E ( P f P) I + P c P, and so z E L n P = Q. Finally, consider any asymptotic d i r e c t i o n c of Q (we wish t o show t h a t c = 0 ) .

y

E

and

L

z

Then 0 + R l c c Qc P,

c

i s an asymptotic d i r e c t i o n of

Since simultaneously

,this

I

c E L

P also, c

E P

c

* P = L.

= 0.

QED

yields

Exercises 1. The r e l a t i o n

defined by t h e concept of e x t e r i o r

normal i s a closed r e l a t i o n :

show t h a t , i f

S c limsup Sk,

a, c + c, and ck i s an e x t e r i o r normal t o k then c i s an e x t e r i o r normal t o S a t a. ak

+

2.

Prove t h a t t h e e x t e r i o r normals t o

S

at

Sk a t a

ak,

form a

closed wedge.

3 . Prove t h a t t h e set of p o i n t s of S a t which c i s an e x t e r i o r normal forms a closed, convex set contained i n 3s and perpendicular t o normal t o

S

c.

Then show t h a t

a t a l l p o i n t s of 112

L

c

n

3s.

c

i s an e x t e r i o r

STROBOSCOPIC AND ISOCHRONOUS CAPTURE

P

0 sets is

4.

Fig. 1 Pontrjagin d i f f e r e n c e P Q, of , P and Q; t h e square with rounded corners (P

Q)

+

Q, properly contained i n

Given a compact, convex s e t

P.

S, assume t h a t e i t h e r

t h e r e i s a unique e x t e r i o r normal a t each boundary point (S

is smooth) o r each supporting hyperplane i n t e r s e c t s one point only (S i s s t r i c t l y convex).

S

at

Prove t h a t t h e

correspondence between boundary p o i n t s of

S

and u n i t exter-

i o r normals i s continuous.

5. Show t h a t asymptotic d i r e c t i o n s a r e independent of + end-point: i f S i s closed and convex, x + R c c S and

+

y E S, then a l s o y + R c c S.

obtain any y + 8c points y

and

(Hint:

r e f e r t o f i g . 2;

a s t h e limit of convex combinations of

x + tc.)

6. Prove t h a t a closed, convex, nonvoid set i s compact iff

0

i s i t s only asymptotic d i r e c t i o n .

find w i n t s

xk

+

m

(Hint:

i f one can

i n t h e set, then any l i m i t of t h e i r

u n i t d i r e c t i o n s i s an asymptotic d i r e c t i o n . )

113

PURSUIT GAMES

Fig. 2 Asymptotic directions a r e independent of end-point (Exercise 5).

7.

Prove t h a t p f ( Q 1 + Q2) = ( P r

8.

Prove t h a t (P1 + P2>

'Q

3

'%.

P1 + (P2

Q),

and obtain simple counterexamples t o equality.

9.

Show t h a t (P

+ u(P f

f o r compact, convex s e t s

10.

Q = (a + 1 ) ( P iL Q)

Q))

P,Q and

Show t h a t t h e set

a 2 0.

P2Q

is unchanged i f P and a r e subjected t o a t r a n s l a t i o n x w x + a . I n exercises ll and 12, t h e (m,n) matrix T i s a l s o interpreted as t h e l i n e a r mapping

ll. For subset8 P and Q (TP)

f

XH

fl

of

Tx. Rn

show t h a t

(TQ) = T ( P + n u l l T ) f Q);

hence,

114

Q

STROBOSCOPIC AND ISOCHRONOUS CAPTURE

(TP) 2 (TQ) = T ( P f Q)

i s one-to-one, rank T = n. t h e n P @ and x E range T.) if

T

4

12.

For subsets

P,Q T'l( P

of

T

and s y m e t r i c .

f

n

range T) j

T-'(Q>

T = m.

maps onto, rank

13. Assume t h a t

P) ik T-l(Q

* Q) = T-'(P>

T-'(P

x + T Q c TP,

if

show t h a t

Rm

* Q) = T'l(

hence,

if

(Hint:

P i s a l i n e a r subspace, and

Prove t h a t

P

f

Q

convex

Q cP, and

P if

is

Q

fl

if

not.

14. Show t h a t ,

P i s a polyhedron or a polytope, then

if

P f Q a l s o has t h e corresponding property, without any assumption on Q.

15. This e x e r c i s e c o n s i s t s of v e r i f y i n g t h e s e r i e s of a s s e r t i o n s below. For t h e c o l l e c t i o n of compact, convex subsets of

R

n

,

a d d i t i o n is a commutative and a s s o c i a t i v e operation; thus one has an Abelian semigroup, with zero element 0.

The operation

i s a l s o c a n c e l l a t i v e according t o Proposition 2; thus t h e r e i s a corresponding ' s u b t r a c t i o n group'.

The elements of t h i s a r e

p a i r s (P,Q) of compact, convex s e t s i n (P,Q) and ( P

+ U,

Q

+

(PI,$>

(8)

Rn,

and t h e p a i r s

U) a r e equivalent; more generally

-

CP2,Q2> iff P1

+

%=

P2 + Q1.

The group operation i s a s indicated i n (PliQl)

+

( P p % , > = (P1

independent of r e p r e s e n t a t i o n .

+

P2, Q1 + Q2>,

The zero element i s (O,O),

t h e element opposite t o (P,Q) i s (Q,P); t h e n a t u r a l i d e n t i f i cation i s

P W (P,O).

115

PURSUIT GAMES

M u l t i p l i c a t i o n by n-square matrices of sets extends t o t h e p a i r s , d e f i n i n g a set of operators on t h e group:

A(P,Q) = Analogously, containment c a r r i e s over, v i a an

(AP,AQ). analogue of

(a),

(P,,%>

2

(P2jQ2) iff P1

+

%2

P2

+

Ql,

again independent of r e p r e s e n t a t i o n j and t h i s behaves w e l l under both t h e group a d d i t i o n and t h e set of operators. (An i n t e r e s t i n g p r o j e c t would be t o t r e a t t h e l a t t i c e - t h e o r e t i c properties. ) There i s a n a t u r a l mapping (P,Q) I+ P

Q independent of

representation, from t h e group t o t h e semigroup ( a l e f t in-

verse t o t h e embedding); it i s isotone but probably not a semigroup homomorphism. There i s a n a t u r a l decomposition (p,Q> = (P,O)

+

(0,Q)

i n t o ' r e a l ' and 'imaginary' p a r t s ; t h i s defines homomorphisms from t h e group t o t h e semigroup (which a r e not i s o t o n e ) . For given n-square matrix

A

and r e a l t, and f o r each

p a i r (P,Q), d e f i n e a generalised reachable set

t h i s i s independent of r e p r e s e n t a t i o n i n t h e group, and t h e corresponding mapping i s an isotone honmorphism (not i n v a r i a n t under t h e n a t u r a l operators). Remarks The notion of t h e Pontrjagin d i f f e r e n c e is p r e - h i s t o r i c : f o r i n t e r v a l s on t h e p o s i t i v e half-axis,

[o~uI

f

[o,PI = [O,U

-

~ l j

and claims f o r p r i o r i t y may (and, probably, w i l l ) be made f o r analogues t h e r e o f .

However, t h e concept (and n o t a t i o n ) f i r s t

appeared i n 116

STROBOSCOPIC A N D ISOCHRONOUS CAPTURE

L. S. Pontrjagin: Linear differential games 2, Doklady Akad. Nauk SSSR 175 (197)721-723. Several further properties of the operation were summerised in E. Rechtschaffen, Unique winning policies for linear differential pursuit games (thesis), Case Western Reserve University, 1973. One of these, our Exercise 11, has the consequence

This remained in the form on the left in Pontrjagin's paper; the second describes the essence of the First Reciprocity Theorem. The idea of the 'subtraction group' in Exercise 15 is due to Mr. S. E. Claws (personal communication),. Experimentation with planar figures suggests that, if P is a polytope, then each face of P Q is parallel to one of P (even though not all faces of P need be utilised thus); (Rechte.g., fig. 1. The assertion is false already in

2

schaffen's thesis).

3.9 Measurable Selection Consider the following problem. Given two sets P,Q in Rn, for every point x E P + Q there exists, by definition, a decomposition x = y + z with y E P and z E Q; this might be interpreted in terms of appropriate 'eelection' mappings P + Q-, P, etc. The question is whether one can choose these mappings so as to preserve measurability: if t n x is measurable, is t w y also such? Apparently the selections will have to be made canonically, in some sense. The definition of measurability of a set-valued mapping t w Ft, from R1

to the collection of subsets of Rn, is that be Lebesgue measurable for every compact {t: Ft u C =/ @} subset of C of Rnj apparently this is a legitmate 117

PURSUIT GAMES

g e n e r a l i s a t i o n of t h e c l a s s i c a l concept. I n t h e r e s u l t below it might seem n a t u r a l t o r e s t r i c t oneself from t h e o u t s e t t o nonvoid s u b s e t s of

curiously

Rn;

enough, t h i s l e a d s t o i r r e l e v a n t t e c h n i c a l d i f f i c u l t i e s . There e x i s t s a mapping

MEASURABLE SELECTION THEOREM S:

on t h e c o l l e c t i o n

9 -+ Rn

5

of a l l closed subsets of

Rn, which i s a s e l e c t i o n ,

S(F) E F

(1)

whenever

and preserves measurability: so i s t H S(Ft). (Proof)

if

$

+F

t w Ft

=

F c Rn,

i s measurable, then

The underlying idea i s extremely simple:

one can

choose a p o i n t i n a given set by f i r s t s e l e c t i n g i t s first coordinate, then an appropriate second coordinate, e t c . Second, one of t h e n a t u r a l choices of a p o i n t w i t h i n a compact subset of

RL

We e x h i b i t Sn.

Here

So

S

i s t h e left-most one. as t h e composition of mappings

So,S1,...,

i s t h e operation of taking points c l o s e s t t o

origin,

s ~ ( F )= {x E F: 1x1 = min J y ) ] ; and, f o r

k

2

coordinate,

(here e.g.,

ek

1, S k

Sk(F) = {x E F: ek'x = min e $1 Y@ is t h e k-th u n i t b a s i c v e c t o r j f o r F =

So(@) = 0 ) .

subset of Sk(F)

YEF i s t h e operation of minimizing t h e k-th

F

Obviously So(F)

whenever

and compact

F

@

4 8.

4

S

i s a nonvoid, compact

Fc Rn,

Since Sk

and s i m i l a r l y f o r f i x e s t h e k-th

S i s single-point1 0 i s a s e l e c t i o n i n t h e sense of (1).

coordinate, t h e composition valued; thus indeed

F =

$ take,

S = Sn...S

The proof i s completed by v e r i f y i n g t h a t each serves measurability; we s h a l l c a r r y t h i s out f o r

118

Sk So

preonly.

STROBOSCOPIC AND ISOCHRONOUS CAPTURE

t w Ft

Take any measurable t h e set either

[t: So(Ft)

Ft = fd and

n C 4 fl.

C c Rn,

and compact

and consider

The requirement unwinds thus:

0 E C, o r

C

contains a point of

Ft

with minimal norm, i.e.,

minlxl s min 1x1 Ft Ftw Now, t h e second condition i s 'measurable' ( f o r t h e first, see Exercise 3), since t h e mappings appearing t h e r e are such. E.g., f o r t H minlxl and r e a l a, a time t s a t i s f i e s Ft minIxJ s a iff F~ i n t e r s e c t s t h e compact a - b a l l about t h e Ft origin.

QED

FILIPPOV'S LEMMA e x i s t s a mapping u

Given continuous

f:

Rn

+

Rm, there

with t h e following properties:

-

n n 1. o(x,F) E R whenever x E Rm, F = F c R . 2.

If x E f(F), then f(u(x,F)) = x; thus a ( * , F ) :

+ F is a r i g h t inverse t o f . t c, x ( t ) and t c, Ft a r e measurable, and a l l

f(F)

3.

If

f(Ft), then t n u ( x ( t ) , F t ) is a l s o measurable; x(t) thus t h e r e is a measurable solution y ( . ) t o x ( t > = f ( Y ( t ) ) , Y(t)

(2)

n

(Proof) Set a(x,F) = S ( f - l ( x ) able s e l e c t i o n from t h e theorem. v e r i f i e d easily.

F)

the s e t s

S i s t h e measur-

For t h e l a s t one need only r e f e r t o t h e

Indeed, f-'(x(t))

C, f(C)

where

Ft'

The f i r s t two assertions are

theorem again, once it i s shown t h a t measurable.

E

compact.

t n f-'(x(t))

n C =/ #

iff

is

x ( t ) E f(C), with

&ED

Example L e t us r e t u r n t o t h e problem described i n t h e

f i r s t paragraph.

We apply Filippov's Lemma, with addition as

t h e continuous mapping f : Rn x Rn .) Rn, and Ft = p x Q. 1 Then f o r any measurable x: R 4 P + Q we obtain measurable 119

PURSUIT GAMES

1 1 y : R + P, z: R

Q

x(t) = y(t) + z(t)

such t h a t

a r e t h e two coordinates of

'y,z

Exercises

a(x,F)

E

(here

Rn x R").

1 1 f o r i n t e g r a l s of (Ft + Gt) = J0Ft + JOGt 0 set-valued mappings, assuming t h a t t h e values Ft,Gt are 1 ~

1. Prove

closed, and, e.g.,

all

Ft

admit a common bound.

(Hint:

it

i s , of course, t h e ' d i f f i c u l t ' i n c l u s i o n t h a t i s t h e h e a r t of

t h e exercise.

R e c a l l a l s o t h a t t h e d e f i n i t i o n of JFt involves

i n t e g r a b l e functions. ) 2.

Show t h a t , i f

M

is an (m,n) matrix, t h e r e i s a

measurable s o l u t i o n x ( * ) of able

a(-)

Mx(t) = a ( t )

mapping i n t o t h e row space of

f o r every measur(mnt:

M.

you may

wish t o apply generalised inverses r a t h e r t h a n Filippov's

Lemma.) Prove t h a t , f o r measurable

3. t

with

F = @

t

i s measurable.

t

H Ft,

t h e set of times

consider i t s comple-

(Hint:

ment, and use compact b a l l s with r a d i i 1,2,.

..)

is measurable, then so i s t H Ft n C ( t h i s i s a d e t a i l needed i n t h e proof of t h e Theorem; Exercise 8 g e n e r a l i s e s t h i s ) . Show t h a t i f t n Ft n C i s measurable f o r each compact C, then so i s

4.

If

t n Ft

f o r fixed compact

t

H

Ft*

t

H

Fk(t)

C

5. Show t h a t t n Fl(t) u F p ( t ) i s measurable i f both a r e such; and s i m i l a r l y f o r

continuous

6.

f : R"

+

t

r)

f(Ft)

with

R".

I n t h e d e f i n i t i o n of mensurability, t h e test sets

C

may, equivalently,be replaced by closed sets. Prove t h e more u s e f u l r e s u l t t h a t they may be replaced by any b a s i s of t h e open sets.

(Hint: write an open set a s t h e union of compact sets, and a compact set a s t h e i n t e r s e c t i o n of open, bounded

sets. ) 120

STROBOSCOPIC AND ISOCHRONOUS CAPTURE

7. Show that if Fk: R1 +

9 are measurable, then so are

(Hint : Exercise 6.) 8. Prove that t Fl(t) n F2(t) is measurable if both Fk are such. (Hint: F1 n (F2 n C) @ iff F1 x (F2 n C) intersects the diagonal.) 9. If X I + f(x,t) is continuous and t H f(x,t) measurable, then t f(C,t) is measurable for fixed compact C. 10. Obtain analogous concepts and results for the case of Borel-measurable set-valued mappings.

+

Remarks Filippov'e Lemma, essentially in the form (2) but for compact-valued maps, appeared in A . F. Filippov, On certain questions in the theory of optimal control, SIAM J. Control 1 (1952)

76-84.

The extension to closed-valued mappings is easily effected by the preliminary operation So in the proof of the theorem; it is needed to treat, e.g., the continuously moving linear subspaces in the Second Reciprocity Theorem. The point of isolating the Measurable Selection Theorem is that is shows, e.g. in (2), that y(t) depends on x(t) only, rather than

on This is implicitly contained in Filippov's proof, and of fundamental importance for our purposes of constructing stroboscopic strategies. The modifications appeared in the proof of Theorem 2 in ~

(

0

)

.

0. djek, Duality for differential games and optimal control, Math. Systems Theory 8 (194) 1-7.

121

PURSUIT GAMES

3.10

Richter's Theorem LEMMA 1

(1)

If

f : [0,1]+ R

lf:

{

n

i s integrable, then the s e t

measurable M c [0,11]

M has compact and convex closure. (Proof) The set i s bounded, since each member has

IJfI

1

JI'I

4

4

J If1 0

M Obviously

M thus i t s closure is compact.

0,

+

1 Jog

1 = J f + J g -

fg

J

[Oll\M

-

UJ

M

1 g = 0

1

"so' +

O

sg

M

1 (1-a)J g . 0

1 1 h E St i s 3€-close t o a x + ( 1 - z ) ~ . The 0 0 a s s e r t i o n follows on t a k i n g € t o 0. QE ,D

Hence, f i n a l l y , COROLLARY

J

If

S

i s compact, t h e n t h e sets 1 $Jtds,

have t h e same c l o s u r e s f o r a l l (Proof)

F i r s t note t h a t

cvx S

124

ss [ t

t

0

s

Sdrds 0

t > 0, namely

cvx S .

is compact, by C a r a t h 6 d o r y ' s

SrROBOSCOPlC AND ISOCHRONOUS CAPTURE

Theorem.

S c t -1 tSds, and t h e r e f o r e 0 I cvx S c closure soSds

Obviously

7

t

by Lemma 1.

t I n t h e opposite d i r e c t i o n , t-'J Sds c cvx S

follows from

0 S i m i l a r l y f o r t h e second moments; a l t e r t s t J i J i S d rd s = I - ( t - s ) S d s . Q,FD 0

Corollary 2 i n 3.8. n a t e l y , one may use

3.11 Reachable Sets

With a l i n e a r c o n t r o l system

i=

(1)

i n Rn

- u j U(t) E u

AX

one may a s s o c i a t e i t s reachable R(t)

=

[/ e-As u ( s ) d s :

sets

i n t e g r a b l e u : R1

+

to

= /;-AsUds.

Properly speaking, one ought t o refer t o t h e mapping t p R ( t ) ( t h e 'performance' of (1)). These sets a r e important f o r two r e l a t e d reasons. For any admissible c o n t r o l u ( * ) , t h e s o l u t i o n t o t h e equation i n (1)with i n i t i a l value x a t t = 0 i s x(t)

=

At

e

(x

- JO e-As u ( s ) d s )

by t h e variation-of-constants formula. p o i n t s t o which

x

Thus t h e s e t of a l l

can be steered a t t i m e

t

within (1)

( t h e set of a t t a i n a b i l i t y ) i s Ax(t)

=

At

e

(x

- R(t));

t h e e f f e c t of l i n e a r i t y i n (1)i s t h a t t h e second term i s

independent of x. Second, t h e i n i t i a l p o s i t i o n x can be s t e e r e d t o 0 a t t i m e t i f f x ( t ) = 0 f o r some u ( * ) i n

125

PURSUIT GAMES

(2); since x

eAt

ia nonsingular, the condition reduces t o

E R ( t ) . More generally, a point x can be steered t o a

target

R C R"

at

time t x

iff

E R(t) +

e-Ati2.

Obviously, i f U i n (1) i s nonvoid, or convex, symmetric or bounded, then so i s each R ( t ) j e.g., i f a. i s a bound on U and ( A 1 a matrix norm f o r A, then each point x i n R(t) has

If

U

i s compact and convex, then, by an obvious weak com-

pactness argument i n ' t h e space of controls, each R(t)

is

compact. I n point of f a c t , convexity i s not needed here (Richter's and Aumann's theorems). I n the next r e s u l t , 'span' denotes t h e l i n e a r span (the s e t of a l l l i n e a r combinations of elements i n the set, o r equivalently, t h e intersection of a l l l i n e a r subspaces containing the given s e t ) . I n (1) assume t h a t U i s convex and For s t r i c t l y positive t the following s e t s coin-

PROPOGITION 1

symmetric. cide :

span R ( t ) -AS

span {e

=

span

u: s E R

k

span {A u: k = 0, L

e

0 1

-As

Uds,

, u E q,

...,n-1,

u E

q,

-As

J,e

span

ds*

The middle two terms do not depend on t > 0; the resulting concept is the c o n t r o l l a b i l i t y space of (1). The t h i r d term provides an a l t e r n a t e characterisation: the Remarks

126

STROBOSCOPIC AND ISOCHRONOUS CAPTURE

s m a l l e s t l i n e a r subspace

which is i n v a r i a n t under t h e

L

c o e f f i c i e n t matrix of (1)and contains

The

U, A L c L 2 U.

l a s t %erm j u s t i f i e s t h e a p p e a l l a t i o n reachable s e t with 'unbounded c o n t r o l s ' .

If

U = {Bv:

t h e n t h e columns of

i s a parallelepiped,

U

i n u n i t cube of

v

span t h e s e t

B

R?,

Uj t h e t h i r d term above

shows t h a t t h e c o n t r o l l a b i l i t y space i s t h e column space of t h e (n,nm) matrix (A"-$,

...,AB,B),

An"%,

t h e ' c o n t r o l l a b i l i t y matrix' of (1). Denote t h e i n d i c a t e d sets by

(Proof)

The only

C1,C2,C3,C4.

obvious inclusions a r e

The subsequent proof proceeds by t h e f a m i l i a r technique Xc Y

orthogonal complementation:

if

and

= [y

x

u

X

A

= span X

(here X

E x)). c

Consider any

e

E

in

Rn,

then

of

Rn: y'x = 0 f o r a l l

,

I

R(t)

c'JOe -As u ( s ) d s = 0 f o r a l l admissible c o n t r o l s t r o l with constant value Then from ( 3 ) ,

(e,t].

e

u

u(-).

e

U

One p o s s i b i l i t y i s a con-

i n [O,e], and value

in

0

f c'e-*'dsu

= 0 , f o r a l l u E U and 0 By continuity, c'e A 8 ~= 0; s i n c e t h e function of

e E [O,t].

e i s a n a l y t i c , we have c'e-ABu = 0 f o r a l l u e U and 1 I I a l l e E R . T h i s shows t h a t C 1 c C2 . Repeated d i f f e r e n t i a t i o n of A

A

C2c C

3'

k = O,l,

If

c

e

C

..., since,

, then

I

3

@

k

c'e-Aeu = 0 y i e l d s

c'A u = 0

for

u

E

U

and a l l

by t h e Cayley-Hamilton Theorem, An

l i n e e r combination of

I,A,

...,An-', 127

,

& . I .

X 2 Y

is a

and hence so a r e a l l A

k

PURSUIT GAMES

for

k

n.

for

u E U

2

From t h i s ,

s E R1. This proves t h e second containment L A L C2. F i n a l l y , C 2 c C4 i s verified e a s i l y .

and A

needed f o r

C

3

=

Thus a l l t h e

coincide; hence so do t h e i r orthogonal

Ci

complements, t h e l i n e a r spaces ci. QED COROLLARY 1 The system (1)i s s a i d t o be c o n t r o l l a b l e

i f e i t h e r of t h e following equivalent conditions obtains : some

R(t)

has nonvoid i n t e r i o r ; a l l

.

(Proof)

t > 0 are

with

0; t h e c o n t r o l l a b i l i t y space coincides with

neighbourhoods of R"

R(t)

Every nonvoid, convex, symmetric s e t

neighbourhood of

within t h e l i n e a r space span

0

merely a p p l i e s t h i s i n t h e preceding r e s u l t . Symmetry i s e s s e n t i a l here: asymmetry, t h e l i n e a r span of space b u t

0

U M M A 1

t,s

0

2

U{R(t):

t

R(t)

2

QED

01

is t h e e n t i r e f o r small t > 0.

we have t h e a d d i t i o n formula

R ( t + s) = R ( t ) + e (Proof)

is a

i n t h e t y p i c a l case of

is on t h e boundary of For

s

S; one

-At

R(s).

The two needed containment r e l a t i o n s a r e obtained by

reading

so so st t

t+s

=

t+s

+

i n t h e two d i r e c t i o n s . COROLLARY 2

If

QED

i f (1) i s c o n t r o l l a b l e , then even

O r t r (Proof)

6.

For

theref ore

E = s

-t

R ( t ) c R(s) f o r 0 c t s s;

0 E U, t h e n

w e have

128

R(t) c Int R ( s ) 0

E

R(E), 0

for

E e-At R(E),

and

STROBOSCOPIC AND ISOCHRONOUS CAPTURE

R(t) = R ( t ) + O c R ( t ) + e-AtR(c) = R(s). Similarly i n t h e second assertion.

QED

According t o Proposition 3 i n 3.8, s e t s which a r e closed, convex, and symmetric may be w r i t t e n a s t h e d i r e c t sum of a l i n e a r subspace and a compact s e t .

L e t us apply t h i s decompo-

s i t i o n t o t h e control constraint s e t of (1): If

PROPOSITION 2

+L

= V

U

with V

a l i n e a r subspace, then

compact and

L

*

q t >= % ( t > + L where

Y

L i s t h e l e a s t A-invariant subspace containing L. (Proof) From Ekercise 1 i n 3.9, % ( t ) = lf,(t) + R L ( t ) * Since L i s a l i n e a r subspace, t t and

RL(t) =

0

e-AsLds = Joe-Asspan IdSj

thus, by Proposition 1 applied t o

L, RL(t)

LIMIT THEOREM ( f o r reachable s e t s )

Y

= L

If

U

.

&ED

i s compact and

convex, then

EM= U.

lim

t+o+ (Proof)

This w i l l follow from t h e estimate

t

r emAsdsUcR ( t ) c f on dividing by

t

and taking

t + 0:

The f i r s t term i n (4) describes points constant controls i n

e

-As

dsU

t-lJoe x

+ o(t2)

-As

+

0 e = I.

steered t o

0 by

U; t h i s establishes t h e f i r s t contain-

ment. For t h e second, f i r s t write 129

PURSUIT GAMES

-As

t

Given

t

+ J0emAs(u(s)-v)ds

-As ds v

e-Asds v +

t

[

0

+ J (e -As - I ) ( u ( s ) - v ) d s .

(u(s)-v)ds

0

v E U, t h e f i r s t term belongs t o J'le-AsdslJ;

u ( * ) and

t h e l a s t i s estimated by

t J' (elAlS-1)6ds = o ( t 2 ) 0

with

6 = diam U

t

0,

as a

a sharper r e s u l t appears i n

Proposition 1 of 4.2.)

5.

Prove

w(2t) c ( R J ~ )+ e

-At

-At 0) ( ( ~ ~ (+t e)

and generate an analogous formula f o r

138

* ~ ~ ( t ) ) ~)

W(3t).

f

~ ( t ) ,

ISOCHRONOUS CAPTURE

*

6. Setting W ( t ) t,s

for

2

= eA%(t),

show

O j conclude t h a t

l i m sup

W*(t+s) it W(t) S

wo+

l i m sup W + ( s ) c p % Q . Ewe+

7.

Prove t h a t , i f

8. Suppose t h a t

P c Q + V with

V

compact, then

P i s t h e square -1 s x,y s 1 i n

R

,

2

Q a symmetric segment w i t h i n P. Construct sets V such t h a t P c Q + V. Observe t h a t t h e r e a r e i r r e d u c i b l e sets and

V, and e x k i b i t a t l e a s t two q u i t e d i s t i n c t ones.

9.

Show t h a t i f

d i r e c t summand of

P3Q

L e t t h e t a r g e t set

10.

a r e both segments, then

Q is a

P. R be t h e a f f i n e manifold

b + N.

Prove t h a t W(t) + e'AtNc

W(t).

Caref'ully prove t h e following a l t e r n a t i v e : W(t) =

$ ,or

R

either a l l

i s a single point, or a l l W ( t )

a r e non-com-

pact. 11. Assume t h a t a winning set W ( t ) contains an e n t i r e + n ray x + R c ( x E W(t), c E R ), R = b + N i s an a f f i n e

manifold, and P,Q c E e- A t N (Hint:

.

by

8

and t a k e

8

a r e compact.

Prove t h a t n e c e s s a r i l y

i n the relation

++

x

+ 8c E W(t) divide

m.)

Remarks A s mentioned i n t h e proof of Lemma 2, t h e i n c l u s i o n (3)

i s a s e t - t h e o r e t i c v e r s i o n of t h e p r i n c i p l e of suboptimality; and t h e r e i s a corresponding statement involving ofiimel 139

PURSUIT GAMES

s t r a t e g i e s and minimal times. Exercise 5 extended t o W(kt) f o r general k = O,l, corresponds s i m i l a r l y t o ‘repeated

...

min-max’ constructions a s i n Section 1.3 of A. Friedman:

D i f f e r e n t i a l Games, Wiley-Interscience, New York, e t c . , 1971.

Returning t o Lemma 2, assume R = 0 t o simplify matters, and c a l l (3) t h e hereditary property f o r the s e t s W(e). According t o ( 6 ) , W ( t ) c Rp 4, R p ( t ) , but the l a t t e r s e t s do not have the hereditary property.

Indeed, f o r the harmonic

o s c i l l a t o r w i t h unit segments f o r excepted) reachable s e t s a t time

P and Q, ( t r i v i a l cases TI coincide with t h e unit

c i r c l e , so Rp(t)” but

RQ(?) = 0

$3

t E (0,n).

R p ( t ) E R ( t ) = fd f o r a l l

Q

4

The isochronous winning s e t s f o r the game

CONJECTURE

(1)are the largest s e t s with t h e hereditary property: t h e set-valued mapping t H St s a t i s f i e s So = 0 and So

then St

+

R (t)

Q

St c W ( t )

c Rp(t)

e

+

for a l l t

2

-AtS

0.

e-t

whenever 0 s t

if

5 -9,

(Note t h a t automatically

c Rp(t) ii R Q ( t ) . )

4.2

Necessary Conditions Here we develop two variants of a necessary condition

f o r isochronous capture.

The first, Proposition 1, i s obtain-

ed by simply omitting the q u a l i f i e r ‘stroboscopically’ i n t h e Necessary Condition from 3.4.

The second, Propositions 2 and

3, i s related t o t h e results of 3.5 on unions of target s e t s . The notation of 4.1, and the underlying game, i s retained.

It w i l l be convenient t o i s o l a t e a technical portion of

t h e reasoning. LEMMA Assume t h a t a strategy u 140

and a terminal time

ISOCHRONOUS CAPTURE

e

5

0

have been fixed.

and time t E [O,e],

For any two points

let

st

q,

90

in

Q,

be the piecewise constant quarry

control,

st

=

90 i n

~t

[o,e-tl,

= q i n (e-t,el;

the pursuer response; and, f o r an i n i t i a l posi-

pt = a [ % ] tion x,

Yt =

(1)

t h e s t a t e response endpoint a t

*

At

where R ( t ) = e P

-

Rp(t)

8.

Then

i s t h e reachable s e t of the control

p, p ( t ) E P. system G = -AX (Proof) The e s s e n t i a l point here i s t h a t

u

is non-anticipa-

tory, so t h a t Pt =

“[%I

u[%I

=

=

po a.e.

[o,e-tl;

the v e r i f i c a t i o n i s then straightforward, using

e

e-t

e

so = s o - se-t

*

QJm

An obvioue but important observation i s t h a t the assertion i s vacuous unless so t h a t

yt

=

yo

8

>

0:

e

if

= 0,

the only possible t = 0,

i n (2).

is compact and convex, and t h e t a r g e t s e t i s contained i n t h e affine manifold [x: Mx = Mb]. If n o n t r i v i a l isochronous capture i s possible, NECESSARY CONDITION

i.e.,

if

W(e)

4

Assume t h a t

f o r some

8

M(Q

P

> 0, then necessarily

- Q) z M(P - P).

141

PURSUIT GAMES

Moreover,

( 31

quarry c o n t r o l order 2 pursuer c o n t r o l order,

and, i f

k

denotes t h e l a t t e r ,

(41

-

MAk”(Q

Q) c MAk-’(

P

- P).

(Proof)

BY assumption, some s t r a t e g y u f o r c e s a point x E W(@) t o R a t time 8. Then, i n ( 2 ) , t h e endpoints belong t o by

R, so t h a t

M(yt

- yo)

=

and l e t t h e q ’ s range Over

M

(5)

t E (o,el. Here divide by

Thus, i f we premultiply

Q,

- Q) c M(R*p(t) - R*p ( t ) )

(Q

Ml:eAeds

0.

yt

for a l l

t

t -+ 0.

and t a k e

Theorem f o r t h e reachable sets

(6 1

M(Q

*

Rp,

- Q ) c M(P - P).

I n case t h e pursuer c o n t r o l order

result (4) p r e c i s e l y .

(7) hence

M(Q

- Q)

For

Returning t o (5),

k = 1, t h i s i s t h e required

k = 2 we have M(P

= 0

Using t h e L i m i t

-

P) = 0,

from (5); i n p a r t i c u l a r , (3) holds. p o i n t s i n t h e second member may be

w r i tten

with t h e first term vanishing according t o (7).

with l i m i t i n

1 lii; MA(P-P).

Hence

A simpler argument f o r t h e f i r s t

142

ISOCHRONOUSCAPTURE

member of (5) t h e n y i e l d s

-12 MA(Q - Q ) i . e . , (4). induction.

C

The case of g e n e r a l k

1

MA(P

- P),

i s obtained by an obvious

QED

The l a s t r e s u l t s t r e a t t a r g e t sets of t h e form

r where each Fk: Rn

%.

hood of

near any Y

+

i s of c l a s s

Rm

Then each

E

Fk

9,

(9 1

F ~ ( x )=

o

has a f i n i t e Taylor expansion

+ D F ~ ( Y ) ( X - Y+)a( [ r - y l ) a s x + y .

is t h e Jacobian matrix of

Here DFk

i n some neighbour-

C1

Fk

(of type (%,n),

e n t r i e s a r e p a r t i a l d e r i v a t i v e of t h e coordinates of

Fk).

L e t us a l s o denote Nk(Y) =

{X:

Fk(Y)x = 0 ) )

t h e tangent space of {Fk = O} a t y. PROPOSITION 1 Assume t h a t P i s compact and convex, and

Q

Fk E C1 near \. If t h e r e x E W(e) with 8 > 0, and y E R

i s a s i n (8) with

e x i s t winning p o s i t i o n s

i s a s t a t e response endpoint t o a constant quarry c o n t r o l € Q, then

r

(Proof) Referring t o (1) and (2), l e t (2),

yt

+

y

I n (2),

t

+

= q, yo = y .

0, so t h a t ( 9 ) w i l l apply if y d i v i d e by t > 0. Obviously t h e term

as

E

From

4.

A s f o r t h e elements of t h e next term there, t h e L i m i t Theorem 143

PURSUIT GAMES

%Y

applied t o t h e reachable sets

t P

=

-

t . + Ot J

such t h a t t h e e n t r i e s converge t o a point of

- y)/t

But then (yt

P.

y i e l d s a sequence

w i l l a l s o have a f i n i t e limit, i n

q,-q+P-P.

belong t o R = u $. Then, j f o r some f i x e d k and subsequence t = t we have a l l i’ y t E Rk whereupon y = l i m yt E a l s o . Then ( 9 ) holds f o r with F (y ) = 0. Divide by t t o o b t a i n x = y t k t Yt-Y IYt-Y o = DF~(Y) + t lJ(l), Finally, the points

yt = yt

,

\

,

I

-

w i t h t h e l a s t term vanishing i n t h e limit.

Thus lim(yt-y)/t

E

r

Now merely l e t Example

q,

Q.

vary over

Q,ED

(Capture with two pursuers)

Consider a game with

p a r t i t i o n e d dynamical equation

= y o r x2 = y, and player con1 U1, U2, V (nonvoid, compact, convex, symmetric).

termination condition s t r a i n t sets

x

We need not assume t h a t t h e dynamical equation i s a c t u a l l y uncoupled (i.e., t h a t

.

is appropriately block-diagonal) The Necessary Condition i s not d i r e c t l y applicable, s i n c e t h e t a r g e t set R = L u L2 i s a union of l i n e a r subspaces, 1 and hence i s not convex ( t r i v i a l cases expected). Enlarging R t o i t s convex h u l l y i e l d s no information a t a l l : cvx R i s A

t h e e n t i r e s t a t e space. I n applying Proposition 1, l e t endpoint t o any f i x e d constant

y E R

v € V.

144

be a s t a t e response

Then

Nk(y) =

4 is

ISOCHRONOUS CAPTURE

independent of

y, so t h a t Proposition 1 y i e l d s

Q (we have l e t

q

-

vary Over

QC

P

-

also).

Q

P + (L1 U L2>

By convexity and

symmetry, then QC P +

(L1 U L2> = (P

+

4)U ( P

+

L2>*

Here

so t h a t , i n t h e n a t a t i o n of our game,

vc

u1

u

u2

i s a necessary condition f o r isochronous capture.

In particu-

l a r , i f U1 = U2 = U, then isochronous capture by t h e two pursuers i s ruled out u n l e s s t h e necessary condition V c U for a single pursuer ( t h e o t h e r i n a c t i v e ) is s a t i s f i e d . F i n a l l y we t r e a t one case where (10) provides no informat i o n because, i n a sense, t h e players have c o n t r o l orders higher than 1. PROPOSITION 2

and

r R = U (bk + $) k=1

Assume t h a t

P i s a compact and convex,

a f i n i t e union of a f f i n e manifolds.

Q - Q cn $ 2 P - P , k=1

then (12)

A(Q

r

- Q ) c A(P - P) + k=l u $

i s a necessary condition f o r n o n t r i v i a l isochronous termination.

(Proof) Choose matrices

l$

so t h a t

whereupon (11)y i e l d s 145

4=

{x: l$x

= 0’)

,

If

PURSUIT GAMES

-

%(Q

Q)

= 0 = %(P

- P)

t = t. + 0 J and an index k such t h a t a l l yt E R belong t o bk + $. Then so does y = l i m yt, and Z ( y t y ) = 0. I n ( 2 ) premultiply by Z, d i v i d e by t2 and t a k e t = t . + 0: J for a l l

k.

Referring t o ( 2 ) , f i n d a sequence

-

(with

k

p o s s i b l y depending on

qe,

and one need only l e t

90

q,,

q).

vary Over

I n any case r Q t o obtain (12).

BED Exercises 1. I n Proposition 1 omit t h e assumption on

R, and con-

clude, i n place of (lo), t h a t Q

where

Ry,

t h e tangent set of

f i n i t e limits of 2.

- q C P - P + RY

%(yk

Prove t h a t

R Y

- y)

R

as

at

y E R, c o n s i s t i n g of a l l

% + + 03. i s such; show t h a t ,

yk + Y, yk E

i s convex i f

R

f o r g e n e r a l t a r g e t s , t h e tangent s e t need not be convex.

3.

Show t h a t , i n ( l o ) , one need only t a k e unions over

those indices

4.

k

f o r which y E

%.

Consider a game with two pursers and one quarry, with

n-dimensional equations of motion

..5 = %, ..y = v,

and termin-

o r y = x2' Show t h a t V c U1 u U2 i s a necessary condition on t h e c o n s t r a i n t s e t s (compact, cona t i o n condition y = x1

vex, symmetric) f o r n o n t r i v i a l isochronous capture. mediate answer:

(Inter-

%A = (0 I 0 0 0 -I) i n t h e n o t a t i o n from

t h e proof of t h e lemma.)

5. Treat games analogous t o t h e preceding, with one o r two of t h e players having c o n t r o l order 1 r a t h e r than 2. 146

ISOCHRONOUS CAPTURE

6 . Consider yet another type of game with two pursuers and one quarry, = Ax

- p1 - p2

+

9; pk(t> E Pk, q(t) E Q,

with x E 4, as the k-th pursuer goal. Assume that 4, are linear subspaces, and Pk,Q nonvoid, compact, convex, symetric. In terms of the first-order necessary conditions (i.e., (4) for k = 1) compare the following situations: the first pursuer decides to treat the other as added nuisance; both pursuers do this; the pursuers cooperate. (Warning: the last implies that both goals are reached simultaneously.) Remarks The exposition in 4.1 and 4.2 is based on two eponymous sections of 0. Ha’jek:

Lectures on Linear Pursuit Games (unpublished) Case Western Reserve University,

1973*

,

-

The reader of these lecture notes should be warned that a questionable assertion appears there: Q c P + R1R is necessary for isochronous capture if P,Q are compact, convex and symmetric, but R is not necessarily convex. If true, this would provide a simpler apparatus for treating the example of several pursuers in 4.2; the proof is certainly false. As for further developments, it would be desirable to treat the situation of (8), i.e., finite unions of C1 targets, for higher player control orders than 1 and 2; even the case r = 1 is interesting. A more ambitious project is suggested by CONJECTURE 1 Isochronous capture is stroboscopic: if a position can be forced to termination at time t > 0, then some stroboscopic strategy has the same effect. CONJECTURE 2 For each non-anticipatory strategy, the 147

ISOCHRONOUSCAPTURE

set of a l l s t a t e response endpoints, a t given time and from g i v e n i n i t i a l p o s i t i o n , i s convex ( o r a t l e a s t has convex closure )

.

For t h e first, it might b e useful t o experiment w i t h affine strategies,

t

~ [ q I ( t )=

U(t>

for g i v e n u, and matrix-valued

148

W.

+

J

dsW(t,s) 0

q(6)

CHAPTER V

I n a game it may happen t h a t some p o s i t i o n s can be forced t o the t a r g e t , but not isochronously.

A p a r t i c u l a r l y simple

case i s t h e one-dimensional rocket chase of 1.3: every point can be forced t o termination, b u t none isochronously (see Exercise 1 i n 3.4 and t h e Necessary Condition of 4.2). Thus, t h e r e i s a need t o study g e n e r a l capture.

That

t h i s i s not a simple extension of previous results is s i g n a l l -

ed by the f a c t t h a t t h e sets of winning p o s i t i o n s a r e not convex even when t h e y n a t u r a l l y ought t o be: compare Lemma 1 of 4.1 with Exercise 7 i n 5.1. Our results here are, necess a r i l y , much more modest. Material analogous t o t h a t of Sections 4.2 and 3.4 i s presented i n 5.1 and 5.2. The remaining two s e c t i o n s concern s p e c i a l cases:

t a r g e t s e t s which a r e e i t h e r very large,

namely e n t i r e half-spaces, 0

a s one p o s s i b l i t y .

o r s p e c i a l l i n e a r subspaces, with

I n both s i t u a t i o n s it t u r n s out t h a t

capture i s n e c e s s a r i l y isochronous, w(o,e) =

u

cktse

w(t>

i n t h e n o t a t i o n of 2.2. The two cases are, i n a sense, opposite extremes: i n the f i r s t i t sometimes happens t h a t W(0,e) = Rn

for a finite time

8; i n t h e second, u s u a l l y

W(0,t) = W ( t ) .

5.1 Necessary Conditions For t h e study of g e n e r a l capture, t h e appropriate concept is t h a t

of winning position, and w i n n i n g s e t W(O,B), 149

PURSUIT GAMES

introduced i n 2.2.

For l i n e a r games

-

= Ax P + qi ~ ( tE )PI q ( t ) E Qi end: x E R: state space: R",

(1) t h e winning s e t

c o n s i s t s of a l l p o s i t i o n s

W(0,B)

x

E

Rn

to

such

which t h e r e corresponds a non-anticipatory s t r a t e g y u that At x ( t ) = e (x

(2)

SO

e

-As

(u[ql(s)

- q(s))ds) E R

q ( * ) and some termination time

f o r each quarry c o n t r o l

t E

-

[o.el.

The attempt t o o b t a i n r e s u l t s p a r a l l e l t o Lemma 2 and Corollary 1 from 4.1 n a t u r a l l y l e a d s to t h e conaept of minimilm time of winning, and t o t h e p r i n c i p l e of suboptimality i n 2.3.

4

x a r e those w i t h x R (usage d i f f e r i n g from t h a t of Chapter 4, where t h e term refer-

The n o n t r i v i a l winning p o s i t i o n s red t o

x € W(e)

LEMMA

with

0

> 0,

but

P,Q

a r e compact, and

Assume t h a t

x

E

R

allowed). R

closed.

I f t h e r e e x i s t n o n t r i v i a l winning p o s i t i o n s , then even l i m (wto,t)\n>

t+O+ (Proof:

4 $.

d i r e c t consequence of Corollary 1 i n 2.3.)

Consider again t h e one-dimensional homicidal chauffeur game; i n p a r t i c u l a r , t h e d i s p o s i t i o n of i t s winning sets W(0,t)

f o r small t i m e s

t

> 0,

f i g . 4 i n 1.3.

We w i l l show

t h a t , i n an appropriate sense, t h i s s i t u a t i o n i s q u i t e genera l ( s e e t h e Corollary, and Exercise

7).

f o r n o n t r i v i a l capture w i l l be developed

A necessary condition

- obviously t h i s must

be weaker than t h e v e r s i o n i n 4.2 applying t o isochronous capture.

Then, by modifying it s l i g h t l y , a s u f f i c i e n t ' l o c a l '

condition w i l l be obtained i n 5.2. THEORFlM

I n t h e game ( 1 ) assume t h a t t h e p l a y e r s '

150

CAPTURE

c o n s t r a i n t sets a r e compact, convex, and symmetric, and t h a t t h e t a r g e t i s contained w i t h i n t h e a f f i n e manifold [x:

Mx = Mb]

.

Then

(3 1

MQc

holds f o r every point

MP

+ [-1,1] MAX

x E l i m (w(o,t)\n).

ho+

(Proof) F i r s t consider any point x E w(o,B)\R, and quarry c o n t r o l value q E Q. On choosing a winning s t r a t e g y , f o r t h e constant quarry c o n t r o l q we have, from (2), t h a t a.

MeAt(,

+ SbemAsds q ) 0

4

R (see Exert E [O,e]j f'urthermore, t > 0 s i n c e x cise 2 i n 2.3). Analogously for the quarry c o n t r o l -9: r MeAr(x Joe -AS ds q) E Mb + MeArRp(r), 0 5 r s 0.

with

-

NOW

t + r > 0 and rearrange, using

s u b t r a c t , d i v i d e by E [0,11:

A = t/(t+r)

+

t

M(eA thF 1 J e-Asas q + eAr(l-h)$ 0 A t Rp(t) M(e A

t - e

&

Jr e-Asas

Rp(r 1 (1-h) -). r

q)

E

0

Now r e t u r n t o t h e a s s e r t i o n ; take a point

Apply (4) t o t h e verge t o

0

5; t h e

since

Ok

+

tk,rk con[O,ll w i l l The Limit

r e s u l t i n g capture times 0.

The weights

X = Xk

have a convergent subsequence h . + 11 E [0,11. J Theorem f o r reachable s e t s then y i e l d s

151

e

PURSUIT GAMES

- (l-P)Ax)

M(W&

+

M(C19

+

(l-P)q)

(&-1)MAx + M q from t h e assumptions on and

0

-1 s

x

- (l-P)P),

MP

and

q

a r e independent,

1 y i e l d s constant bounds f o r t h e s c a l a r factor,

g CL L

a-14

Here

P.

E

E M(WP

1. &ED

NECESSARY CONDITIONS

Under t h e assumptions of t h e

theorem, [ - i , i ~ c f o r some

M Q CMP +

aim

MQ s 1

E

c

R ~ ,

+ aim MP

a r e necessary f o r n o n t r i v i a l capture.

(Proof) I f some W(0,e) points

x

=/

E l i m (W(O,e)\R,

Example

R, then, by t h e lemma, there e x i s t

and we may take

c

=

MAX.

QJ3D

The n-dimensional equations of motion a r e

..x + A ~ + B x = u ,

with pursuer's constraint s e t

U

$=

Q-V,

and quarry's

The t e r -

V.

mination condition i s perfect capture, x = y. The appropriate s t a t e space i s X,k,y)j t h e matrix

M

(with coordinates

F?"

is then ( I 0 -I); P consists of vect-

ors with components O,u,O,

and

Q

of those with

O,O,v.

MQ = V, MP = 0, so t h a t by t h e condition above, non-

Thus

t r i v i a l capture cannot occur unless

V

i s a segment.

A very

.

s p e c i a l case of t h i s i s t h e i s o t r o p i c rocket game

..

x + a x = u, y = v in

R2, with t h e d i s c f o r quarry's constraint s e t :

perfect

capture i s impossible. COROLLARY

l i n e a r subspace tives:

I f , i n t h e theorem, t h e t a r g e t s e t i s t h e [x: Mx = O}, w e have t h e following alterna-

e i t h e r t h e necessary condition M Q c MP f o r iso-

chronous capture obtains, or, f o r small capture times, t h e r e a r e no n o n t r i v i a l winning positions near t h e origin: 152

CAPTURE

(5)

nG

W(0,c)

f o r some

c

(Proof)

>

0

and neighbourhood

Indeed, e i t h e r

x = 0

c R

G

of

0.

i s allowed i n ( 3 ) , o r

so t h a t ( 5 ) holds.

not belong t o l i m (W(O,t)\R),

does

0

Q,ED

Exercises 1. Show t h a t , i f

W(0,t)

and

P,Q,R and

W(O,t)\R,

a r e symmetric, then so a r e If

T(x) = T(-x).

P,Q,R

a r e com-

i s bounded, and an e x p l i c i t bound i s

pact, then each W(0,t) e a s i l y exhibited. Assume t h a t

2.

{x: M x = Mb], j N = {x: Mx = 01, L = {x: MA x = 0 f o r j = 0,1, 3.

and set

Prove t h a t W(s)

=/

i s t h e a f f i n e manifold

R

W(0,t) + L c W(0,t); i f

W(0,t)

...

contains some

$, t h e n even W(0,t) + e - A s N c W(0,t).

3.

Let

R,N

be a s i n Exercise 2, and

P,Q

+ R+c

Show t h a t , i f W(0,B) contains a r a y x -At then c E e N f o r some t E (O,e]. (Hint:

compact.

with

x

4

R,

Exercise 11 i n

4.13 be r a t h e r c a r e f u l i n checking t h a t t > 0 . ) 4. Obtain a version of Lemma 2 from 4 . 1 applying t o g e n e r a l capture: each point

x

P,Q

if

i n W(O,e)\R X

f o r small t

5. that, i f Md = 0

6.

2

0.

+

(Hint:

a r e compact and

R

closed, then

satisfies

RQ ( t ) c Rp(t) + e-

At

W(0,e-t)

Exercise 2 i n 2 . 3 )

Discover unused assumptions i n t h e theorem, and prove Q contains a r a y with d i r e c t i o n or a U W(O,t) = R.

d, then e i t h e r

Sharpen t h e second a l t e r n a t i v e i n t h e c o r o l l a r y t o :

t h e r e a r e no winning p o s i t i o n s c l o s e t o t h e set

7.

Prove t h a t , i f t h e f i r s t a l t e r n a t i v e i s t h e c o r o l l a r y

does not apply, t h e n t h e winning sets

153

W(0,t)

cannot be

PURSUIT GAMES

convex f o r small t

>0

unless t h e y a r e t r i v i a l .

(Hint:

Exercise 1.) Remarks The elements of t h e a e t

l i m (W(O,t)\n),

hc+

which appeared

n a t u r a l l y i n t h e course of developing t h e Necessary Condition,

w i l l be s t u d i e d i n g r e a t e r d e t a i l and g e n e r a l i t y i n Chapter VII

.

I n t h e game with n-dimensional equations of motion

..x = y

- q,

y = p; end: x = y

a l l p o s i t i o n s a r e winning ones i f

n = 1 ( t h e one-dimensional

homicidal chauffeur), but, according t o t h e example, f o r n

>1

t h e r e a r e no winning p o a i t i o n s o t h e r t h m t r i v i a l ones. Though t h i s might be blamed on t h e n a t u r a l p e r v e r s i t y of t h e number 1, probably a b e t t e r reason i s t h a t t h e t a r g e t set

n = 1 but not otherwise; t h e case of s e p a r a t i o n w i l l be t r e a t e d i n 5.3. (For a nonlinear s i t u a t i o n see Example 1 i n 7.3 and 7 . 2 . )

s e p a r a t e s t h e s t a t e spece i f

The second a l t e r n a t i v e of t h e c o r o l l a r y requires t h a t , i n t h e t e r m i n a l s t a g e of s u c c e s s f u l capture, t h e s t a t e v a r i a b l e b e r e l a t i v e l y large; e.g.,

i n one-dimensional rocket chase,

t h e speed is a t l e a s t 1 terminally.

Actually, t h e phenomenon

r e c u r s i n proposed winning s t r a t e g i e s f o r specific games. See, e . g . , references under 'swerve manoeuvre' i n R. Isaacs:

D i f f e r e n t i a l Games, Wiley, New York, e t c . ,

1967

CONJECTURE

I n t h e game (1)assume t h a t

P,R a r e comis a l i n e a r subl i m (W(O,t)\n), then so

pact, convex, and symmetric, and t h a t space.

If a point

x belongs t o

does t h e e n t i r e r a y [l,+m)x.

R

ho+

A d i f f e r e n t type of necessary condition was presented i n

154

CAPTURE

L. S. Pontrjagin, E. F. Migzenko:

The problem of escape of one c o n t r o l l e d object from another, Doklady Akad. Nauk SSSR 189 (1$9), 721-723;

i n our n o t a t i o n it can be formulated a s follows.

I n (1)l e t

P,Q be compact and convex, and t h e t a r g e t t h e l i n e a r subspace Ex: Mx = O}, with M of type (m,n) and m 2 2. If

t h e n n o n t r i v i a l capture is impossible (furthermore, t o every position

x w i t h small Mx

t h e r e corresponds an i n d i f f e r e n t

evasion strategy, and lower estimates of Note t h e unexpected order of

and

Q

P

IMx(t)

I

a r e given).

i n (6); and a l s o t h e

innocuous b u t s i g n i f i c a n t assumption m 2 2: i n t h e case m = 1 n o n t r i v i a l capture i s u s u a l l y p o s s i b l e according t o 5.3.

5.2 S u f f i c i e n t Conditions I n t h e preceding s e c t i o n a necessary condition f o r nont r i v i a l capture was obtained, applying t o t h e l i n e a r game (1)

= Ax

-P

+ q; ~ ( t E) P, q ( t ) E Q Mx = Mb; s t a t e space: Rn

end: (M i s an (m,n) matrix).

The condition may be r e w r i t t e n as

M Q c M(P

with x

d =

-

Ax

+

[0,2]Ax) c M(P

- Ax - R'd)

-AX; t h i s i s t o hold f o r a l l p o i n t s

E lim(W(O,t)\n).

The t r a n s i t i o n from t h i s t o a s u f f i c i e n t

condition is modelled on t h e f a r simpler case of Corollary 2 i n 3.4.

The b a s i c idea i n t h e proof i s 'constant bearing

navigation,'

properly formulated i n Corollary 1.

THEOREM I n t h e game (1)l e t t h e p l a y e r s ' c o n s t r a i n t sets be nonvoid and compact. If a point x E R s a t i s f i e s (2)

M Q c I n t M(P

155

- Ax - R"d)

PURSUIT GAMES

( i n t e r i o r i n Rm) f o r some vector t r i v i a l winning positions near Md

4 0,

d E Rn,

then t h e r e a r e non-

More precisely, i f a l s o

x.

then

(3)

+ ( 0 , 6 ) d c I n t (W(O,e)\R)

x

holds f o r small (Proof)

~

>6 0.

9

is compact, t h e term R+d

Since MQ

i n ( 2 ) can be

replaced by [a,P]d w i t h 0 C a C p + 0 0 . For t h e same reason, t h e containment i s preserved i f t h e e n t r i e s a r e varied slightly:

x

replaced by

y

small; note t h a t then perturbed :

y

6

t

4

x + cd

R ) j and t h e term

- Ay) -

MeAeQcMeAe(P

[0,9] and 7\ >

(for 0 < 5 < 6 Q

and

P

-

Ay

[a,P]Md

for a l l

8 E

small enough.

is t h a t = 0.

8 H eAe i s continuous, w i t h i n i t i a l value

0

with

The point here I

at

e

Thus, f o r each

q E Q

p E P and h E [a,P]

and

8 E [0,91,

such t h a t A8

Me

A8

q = Me

(p

-

Ay)

there e x i s t s

- AMd.

Now use Filippov's Lemma from 3.9 t o describe these solutions p,k

as values of suitable mappings 0:

Qx R

1

4

P,

which preserve measurability.

(9:

1

Q x R + [a,Pl

We propose t o show t h a t o

is

a winning strategy (stroboscopic but not isochronous) f o r y as i n i t i a l position. If

0L s L t

L

t

7

and

Now integrate, JOdS, using

y = x + Cd, Mx = Mb

q(-)

J

t

i s any quarry control, then

eA(t-S)(-Ay)ds = eAty

0

t o obtain

156

-y

and

CAPTURE

= Mb

- J0~(q(S),t-s)dS)Md.

+ (5

This has been assembled i n t o t h e form

(4 1

w ( t ) = Mb + Jl(t)Md, y ( * ) i s t h e s t a t e response f’rom i n i t i a l p o s i t i o n yo

where and

Jl

Obviously JI

denotes t h e bracketed term.

=/

valued, continuous s i n c e y(.)

i s such and

remains t o show t h a t

f o r small t > 0

pending on t(t) = 5

-

It

0.

(possibly de-

q(0)). Since u g c p ( - ) 5 p, t h e function t J’cp(*) s a t i s f i e s $(o) = > 0, $ ( t ) 5 5 at.

Thus indeed

7).

$(t) = 0

Md

i s real-

0

-

$(t) = 0

for some t E (O,E], where

This shows t h a t t h e p o i n t s

i n W(0,E) and o u t s i d e R.

x

Letting

+ d

sd

with 0

E

= min(6/u,

l s 1.1. 4 The s t a t e space i s R , t h e observation matrix M = (I,-I) With t h e notation from the theorem, the sets has t y p e (2,4). MP and MQ a r e as indicated i n f i g . 2 j i n particular, isochronous capture i s impossible (see 4.2). Next, MA=

(

1

o

0

1

-1

o)(,o 0

0 - 1

1

o o)=(o

0

0

0

0

1

o

0)j

0

0

0

0 0 0 0 0 0 0 0

thus a point 39x4

x

i n the t a r g e t set, w i t h coordinates x1,x2,

has

Fig. 2 I l l u s t r a t i o n of necessary, sufficient conditions f o r capture. 159

PURSUIT GAMES

For the Necessary Condition of 5.1 we investigate all the shifts of MP by vectors between MAX and -MAX. Referring to fig. 2 again, these will cover MA if Ix21 2 1: the necesary condition for capture. The sufficient condition from the theorem is also illustrated there; 1x21 > 1 is needed, and the directions d are obvious. Mere presence of nontrivial winning positions is easiest obtained from Corollary 2. Exercise For the game with n-dimensional equations of motion

x

=

Ax

- p,

= By

- q; end:

x

=

y

find conditions for presence of nontrivial winning positions. (Remark: the case A = B is treated separately as Exercise 2 of 3.1.)

Remarks This section is based on 0. Ha'jek:

The principle of constant observation (submitted)

The form of the sufficient condition in the theorem suggests that the necessary condition from 5.1 is reasonably sharp, that there are no further independent necessary conditions (for presence of nontrivial winning positions; if one also demands complete capture, further requirements will appear). The geometric interpretation (fig. 1) is that, in the terminal stage of capture, the pursuer adopt8 constant bear3navigation, of maintaining constant direction of the observed velocity vector &(t). E.g., (3) in 5.1 may be read as I&=

M(Ax

160

- p + q) E dM'R

CAPTURE

for

a

=

-AX.

The p r i n c i p l e extends t o nonlinear games: see 7.3 and 7 . 2 j t h e formulation applying t o games without separation of t h e players is somewhat unexpected (Theorem 1 i n 7.3). Neither result a p p l i e s w e l l t o second-order games, f o r instance t o t h e example i n 3.4. A necessary condition follows from more g e n e r a l considerations i n 7.3: f o r t h e l i n e a r game

(5)

..x + A .x + B x = p - q j e n d :

with assumptions on

P

and

Q, t h i s reads

M Q c MP + [-l,l](&+Bx)

f o r points ( x , : )

in

Mx=Mb,

lim(w(o,t)\n);

+R

k

i n particular,

dim MQ < 2 + dim MP. A s u f f i c i e n t condition f o r t h i s game i s attempted i n t h e re-

ference.

It would be very d e s i r a b l e t o obtain usef'ul condi-

t i o n s f o r a r b i t r a r y player orders, a s was c a r r i e d out i n t h e case of isochronous capture. A second p r o j e c t concerns 'bracketing' of quarry by

Thus, l e t t h e n-dimensional (n

s e v e r a l weaker pursuers.

> 1)

equations of motion be ;(k =

%

(1s k

5

r),

$

= v,

I \ ( t ) l s 1, I v ( t ) ( s 2, and some 5 = y Are t h e r e any n o n t r i v i a l winning p o s i t i o n s ?

say with c o n s t r a i n t s

a t termination.

Isochronous capture has been r u l e d out by t h e example i n 4.2. One conclusion from 5.1 and 5.2 i s t h a t many one-dimensional ('rectilinear',

i . e . , with rank M

= 1) games a r e

winn-

able, while most planar games a r e not unless they a r e of s p e c i a l form such a s (5).

This might be made more p r e c i s e by

defining an appropriate manifold, whose elements a r e games i n

fixed dimensions, and showing t h a t t h e winnable games constit u t e a set which does, o r does not, have nonvoid i n t e r i o r . 161

PURSUIT GAMES

5.3

Large Targets The games t r e a t e d i n t h i s section, = AX

(1)

- P + qi

have r a t h e r large t a r g e t s :

P ( t ) E P, q ( t ) E Q;

e n t i r e half spaces of the s t a t e

space. Obviously t h e r e is a closely r e l a t e d s i t u a t i o n where the t a r g e t i s the boundary hyperplane {x: c'x = u} (or the s l a b described by

Ic'xI s

C,

etc.)

Indeed, every n o n t r i v i a l

position is forced t o t h e hyperplane; conversely, forcing t o the hyperplane i s equivalent t o forcing t o one of t h e halfspaces {x:

c'x

5

u}, {x:

c'x

2

u]

depending on t h e i n i t i a l position. I n the terminology of 6.1 t h e hyperplane t a r g e t s correspond t o game spaces of dimension

5.1 i s no r e s t r i c t i o n no information). It w i l l be convenient t o introduce a t the outset some common assumptions and notation. 1; f o r these t h e Necessary Condition of

. ., provides

(i e

ASSUMPMONS

I n (1) the player constraint sets are non-

void and compact; t h e normal vector

c

4 0.

Using Filippav's Lemma, choose player controls :(a), 1 q(.) which maximise i n t h e sense t h a t , f o r a l l t E R AtAt t At(2) c'e p ( t ) = max c t e p, c e q ( t ) = max cteAtq. PEP qEQ 1 Define ct 0 i n Rn and ut E R by t At c' = c'e t ' "t = u + ~ o c : ~ ( s ) ';i(s))de.

-

,

-

,

eAsr not our usual e-AS. one reason f o r this is t h a t t h e condition f o r s t a t e response endpoint x ( t ) E R may usef'ully be rewritten 8s

Note t h R t t h e integrand contains

162

CAPTURE

t

(4)

t

I n t h i s notation, ( 2 ) i s

(5)

LEMMA 1 For each t

2

0, the isochronous winning s e t

W(t) = {x:

CCX

5;

ut’l;

f’urthermore, u ( s ) = p ( t - s ) defines an indifferent (hence, stroboscopic) isochronous winning strategy, independent of positions x E W(t). (Proof) F i r s t take any x E Rn i n t h e indicated half-space, and use t h e strategy u . Then the s t a t e endpoint x ( t ) , responding t o any qunrry control q ( * ) , has t

t

(we have estimated the l a s t term via (5)); continue, using ( 3 ) , t o obtain = u

+ (c{x

- at) s u +

0.

R a t t. This establishes one inclusion i n t h e assertion. For the other, take any x E W(t), and consider any winning isochronous strategy u . Then x ( t ) E 0, so t h a t (4) holds with p = u[q] f o r each admissible quarry control. I n particular, we may use q ( s ) = T(t-s), and estimate the second term v l a (5) t o obtain Thus indeed

u

forces x

T h u s indeed

ut

2 CCX.

to

163

PURSUIT GAMES

LEMMA 2

For each winning p o s i t i o n x

outside

R, t h e

m i n i m t i m e T(x) concides with t h e first time t 2 0 f o r which c'x = a t' t (Proof) We s h a l l e s t a b l i s h t h e two i n e q u a l i t i e s T(x) s min t T(x). T h e ' f i r s t follows quickly from Lemma 1: i f c'x t = ut'

4

x E W(t), so t h a t T(x) s t . The proof of t h e second i s considerably more complicated, s i n c e we cannot assume t h a t t h e t e r m i n a l times a r e independent

then

of quarry's choices.

Choose any point

x

4

win, and t h e n a s t r a t e g y u which f o r c e s x Then (4) holds w i t h

bounded time i n t e r v a l .

t

=

i n position t o

R

T(x,p,q) s T(x,u)

to p

=

R

within a

u [ q ] , and

n/4 up t o the time to n when x ( t ) meets the arc; then T(x(tO)) C value 107

PURSUIT GAMES

depending on which tangent, i . e . , aW(s), i s reached.

n

Fig. 1 Forcing t o half-plane. The piecewise smooth curve i s t h e boundary of W(O,n/4); t a n g e n t s t o i t s c i r c u l a r a r c a r e boundaries of isochronous W(t) f o r 0 s t < IT/&. Returning t o (lo), W ( t )

x cos

(11) At

t = n we obtain x

i n (10) has mean value point i n

i s c h a r a c t e r i s e d by

R2

2

0

0

t + Y s i n t s at.

(note t h a t indeed t h e integrand

over [O,nI).

belongs t o some W(t) 168

for

It follows t h a t every

0 s t s n:

and t h e

CAPTURE

conclusion

sup T(x) =

IT

i s q u i t e p l a u s i b l e geometrically.

The result f o r W(O,n/k)

suggest looking f o r c i r c u l a r

enveloping curves. The condition t h a t t h e boundary l i n e of 2 2 (11)be tangent t o t h e c i r c l e ( x xo) + (y yo) = r2 i s

-

that

r

-

be i t s d i s t a n c e from t h e c e n t r e .

-

x0cos t + yo s i n t - a t = + r The c i r c l e w i l l be connuon t o t h e l i n e s i f f t h e expression on t h e l e f t i s constant; s e t t i n g t h e d e r i v a t i v e equal t o zero,

%

(12) -x 0s i n t + y 0cos t = t = Isin t-cos c 1 d

-

l s i n t+cos

tl.

t !s IT need be t r e a t e d , and t s n/4 already has; n a t u r a l l y w e simplify by breaking up a t t = 3 1 ~ / 4 . For n/4 s t !s 3n/4 w e have dat/dt = -2 cos t, (12) becomes -xo s i n t + (yo + 2)cos t = 0, and t h e c e n t r e i s Now, only

(xo,yo) = (0,-2);

analogously, f o r

i s (-2,o). Example 2

3n/4 s t
0 f o r small

8

> 0. Thus,

f o r small t > 0, (14) is

where we have used

The e s s e n t i a l points, i n the sense of Exercise 8 or 7.1, s a t isfy q1

=

E,C2 = 0.

Exercises 1. I n Example 1 of the main text, show t h a t points (x,y) with

y

2

close t o

t

-b

0 x.

and s m a l l x > 0 have IUiniMUIu time6 a r b i t r a r i l y (Hint:

check the positions of aW(t) with

n-.)

2.

Again i n Example 1, show t h a t t h e two s e t s a{x: T (x ) s x/4},

{x: T(x) = n/4}

do not coincide. 3 . Treat t h e game i n R2 with equations k = y + v, = -x -u, constraint s e t s [-1,11, and x s 0 a t termination. are conI n particular, show t h a t sup T(x) = n; b o t h

1,

i s continuous,

may be discontinuous.

6 . How do t h e p r i n c i p a l a s s e r t i o n s of t h i s s e c t i o n change i f t h e t a r g e t i s a hyperplane r a t h e r than a half-space? The i n t e r e s t i n g point i s whether t h e s t r a t e g i e s a r e isochronous, stroboscopic, i n d i f f e r e n t .

7.

If t h e hyperplanes

aW(t)

with

t

near some

have no envelope, they must be p a r a l l e l but d i s t i n c t . t h a t necessarily

c

is an eigenvector of

t o a r e a l eigenvalue, and

8.

c’(5

Prove t h a t , i f p o i n t s

of

&l and

xo

E an (cf.

Show

corresponding

A’,

- 4) i s nonconstant near

xt

to.

are i n the intersection

aW(t), then every f i n i t e accumulations point e s s e n t i a l points i n 7.1) a s

t

+o

satisfies

where t h e r i g h t hand s i d e may equivalently be replaced by

Verify t h i s f o r both examples and Exercises 2 and 3 .

9.

More generally, i f

x

and an i i i f i n t e s i m a l l y c l o s e c‘e (Hint: 10.

At

is i n the intersection

aW(t)

aW(t + d t ) , then (AX

- p(t)

+ q(t))

=

0.

you may wish t o use Exercise 5.) Assume t h a t

d i s c o n t i n u i t y of

:(*

Q

):

i s a polytope, and l e t thus, f o r a constant 172

A > 0 be t h e q E Q,

CAPTURE

At cteAt y ( t ) = max c’e q f o r 0 s t s A. Prove t h e following qEQ ( f o u r t h ) r e c i p r o c i t y theorem f o r (1): A point x E Rn can be

forced t o

within [O,A] iff

R

x

can be s t e e r e d t o

a t i m e i n [O,A] within t h e c o n t r o l system

;= Ax

- u;

11. Show t h a t t h e mapping

t

u(t) E P

H at

R

at

- -q.

defined i n ( 3 ) i s

i n v a r i a n t under nonsingular l i n e a r transformations.

Assume t h a t both

12.

a r e polytopes.

c - q(O+)) s 0.

A’

and

c’(F(O+)

Prove t h a t a l l

i s a r e a l eigen-

i f , and only i f , t h e normal

W(0,t) = R v e c t o r of

P,Q

(Hint:

For small

t > 0 n e c e s s a r i l y c’x z; a implies c’x 4 a.) t t 13. Formulate and prove an a s s e r t i o n analogous t o t h e preceding i n t h e case t h a t t h e t a r g e t i s a hyperplane. 14. For a game with dynamics1 equation = f(x,p,q), t h e I s a a c s ’ (main) equation i s min max D ~ ( x f(x,p,q) ) = o P Q i t s solutions define t h e

(15) (here

Do =

(k axl

9 * * . ,

o(*)

k));

axn ‘semipermeable s u r f a c e s ’ 4 ( x ) = const.

I n t h e game from

Example 1 f i n d t h e semipermeable surfaces through t h e o r i g i n . (Hint:

from (15) a t

0, one of

a , ay ax

is

0, and t h e other

i s p o s i t i v e or negative; t h e four cases reduce t o two.

Answer:

15.

q u a r t e r c i r c l e s with c e n t r e s a t (O,+

2). )

Prozeeding a s i n Example 2, t r e a t t h e game induced

by two s c a l a r equations

1

with c o n t r o l c o n s t r a i n t s l u ( t ) I s p, l v ( t ) g v, and terminat i o n condition Ix yI s c . I n p a r t i c u l a r , describe t h e

-

e s s e n t i a l points.

16.

Apply t h e Second Reciprocity Theorem t o (1);i n

173

PURSUIT GAMES

p a r t i c u l a r show t h a t t h e c o n s t r a i n t set of t h e associated c o n t r o l system i s

if

P,Q 17.

a r e compact, convex, and symmetric. Show t h a t t h e case t h a t

i s q u i t e exceptional i f

n

2

3.

W(t)

increase monotonically

(Hint:

compare f i g . 1 with

f i g . 4 i n 1.3j show t h a t t h e i n t e r s e c t i o n s be p a r a l l e l , and then argue

RS

i n Exercise

W(t)

7.)

n W(0)

must

Remarks A s suggested by t h e examples and exercises, f o r applying t h e method it i s c r u c i a l t h a t max c I eA t p be simple t o comP pute. The geometric i n t e r p r e t a t i o n i s t h a t of a hyperplane

with normal v e c t o r

moving ( a s t v a r i e s ) but remaining

eA'tc

i n contact with t h e set

P:

t h e p o i n t s of contact maximize,

measurable s e l e c t i o n provides -Pand( * >any . If

P

i s a polytope, p( )

a maximizing mapping

may be taken piecewise con-

s t a n t , w i t h d i s c o n t i n u i t i e s having no f i n i t e accumulation p t c) at i s C1-smooth and piecewise ana-

point (whereupon lytic).

This a s s e r t i o n i s an i n t e g r a l p a r t of c o n t r o l

t h e o r e t i c l o r e ; see, e.g., N. Levinson: Minimax, Liapunov and "bang-bang", J. D i f f . Equations 2 (1966) 218-241,

(where our present a n a l y t i c matrix).

i s even replaced by a piecewise More information is a v a i l a b l e about t h e creAt

d i s p o s i t i o n of d i s c o n t i n u i t i e s of assumptions t h e r e e x i s t s a constant has less than €

n

F(

) : under f u r t h e r E

>

0

such t h a t

p(*)

d i s c o n t i n u i t i e s i n each i n t e r v a l of length

(counting m u l t i p l i c i t i e s j t h e r e a r e no f u r t h e r r e s t r i c -

tions).

See

174

CAPTURE

0. Hijek:

Terminal manifolds and switching locus, Math. Systems Theory 6 (1973)289-301;

D. S. Yeung: Synthesis of time-optimal c o n t r o l ( t h e s i s ) , Case Western Reserve University,

1974j

E. N. Chukwu and 0. d j e k :

Optimal c o n t r o l and disconjugacy (submitted).

That t h e number involved i s p r e c i s e l y

- 1 i s of fundamental

n

importance f o r construction of optimal feedback i n c o n t r o l systems:

it i s t h e c o r r e c t number of independent parameters

f o r describing t h e piecewise smooth boundaries of reachable

sets

x in

R".

It w o u l d be most welcome t o have a simple method f o r obtaining enveloping hypersurfaces f o r t h e s e t s c l a s s i c a l one i s r a t h e r unpleasant. CONJECTURE t h e hyperplanes

For aW(t)

t

i n an i n t e r v a l where have t h e envelope

aW(t):

-p

F(x) =

the

i s constant, 0, where F

i s of degree 2: F ( x ) = X'WX

+

W'X

+

u).

This i s t h e obvious f i r s t guess from t h e examples t r e a t e d ; Exercise 9 might be useful; possibly A(x

- xo)

-P

for suitable

PROBLEM

xo

Decide whether

E

Rn, P

F(x) = (x 2

- %)'A'

0:

f ( x ) = T(T(x)) i s a feedback

pursuer s t r a t e g y , possibly even optimal. I n Chapter 8 of R. I s a a c s :

D i f f e r e n t i a l Games, Wiley, New York, etc.,

1967

t h e r e a r e proposed two procedures f o r finding 'game b a r r i e r s ' ( t h e s e a r e put forward as working conjecturesj with counterexamples, and not a s u n i v e r s a l l y v a l i d r e s u l t s ) :

that the

b a r r i e r i s t h e envelope of t h e sets [x: T ( x ) = 81 ( t h e Envelope P r i n c i p l e p. 260); and t h a t it may be obtained by 175

PURSUIT GAMES

'passing a semipermeable s u r f a c e ' through t h e 'boundary of t h e useable p o r t i o n ' of t h e t a r g e t set (p. 215).

As consequence

of t h e results i n t h i s s e c t i o n , n e i t h e r p r e s c r i p t i o n works f o r Example 1 ( t h e envelope

l i n e a r games with half-space t a r g e t s :

of t h e isochrones has a t o t a t l l y d i f f e r e n t function), and Exercise 14.

5.4 I n v a r i a n t Targets We s h a l l now t r e a t games

- p + q;

= Ax

(1)

E Q;

p ( t ) E P, q ( t )

end: x

E

R

which s a t i s f y t h e following o v e r - a l l The player c o n s t r a i n t sets

ASSUMPTIONS

P

and

Q

are

nonvoid, compact, convex, and symmetric; more important, t h e t a r g e t set i s a l i n e a r subspace i n v a r i a n t under t h e coeffic i e n t matrix

A: A R C R = {x:

(2)

Mx = 0).

Subspaces i n v a r i a n t u n d e r ' t h e c o e f f i c i e n t matrix have already appeared, i n 3.7; as mentioned t h e r e , t h e r e a r e obvious s p e c i a l cases: R.

The condition

R = 0

M Q c MP

and a l l A, o r

A = 0

and all

i n t h e following theorem coincides

with a necessary condition f o r isochronous capture ( s e e 4.2). THEOREM 1 isochronous. W(0,t) = W(t)

Under t h e assumptions on (l), capture i s

F'urthermore, e i t h e r

MQ c MP, whereupon a l l

and p o s i t i o n s forced t o

subsequently; or, i n t h e t r i v i a l case W(t) = @

(Proof)

for a l l t

> 0,

can be held t h e r e

MQdMP,

we have

and W(0,t) = W(0) = R.

F i r s t assume t h a t some W ( 0 , B )

and Exercise 6 i n 5.1,

@

R

4 R.

By t h e Lemma

l i m (W(O,t)\R) c l i m W(O,t) = R.

hO+

Thus t h e r e e x i s t s a point

hO+

y

which may be used i n t h e

176

CAPTURE

p r o p o s i t i o n of 5.1: l a s t term has

M Q c MP + [-1,1]MAy; M Q C MP, Q C P

(3 )

( 3 ) holds.

Now, assume t h a t

preserving mapping u : Q

+

and, a s

E

R, t h e

+ R.

There e x i s t s a measurabilityq € u ( q ) + R.

such t h a t

P

y

Thue

MAY E M A R C Ms2 = 0.

Every

x i n W(0,B) i s forced t o R w i t h i n [O,Blj once t h e r e , it can be held i n R until t i m e e (and a l s o subsequently) by u used a s stroboscopic s t r a t e g y :

point

At e (z a s soon a s

- ;1

z E R.

isochronous.

-As

Thus

(u(q(s))

- q ( s ) ) d s ) E eAt(z - R )

= R

W(0,B) = W(e), and capture i s

This shows t h a t

W(0,e) = R

e if (3)

for a l l

f a i l s ; and, i f it does, then a l l W ( t ) =

for

t

>0

by t h e

Necessary Condition from 4.2. Second, assume t h y t ( 3 ) does not hold.

Then a l l W(t) =

t > 0 by t h e Necessary Condition from 4.2; and, a s t h e

for

counterpositive t o t h e f i r s t argument, a l l W(0,B) = R . t u r e i s again isochronous, but only t r i v i a l l y : in

can be captured, a t time

Q

LEMMA 1

For a l l

t a 0

0.

where

V

= (p

+ a)

we have +

0) iL RQ(t)

i s t h e s e t of

Q. Referring t o (l), % ( t )

winning p o s i t i o n s f o r stroboscopic capture a t if

only p o s i t i o n s

QE3

%(t> c W ( t ) c (Rp(t)

(4)

Cap-

t j

W(t) = W(0,t)

M Q c MP; and t h e l a s t term i n ( 4 ) i s t h e set of winning

positions, a t (Proof)

t, f o r capture with a n t i c i p a t i o n .

From t h e Assumptions, eAtn = R

for a l l

t j

we w i l l

use t h i s without f u r t h e r mention. The second i n c l u s i o n i n (4), and t h e i n t e r p r e t a t i o n of t h e l a s t member, i s i n Corollary 2 from 4.1.

Naturally, W(t)

contains t h e points which can be forced t o R stroboscopically a t time

t; t h a t t h i s set is p r e c i s e l y % ( t ) 177

follows from

PURSUIT GAMES

t h e Second Reciprocity Theorem i n 3 . 3 (or, preferably, from Exercise 6 t h e r e ) .

QE .D

MQdMP

I n t h e case

COROLLARY 1

t h e r e i s a constant

evasion s t r a t e g y f o r a l l p o s i t i o n s outside t h e t a r g e t (even i f pursuer i s allowed t o a n t i c i p a t e ) . Using t h e L i m i t Theorem f o r reachable s e t s , it i s

(Proof)

e a s i l y shown t h a t t h e r e e x i s t s a point

90 E

and

Q

E

>

0

such t h a t

+

e -AS ds c+,

4

M Rp(t) whenever 0

0 such E [o,el:

Thus some

t h a t t h e vectors c'e

It follows t h a t

c

-AS

c

e

q

Rp(E) y + x

has non-

Therefore t h e r e e x i s t s

eA'sc

a r e e x t e r i o r normals f o r

c

P, s

c [o,~I.

If

R ( e ) = 0.

E

Q

Rp(E).

c'(y + x) < c'x, c

P at

i s an e x t e r i o r normal t o R p ( € )

then i n p a r t i c u l a r

let

in

remains an e x t e r i o r

Rn

p s c'e-ASq f o r p

us now show t h a t

By symmetry, a l s o

4 p2

P, so t h a t

i n p a r t i c u l a r , has t h e pro-

q

normal even i f perturbed s l i g h t l y . E

is a ver-

0.

P, and

Rn.

a segment for a l l

of

p e r t y t h a t t h e set of e x t e r i o r normals t o void i n t e r i o r i n

Q

P 5 Q = 0, then W(t) = 0

If

c'(-y) s 0, so t h a t

a t the

y = R

Thus

Q

(E)

c'y s 0. c'y = 0.

I f we now

vary over t h e open s e t ( t h e i n t e r i o r of t h e collec-

t i o n of e x t e r i o r normals t o

P at

q), it follows t h a t y = 0.

From Lemma I, W(c) = 0; using Lemma 2 i n 4 . 1 ( a l s o see Exercise 5 t h e r e ) one obtains

W(2C) = 0, W(3c) = 0, etc.

Finally, from Theorem 1 and monotonicity of W(t) = W(O,t), a l l W(t) = 0.

QED

Exercises 1. Verify t h a t , i f t h e coefficient matrix

180

A

is scalar,

CAPTURE

A

=

aI, then Rp(t) =

(with t h e obvious modification i f

1 - ea t Q,

P

a = 0).

Prove t h a t capture

is isochronous and stroboscopic f o r a r b i t r a r y Qc P. 2. Assume t h a t A is simple with a l l eigenvalues r e a l ; and t h a t P i s a symmetric p a r a l l e l l e p i p e d with edges paral l e l t o eigenvectors ak of A : =

[ C”kak:

lakl

pk]

t 2 0. Prove t h a t Rp(t) = e-ASds P, and conk 0 clude t h a t capture i s isochronous and stroboscopic f o r

f o r given

IJ.

arbitrary

Q cP.

3.

Consider t h e game with uncoupled players,

x = y ; = ~ x - p , i = ~ y - q end: ; Prove t h a t t h e t a r g e t i s i n v a r i a n t ( i n t h e sense of t h i s section) iff

A = B.

W(t) = Rp $, Q ( t ) i f Q i s a segment cont a i n e d i n t h e polytope P (without assuming P % Q = 0). CONJECTURE

181

This page intentionally left blank

CHAPTER V I ALGEBRAIC THEORY

This chapter contains t h e elements of an a l g e b r a i c theory of autonomous l i n e a r p u r s u i t games

-

a l g e b r a i c i n t h e sense of

depending on t h e shape of t h e c o n s t r a i n t sets b u t not on t h e i r size. The f i r s t s e c t i o n contains, primarily, d e f i n i t i o n s of t h e notions of game space, pursuer and quarry c o n t r o l orders, and uncoupled dynamics.

I n 6.2 we e x h i b i t an a l t e r n a t e i n t e r p r e t a -

t i o n of one of t h e secondary concepts, min-max c o n t r o l l a b i l i t y ; and e s t a b l i s h two necessary a l g e b r a i c conditions f o r winning. The l a s t two s e c t i o n s t r e a t games proper, a s distinguished from t h e i r various s t a t e space d e s c r i p t i o n s .

I n particular,

it i s shown t h a t t h e game space dimension, and t h e player c o n t r o l orders, a r e independent of p a r t i c u l a r s t a t e space representations.

The concept of equivalence between game

representations i s rather rich:

i n c o n t r a s t with algebraic

isomorphism, it i s highly s e n s i t i v e t o changes of t h e t a r g e t s e t ( t h e desired outcome of r e g u l a t i o n ) . The reader who i s not i n t e r e s t e d i n t h e s e t o p i c s may choose t o omit a l l of t h i s chapter, with one exception:

the

notion and formal concept of player c o n t r o l order is needed a t s e v e r a l places i n t h i s book.

6 . 1 Game Space, Control Order Consider t h e s t a t e d e s c r i p t i o n of a l i n e a r game, end:

x E R j s t a t e space: 183

RL’

PURSUIT GAMES

(occasionally not a l l of t h e data specified).

n, A, P, Q, R

need be

The example w e s h a l l r e f e r t o repeatedly is one-

dimensional rocket chase (see l.3), defined by one-dimensional equations

i

? = u,

= v.

2,t h e

Here t h e s t a t e space i s

dynamic81 equation i s

t h e c o n t r o l values a r e i n t h e i n t e r v a l [-1,1], and t h e termina t i o n condition i s

'5'

xl= With t h e game (1)t h e r e i s n a t u r a l l y associated t h e

pur-

s u e r ' s c o n t r o l system = AX

(formally:

replace

p; p ( t )

E

P; end: x

E

R

i n (1)by 0), and analogously f o r

Q

quarry's c o n t r o l system. into the picture.

-

One can t h e n bring c o n t r o l theory

I n particular, there are available the

concepts of p u r s u e r ' s reachable sets R p ( t ) = [Joe -As p ( s ) : i n t e g r a b l e p: R 1 +

q,

pursuer c o n t r o l l a b i l i t y space (see 3.11) Cp = span Rp(t)

etc.

( t > 0),

One i n t e r e s t i n g numerical c h a r a c t e r i s t i c may be i n t r o -

duced here:

pursuer's controllability defect dim Rn

-

dim C

P

= n

- dim Cp.

For t h e example, i n t h e t r s n s i t i o n t o s t a t e space one v i t a l piece of information has been obscured:

R3,

that, i n

some sense, t h e game i s played ( o r , t h e players move) i n a one-dimensional s t r a i g h t l i n e .

To recapture t h i s notion we

make t h e following d e f i n i t i o n :

r e f e r r i n g t o (l), t h e game

space i s ( 0 5 $2) subspace of

, the

I

orthogonal complement (hence a l i n e a r

Rn) of t h e Pontrjagin d i f f e r e n c e of t h e t a r g e t

set with i t s e l f ( c f . Proposition 3 i n 3.8). 184

V e r i f i c a t i o n of

ALGEbltAIC THEORY

t h i s on our example i s a s p e c i a l case of t h e following result. LEMMA 1

Assume t h a t t h e nonvoid t a r g e t set i s of t h e

form R = {x: Tx E

El,

i s a compact, convex s e t i n

where

i s an (m,n) matrix; denote by t h e n u l l space of T. Then

Rm, and

T

01

N = {x: Tx =

RE,=,;

(2)

t h u s t h e game space i s (i.e.,

, and

I

N

i s isomorphic t o t h e range

column space) of T.

(Proof) One i n c l u s i o n i n ( 2 ) i s simple: i f x E N, then x + y E R f o r a l l y E R, s i n c e T(x + y ) = Ty E E ; hence R . For t h e other, assume x 4 N, Tx =/ 0; x + Rc R, x E R and aim t o prove x R E R, i . e . , x + y 4 R f o r some y E R. Consider t h e compact, convex set fl range T; it i s nonvoid, s i n c e it contains TR = 8. One can then f i n d a point (with y E R ) a t which Tx =/ 0 i s an e x t e r i o r normal. Ty E so t h a t indeed x + y 4 R. Then Ty + Tx 4

4

c

c,

Then ( 2 ) follars by t a k i n g orthogonal complements; and t h e l a s t a s s e r t i o n by elementary l i n e a r space theory: dim(row space T ) = rank T = n

-

L

dim N = d i m N

.

QED

I n one-dimensional rocket chase, t h e primary equations

..

of motion were x = u,

= v; one i s n a t u r a l l y l e d t o say t h a t

t h e pursuer's order of c o n t r o l i s 2, s i n c e he c o n t r o l s t h e second-order d e r i v a t i v e of h i s s t a t e variable, while t h a t of quarry i s 1. This provides a rough measure of t h e p a r t i c i pants' maneuverability:

y ( *)

is continuous, while x( )

even has continuous f i r s t derivative, and so cannot negotiate sharp corners. For t h e formalization, denote by

L = span R

(3 1

A j( P

- P)

E

185

R

the

Then t h e pursuers'

orthogonal complement of t h e game space. c o n t r o l order is t h e l a r g e s t i n t e g e r k

f

4

+

m

such t h a t

L for 0 s j s k

-

2.

PURSUIT GAMES

S i m i l a r l y f o r t h e quarry c o n t r o l order, with of

The concept

P.

WRS

i n place

Q

designed so t h a t Exercise 4 would

It is t r e a t e d beluw and i n t h e theorem of 7.4; and

hold.

applied i n Chapters I11 t o VII. LEMMA 2

P contains

If

and

0

i s a s i n Lemma 1,

R

then ( 3 ) i s equivalent t o TAJP = 0 LEMMA

3

0 s j s k

The pursuer c o n t r o l order

games with s t a t e space (Proof)

for

Rn,

either

- 2.

k z 1; and, f o r

k s n

or

k = +

The condition i s s a t i s f i e d vacuously f o r AJ(P

t h e second a s s e r t i o n , i f then a l s o f o r also for a l l

j = n

j

>

n.

-

P) c L

m.

k s 1. I n

holds f o r

j c n

-

1,

by t h e Cayley-Hamilton theorem, and thus QED

I n t h e rocket chase each p l a y e r ' s c o n t r o l a f f e c t s s o l e l y t h e appropriate component of t h e s t a t e variable; t h e only connection between t h e components i s by means of t h e terminat i o n condition.

I n o t h e r games (e.g.,

1.1)t h e p l a y e r s ' con-

t r o l s a r e i n e x t r i c a b l y connected w i t h t h e s t a t e v a r i a b l e . Some i n s i g h t i s provided by a Ke'h'n-type

decomposition

theorem: DECOMPOSITION THEOREM

A=:(

Referring t o t h e game (l), t h e r e

i s a nonsingular l i n e a r coordinate transformation and a partitioning

x

of t h e new s t a t e v a r i a b l e

x

i n which

\x4/

t h e dynamical equation decomposes a s follows:

$=

A

x

33 3

+ A

x

34 4

A44X4

x4 =

186

+

329

ALGEBRAIC THEORY

f o r appropriate matrices

Furthermore,

Bij.

Aij,

3 = 0 = x4

c h a r a c t e r i s e a t h e pursuer's c o n t r o l l a b i l i t y space, and

x 2 = o = x4 t h a t of t h e quarry. (Proof) and

Begin w i t h t h e players ' c o n t r o l l a b i l i t y spaces, Cp

By augmenting a b a s i s i n

CQ.

bases f o r

and

Cp

Q'

one obtains n and R Re-

Cp fl CQ

and then f o r

C

Cp

+

C

.

Q

l a t i v e t o t h e l a t t e r , t h e r e i s a p a r t i t i o n i n g such t h a t and

C

cribes

Q

a r e c h a r a c t e r i s e d a s asserted; t h e n

+ CQ' and x2

Cp

=

5 = x4 =

0

Cp

x4 = 0 des-

c h a r a c t e r i s e s Cp

n CQ'

(For t h i s one must, of course admit t h e p o s s i b i l i t y of zerodimensional vectors:

e.g.,

Now, observe t h a t

x 6 Cp Cp

+ CQ'

and

p

e

(x

maps i n t o

-

t o i n i t i a l position

p

x

Cp

iff

E

e

n CQ

= 0.)

A, hence a l s o

-As

p ( s ) d s ) E Cp whenever SO P; and analogously f o r C and

Consider any p o i n t

and constant c o n t r o l s

- 0

1i s i n v a r i a n t under

Cp

At

eAs; thus

under a l l

dim x

x

in

P, q

Q.

Cp

+

C

Q

(thus, x4 = 0),

Q

Then t h e s t a t e response,

and player c o n t r o l s

p,q, w i l l remain

(hence, x 4 ( * ) 3 0); furthermore, it is d i f f e r e n in C + C P Q t i a b l e and thus s a t i s f i e s t h e d i f f e r e n t i a l equation everywhere (hence,

i4(*) e 0): 4

At

=

4

C A4 x 1

t = 0 t h i s yields that

j j

- B41P

41x1 + A42x2 + A 4 3 3 + A440

0 = A

Here

x1,x2,5,

choice

x = 0

and a l s o

-

Bk1p

are quite arbitrary.

+

Bk2q.

The

shows t h a t

(4 1 so t h a t

p,q,

+ B429-

-BJ+~P + B42q = 0

(all P

E

P, q E Q),

3

Akjxj = 0 f o r a l l x and hence A41 = 0, 1 j' A42 = 0, A43 = 0; f i n a l l y , from (4), t h e game i s not changed i f we replace

B41

and

B42

by zero matrices.

187

This y i e l d s

PURSUIT GAMES

G4;

t h e desired form of equation f o r simplifies those f o r

"$5.

a s i m i l a r procedure

QEII

%

I n t h e s i t u a t i o n of t h e preceding a s s e r t i o n , l e t t h e dimension of

5

for

k = l,2,3,4;

determined uniquely i s shown i n Exercise

\

t h a t these

7.

be

are

For reasons t h a t

a r e probably obvious we w i l l say t h a t t h e game ( o r t h e players' n1 = 0; i n g e n e r a l

dynamics) a r e uncoupled i f d i m Rn

- dim(Cp n CQ ) = n - "1

i s t h e degree of u n c o u p l i q .

5

If

= 0, i . e . , CQ'

game i s s a i d t o be min-max c o n f r o l l a b l e .

Cp

the

As i n c o n t r o l theory,

w e w i l l say t h a t t h e pursuer c o n t r o l system i s c o n t r o l l a b l e ... if C = R", i . e . , n1 + n2 = n, o r = n4 = 0. The game i s P Finally, t h e c o n t r o l l a b l e i f n4 = 0, i . e . , n1 + n2 + = n.

5

5

pursuer's controllability defect is

- dim Cp = n -

d i m Rn

(nl + n2),

and analogously f o r t h e quarry. Exercises 1. For one-dimensional rocket chase, e x h i b i t e x p l i c i t l y R3 (thus, f i n d t h e (3,3)

t h e s t a t e space d e s c r i p t i o n i n matrix

A, and t h e t h r e e subsets

P,Q,R

of

2).Determine

t h e game space and player c o n t r o l orders according t o t h e d e f i n i t i o n s , and v e r i f y t h a t they coincide with t h e i n t u i t i v e notions.

Find t h e p l a y e r s ' c o n t r o l l a b i l i t y spaces, t h e i r

i n t e r s e c t i o n and sum; determine t h e degree of uncoupling and c o n t r o l l a b i l i t y defects.

( P a r t i a l answer:

t h e game i s un-

coupled. ) 2.

The same assignment f o r t h e reduced d e s c r i p t i o n x = g -

in

R2.

( P a r t i a l answer:

V,

i

= u; end: x =

o

t h e game i s min-max c o n t r o l l a b l e ,

188

ALGEBRAIC THEORY

with 1 a s degree of uncoupling.)

3 . Consider an n-th order one-dimensional equation

(5) with t h e forcing term u

a s c o n t r o l v a r i a b l e , and take

1x1 s e a s termination condition (0 s E < + m ) . Show t h a t , i n terms of t h e corresponding f i r s t - o r d e r equation i n RAA, t h e control order i s precisely

n.

Extend t h e preceding r e s u l t t o t h e case of k-vectors

4.

x and k-square matrices A B e s p e c i f i c about t h e c o n t r o l 3' c o n s t r a i n t set. 5. For a game involving k-vectors x and y, governed by ( 5 ) a d , say,

Y (m) with termination condition t h a t t h e game space i s

6.

I n (1) l e t

m-1

+ c o BjY ('1

Ix

- yI

Rk.

R = [x: Tx

E

m

(0

e

3

,

=

V)

e

0).

x E Cp, then

f o r any t > 0, since c i s and the right side, but not t o t h e q =

QED

Under the assumptions of

Lemma 2, i f a

i s i n position t o win, then necessarily

cQ c c p 3 x .

( 31 C0ROI;LARY 2

I n the same situation, unless the pursuer

control system i s controllable, t h e s e t of winning points has

.

empty i n t e r i o r (Proof of Theorem) Having the two lemmas, it only remains t o show that, i f (3) holds, then f o r a r b i t r a r i l y s m a l l t > 0 192

ALGEBRAIC THEORY

and any quarry control

Remark

Cp = R ( t )

P

Q

t > 0.

QED

q( )

P

3

c)

p( ); here we present a

The assumptions a r e as i n t h e theorem,

constructive procedure. C c C

is a

P

since e.g.

independently of

p(.)

The preceding proof e s t a b l i s h e s t h e existence

of an appropriate mapping and

t h e r e i s a pursuer control

This is immediate:

such t h a t ( 2 ) holds. l i n e a r subspace,

q(*)

X.

Without e s s e n t i a l l o s s of generality, l e t t h e terminal

t i m e be 1. Choose bases, q l , . . . , s Cp.

Since

integrable

Cp = Rp(l), each

p

*

j'

[0,11 a

pxist c o e f f i c i e n t s

(and a c t u a l l y t h e

P.

1J

Pi ( )

3

=

?At

in

j

q E C c Cp,

Q

80

there

Ca. (t)c, 15 3

a r e continuous on [0,11).

Now con-

[0,11 +

q:

.

s

Also, e

such t h a t -At e qi =

a. . ( a )

s i d e r any integrable with i n t egrable

+

c

i n Q and c19 ...,c,, 1 edsp (s)ds f o r sane

Finally, 1

1

1

with t h e bracketed t e r m being an integrable function, having values i n

p(*),

P.

A l l this was independent of x. Since x E Cp' we have 1 x = ~oe-Aspo(s)ds f o r s u i t a b l e integrable po(*) (indepen-

dent of

q(0)).

Then (2) holds with

p =

5

f

po

and

t

= 1.

Exercises 1. Carry out t h e construction described i n t h e remark f o r t h e game

193

PURSUIT GAMES

2

in

R

ul2

3

.

.

x = y ( P a r t i a l answer:

- v,

y = u

f o r t h e obvious bases, ull

0; and one may choose, e.g.,

p,(*)

as

4 in

3

1,

[0,1/21,

-4

i n [1/2,11.) 2.

so t h a t

3.

Show t h a t , i f t h e bases a r e taken orthonormal, then

f;(*)

q( )

i s e f f e c t e d by a bounded l i n e a r operator.

Find an e r r o r i n t h e following argument, purporting

t o show t h a t , i n t h e min-max c o n t r o l l a b l e case, t h e s t r a t e g y may be taken non-anticipatory. Again choose bases q i, C j ? c o e f f i c i e n t s a. .( * ) and pi(.) a s b e f o r e j now, however, 1J

f o r each

'I

t < 1 express

I

cJ = Ste

-AS

pj(t,s)ds.

Then an in-

terchange i n t h e order of i n t e g r a t i o n y i e l d s

Joe

-At

q(t>dt=

l0e-As ( ci j

S

Pi(t)Uij(t)pj(tt")dt)ds. 0

and t h e bracketed term depends only on values

q ( t ) with

t < s.

4.

I n Lemma 1, Cp

contains t h e l i n e a r span of t h e set

of winning p o s i t i o n s . Show t h a t e q u a l i t y does not hold. 1 k = p + q with I p ( t ) l s 1, I q ( t ) / L 2.) (E.g., i n R

,

Remarks The term min-max c o n t r o l l a b i l i t y was introduced i n M. Heymann, M. Pachter, R. J. Stern:

Max-min c o n t r o l problems : a system t h e o r e t i c approach ( t o appear)

where a l a r g e p a r t of t h e theorem was a l s o presented; f u r t h e r

results concern optimisation of

194

slpl2

-

lqI2

as cost.

ALGEBRAIC THEORY

6 . 3 Equivalent Descriptions L e t us r e t u r n once again t o one-dimensional rocket chase.

Two s t a t e space d e s c r i p t i o n s of t h i s have been given: obtained by a n a t u r a l procedure i n description i n

R

2

.

2,and

one

then a reduced

The p o i n t t o be made here i s t h a t a game

may have s e v e r a l d i f f e r e n t b u t equally adequate descriptions, e f f i c i e n t o r redundant t o various degree:

but t h e game i t s e l f

should be independent of t h e d e s c r i p t i o n s . This i s resolved i n t h e u s u a l manner, by defining a game proper as an equivalence c l a s s of an appropriate equivalence relation.

To begin w e formalize t h e notion of 'reduced des-

cription'. Given games described by = Ax

-P

+

E

end: x

i

(2)

=

q i ~ ( t E) P, q ( t ) E Qi R; s t a t e space: Rn,

-

v + v j u ( t ) E U, v ( t ) E vj ; s t a t e space: Rm, end: y E

EY

we w i l l say t h a t ( 2 ) is a reduction of (1)i f t h e r e i s a linear

onto mapping

(3 1

XH

y = Tx: Rn

-t

Rm

of t h e s t a t e spaces, such t h a t TA = BT, TP = U, TQ = V, R = T-lCC>.

(4)

I n (4) t h e f i r s t t h r e e conditions r e q u i r e t h a t t h e s t a t e responses and c o n t r o l s of (1)map appropriately i n t o those of (2);

t h e l a s t i s t h e requirement t h a t

to y E

whenever

y = Tx.

x E R

be equivalent

(Occasionally t h e mapping T

w i l l a l s o be termed a r e d u c t i o n . ) With t h i s s e t t l e d it i s simple t o formalize o t h e r i n t u i t i v e notions.

Thus, ( 1 ) and ( 2 ) a r e l i n e a r l y isomorphic

i f t h e onto mapping

( 3 ) can a l s o be taken one-to-one. 195

The

PURSUIT GAMES

game (1)i s a minimal representation i f each of i t s reductions

i s l i n e a r l y isomorphic t o (1). Finally, (1)and ( 2 ) are equivalent representations ( o r descriptions) i f t h e r e i s a f i n i t e sequence of games, beginning with (1) and ending with (2),

such t h a t neighbouring terms are r e l a t e d by reduction; and

a game proper is any equivalence class (modulo t h e r e l a t i o n of being equivalent representations). A l l t h i s is n a t u r a l enough, and actually yields some

r e s u l t s (e.g.,

existence of minimal representations).

Never-

theless, without applicable r e s u l t s , it i s merely a top-heavy p i l e of definitions; thus it i s not a t a l l c l e a r whether the minimal representations of a game proper are a l l l i n e a r l y isomorphic, nor whether minimal representations are controllable ( t h e former i s t r u e , t h e l a t t e r f a l s e ) . The key appears i n focusing on one or t h e other of t h e games i n t h e r e l a t i o n of reduction. PROPOSITION 1

i t s n u l l space

If

T

is a reduction of (1)t o (2), then

N = {x: Tx = 01

satisfies

ANc N, R

(5)

Conversely, i f a l i n e a r subspace

+

N c R.

of

N

Rn

t o a game ( l ) , then t h e r e e x i s t s a mapping T

has ( 5 ) , r e l a t i v e which reduces

and ( 2 ) i s then determined uniquely up t o

(1)t o a game (2),

l i n e a r i s omorphism. (Proof)

F i r s t note t h a t , i f

(61

R = T'lE)

Now, i f

T

i s t h e n u l l space of

N

i f f TR =

T, then

c and R + N c R.

i s a reduction, then ( 5 ) follows &om t h e f i r s t

and l a s t condition6 i n ( 4 ) .

(5) holds, then

i s determined by N uniquely up t o nonsingular l e f t f a c t o r s (since it i s t o be an Conversely, i f

onto mapping). that

T

Next, Tx = 0

T

implies

i s a r i g h t f a c t o r of

TAX = 0 by ( 5 ) , so

TA, i . e . ,

196

TA = BT

for suitable

ALGEBRAIC THEORY

The s p e c i f i c a t i o n of ( 2 ) i s then completed by s e t t i n g m R = range T, U = TP, V = TQ, TR (7) (where we then use (6)). Q,ED

B.

c=

COROLLARY 1

(1)i s a minimal r e p r e s e n t a t i o n i f , and

i s t h e only subspace N

only i f , 0

of

which s a t i s f i e s

Rn

(5); and also

i f , and only i f , none of t h e minimal A-invariant

subspaces

(thus, d i m N i s 1 o r 2) s a t i s f i e s

(Proof:

N

t h e conditions a r e p r e c i s e l y t h a t

COROLLARY 2

If t h e t a r g e t set

T

R + N c R.

be one-to-one.)

of (1)i s compact, or,

R

more generally, contains no s t r a i g h t l i n e s , then (1) is mini-

mal. Example (n-dimensional rocket chase)

€?" a s

with

s t a t e space, n-dimensional p a r t i t i o n e d equation

G1

(8)

- 5I

4

e

(here 0

5

in

Rn,

e 0; hence cp(t d ) > 0 and t2d

4R

f o r small t > 0.

N e x t , f i n d a point

for

Mx

5E

i n (4) i s a t t a i n e d :

P a t which t h e o u t e r minimum

214

NONLINEAR GAMES

(6 1

s Mx < 0 f o r 9 E Q,

d'f(O,F,q)

5

We propose t o i n t e r p r e t patory) s t r a t e g y .

a s a constant (hence, non-antici-

For any quarry c o n t r o l

q(*)

let

x(*)

b e t h e corresponding s t a t e response with i n i t i a l p o s i t i o n 2

t 2d E W(0,t) it i s s u f f i c i e n t t o v e r i f y t h a t cp(x(t)) < 0 f o r 0 < t < E, with E > 0 independent of q( ). Again by Taylor expansion a t 0, x ( 0 ) = t d.

The point

To shod t h a t

x(t)

2

i s a t t a i n e d from i t s i n i t i a l value x ( 0 ) = t d

t; from t h e L i m i t Theorem f o r a t t a i n a b l e sets i n 3.11, f o r small t > 0 t h e p o i n t s at time

x(t>

&=

t

2

+

td

t a r e c l o s e t o t h e compact set cvx f(O,p,Q) + 0. I n p a r t i c u l a r , x ( t ) / t i s bounded, so t h e remainder term i n (7) i s o ( l ) o ( l ) + 0 a s t + 0. Again from t h e L i m i t Theorem, t h e accumulation p o i n t s

i n (7) belong t o d'cvx f(O,p,

cp(x(t))/t

Q), i.e., l i e i n (-m,MX] c (-00,O). Since t h e estimates a r e uniform ( c f . Corollary 4 i n 3.11), we conclude t h a t cp(x(t))/t < 0 indeed holds f o r small t pendent of q(*). For t h e necessary condition, assume t h e r e e x i s t sequences of p o i n t s

x =

5

> 0, 0

estimate inde-

E ess

R; thus

and times

t = tk

with 0

(indices

k

x

E

W(O,t)\R,

t +

ot

w i l l be suppressed where p o s s i b l e ) .

shaw t h a t (5) holds. maximum f o r

4-

px

Find a point

i s attained i n ( 5 ) .

corresponds a s t r a t e g y uk

We wish t o E Q a t which t h e outer

To each

which f o r c e s 215

x =

x to

R

%

there

within

PURSUIT GAMES

p = + = uk[T], and l e t xk( ) be t h e corresponding s t a t e responses. Thus Consider t h e pursuer c o n t r o l s

[O,t]. x(. ) =

(8)

x = x(0)

for suitable

4R

> 0,

and v ( x )

s = sk E (O,tk].

x ( s ) E R and cp(x(s)) s 0

5

The Taylor expansions a t

are ID(x(s) = d x ) with

di =

s > 0,

+

w(5) + m(0) = d';

(9)

a;

0 2

d;l(x(s)-x>

*

+

Ix(s)-xI>,

f r o n (8), a f t e r d i v i d i n g by

+

1-

a(1).

From t h e L i m i t Theorem f o r a t t a i n a b l e sets, t h e terms (x(s)-x)/s

a r e i n a neighbourhood of

cvx f(O,P,q);

thus they

a r e bounded, t h e remainder term i n ( 9 ) i s 0(1)~(1) + 0, and t h e nonpositive accumulation p o i n t s a r e i n

d'cvx f(O,P,q).

Hence indeed 0

min d'f(O,p,';i)

'L

= pk

.

Q D

Condition (4) w i l l b e used f o f i n d e s s e n t i a l points, and t h e negation

pk

of ( 5 ) f o r t h e i n e s s e n t i a l points; they

>0

a r e e a s i l y v i s u a l i s e d , as i n f i g . 1. The t e s t i n d i c e s

Mx,

px

depend continuously on

x.

Always pk s Mx, with e q u a l i t y i f t h e p l a y e r s ' c o n t r o l

effects s e p a r a t e a s i n t h e l i n e a r case with

ux

f(x,p,q) = g(x,p) + h(x,q);

Lrp(x) = d',

e.g.,

-

- max

in

d'p + max d'q = d'Ax d'(p-q) P€p qcQ i n t h e n o t a t i o n of 5.3. The a s s e r t i o n of t h e theorem i s t h a t M

X

(10)

=

= d'Ax

[x

E an: Mx < 01 c ess Rc {x E R: uk s

thus, t h e boundary of

ess R

* a ess

relative t o a R

R c {x E

an:

0);

satisfies

px s 0 s

Mx]

.

The results of t h e theorem may be sharpened s l i g h t l y : 216

NONLINEAR GAMES

Fig. 1 Geometry of e s s e n t i a l p o i n t s . The e s s e n t i a l point has some quarry vectoqram i n t h e t a r g e t s i d e of t h e tangent hyperplane (upper figure); t h e i n e s s e n t i a l point has a pursuer vectogram i n t h e opposite s i d e (lower figure).

’/

217

n

PURSUIT GAMES

M Y

IJ.

< 0 and y k + x, then x E ess R. s 0 f d r some sequence y = y + x

outside R. k The f i r s t a s s e r t i o n follows from t h e theorem, s i n c e

Y (Proof)

ess R

y = y on an with k If x E ess R, then

If t h e r e i s a sequence

COROLLARY

i s closed.

7.1 with

%=

The second follows from Corollary 2 i n

{x: cp(x)

5

k-7

and t h e theorem, s i n c e t h e

WD a d d i t i v e constant does not e f f e c t 6 q nor FL Y' Example 1 We w i l l examine t h e e s s e n t i a l points f o r t h e game i n d i c a t e d by uncoupled n-dimensional equations,

pursuer's and quarry's c o n s t r a i n t s u ( t > E U, I v ( t ) l

1;

and c-capture a s t h e termination condition, lx-Yl (here

n

2

1, U

4 $,

5

c

c > 0, and f,g

s a t i s @ appropriate

versions of t h e standard assumptions from 2.1).

Sna s

.

There i s an i n t e r p r e t a t i o n i n coordinates p a r t i t i o n e d i n t o x,x,y.

s t a t e space, with

An obvious choice f o r

describing t h e termination condition by cp Q(X,G,Y) =

1

0

5

has

- Y I 2 - c 2 1.

(1.

Then

The necessary condition describes t h e boundary of t h e Corollary: adjust

x

since

t o obtain

point s a t i s f y i n g u

5

i s also s u f f i c i e n t , and p

IJ. 5 0

ess R

>0 FL < 0

in

and thus

E

aR.

This follows from

x-y

f o r p o i n t s on

=/

0, one can

aR

c l o s e t o any

0.

To s i m p l i f y f u r t h e r , f o r

z

218

E

Rn

= 0

with

(21

= 1 set

NONLINEAR GAMES

x = y

-

CZj

then

z,;,y

(in

coordinates describing

Sn'l

x Rn x Rn) a r e independent

an, and t h e condition f o r e s s e n t i a l i t y

is

(9)

z'(k

- g(y))

1.

5

Thus t h e e s s e n t i a l point set w i l l c o n s i s t of e n t i r e half rays z,&,y

w i t h a l a r g e enough ( p o s i t i v e i f z';

if


0,

t h e point

t 2 d E W(O,t)\nj expand s l i g h t l y t o prove t h a t an e n t i r e neighbourhood of t h e

point i s i n t h e s e t . 2.

For allonornous games w i t h dynamical equation

&

= f(t,x,p,q) and termination condition cp(t,x) s 0 expression analogous t o M i n (4) is

a + at

the test

max D cp(t,x) f(t,x,p,q), P q x Formulate t h e appropriate assumptions and p r w e t h i s . (Hint: a short-cut

min

i s t o reduce t o an autonomous s i t u a t i o n . )

3. I n t h e reduced v e r s i o n of t h e one-dimensional 220

NONLINEAR GAMES

homicidal chauffeur game, f i n d a l l pursuer and quarry vectograms a t t h e point (-€,1.2) and a l s o e t (-e,o.8).

4. point

I n t h e theorem omit t h e assumption on

x E aR

and a b a l l

B

about

cont8ins a convex cone with v e r t e x and i f if

R

x

min nmx d'f(x,p,q) < 0, t h e n P 9 i s convex, x E ess R and

nB

max min d'f(x,p,q) 9 P For t h e game i n R2

mal, t h e n

5. = Y

x.

-

qi

j, =

pi

R, consider a

n

Prwe t h a t i f R

B

and e x t e r i o r normal d, x d

E

e s s R.

Prove t h a t

i s an e x t e r i o r nor-

s 0.

I ~ ( t ) 5;l 1 2 I q ( t ) I j end:

1x1

2 2

IYI

use t h e geometric method suggested by Fig. 1 t o f i n d t h e e s s e n t i a l and i n e s s e n t i a l portions of t h e t a r g e t set boundary. (The answer i s summarised i n f i g . 2 . )

n

Fig. 2 The L-shaped sets form t h e e s s e n t i a l p o r t i o n of t a r g e t i n t h e game of Exercise 5.

6. Treat (4) of 1.6 f o r

t h e two-dimensional game of two cars, with t h e dynamical equation. 221

Show t h a t , i n aqf

PURSUIT GAMES

configuration of the ( s t r i c t l y positive) parameters, there always

exist

nontrivial winning positions.

( P a r t i a l answer:

.

69 is independent of controls ) 7. Treat e s s e n t i a l points i n t h e case of an n-dimenaiona 1 equation of order k 2 2

,...,x(kG1!,p,q)

x(k) = f ( x

and cp(x) s 0 as termination condition; t o simplify you may assume t h a t Rp(x) 4 0 on t h e boundary of the t a r g e t set. (Check: as f a r as e s s e n t i a l i t y goes, = 0, cp(x) s 0 is an equivalent problem. ) 8. I n the special case cp(x) = X'WX

-e

of the preced-

( positive definite, e > 0 ) find the boundary of the ing W e s s e n t i a l point s e t on t h e surface of the t a r g e t .

9. Consider t h e l i n e a r game with uncoupled players i n n-space,

.

.

x = A x - p , y = By

-

q

-

player constraints I p ( t ) l s 5 , I q ( t ) l s p, and Ix yI 5 E termination (a > 0, p > 0, e > 0 ) . Prove t h a t nontrivial

at

winning positions exist i f A 4 B. (Hint: i f there are no e s s e n t i a l points, then p > 0 a t ell points of t a r g e t boundary.) 10. For t h e paradoxial game of 1.7 i = (p-qI2 g1; o s p ( t > , q ( t )

-

find px point.

and Mx; prove d i r e c t l y that

0

4

1; end: x c

i s an e s s e n t i a l

Remarks The notion of e s s e n t i a l point appears i n R. Isaacs:

Differential Games, Wiley, New York, e t c . ,

1967 (PP. 839 215)

222

o

NONLINEAR GAMES

a s point i n t h e 'useable' p a r t ; i n our notation, t h e d e f i n i -

Mx > 0 (and Mx < 0 f o r t h e 'nonuseable'

t i o n adopted t h e r e i s points).

The disadvantage of t h i s approach i s t h a t Proposi-

t i o n 2 from 7.1 does not hold.

A formulation on p.

55 of

A. Blaquigre, F. Ggrard, G. Leitman: Quantitative and Q u a l i t a t i v e Games, Academic PTess, New York and London, 1969

seems t o suggest t h a t t h e endpoints of t i m e - o p t i m a l t r a j e c t o r i e s a r e 'useable'. A r e l a t e d concept, t h e usable p o i n t s , appears on p. 88 of A. Friedman:

D i f f e r e n t i a l Games, Wiley-Interscience, New York, etc., 1971.

The d e f i n i t i o n can probably be deciphered; a subsequent s t a t e ment i s t h a t

u X < 0 i s a s u f f i c i e n t condition i n t h e case

t h a t t h e p l a y e r s ' dynamics a r e separated.

This should not be conf'used with what we have c a l l e d t h e a c t u a l l y used p a r t of t a r g e t ( f o r which a s u f f i c i e n t condition min min 6rq < 0 ) . P 9 PROBLEM Is ux < 0 s u f f i c i e n t for x

would be

t o be an

e s s e n t i a l point?

7.3 Necessary Conditions for Small Targets

our game i n R"

i s again

with t h e standard assumptions from 2.1.

The t a r g e t sets w i l l

be described by F(x) =

R = {x:

(2)

where

n m F: R + R

F

x

at

i s of c l a s s

i s denoted by

223

C

.

1

01

The Jacobian matrix of

PURSUIT GAMES

Every closed set taking as

can be described i n t h i s manner, e.g. by

R

F(x)

t h e distance-square of

t o be t r e a t e d have s p a r s e t a r g e t s :

x

from

R.

The games

-

t h e results f o r points

E a R become t r i v i a l i f DF(x) vanishes (e.g., x E I n t On t h e o t h e r hand, m > 1 i s allowed.

x

R).

I n game ( 1 ) l e t t h e standard assumptions

THEOREM 1

apply, with t h e t a r g e t set a s described above.

For a point

x

+

t o be e s s e n t i a l it i s necessary t h a t t h e r e e x i s t a ray L = R d in

Rm

such t h a t

cvx DF(X) intersects

f o r every

L

Remark

q

E

f(x,P,q)

Q.

Neater versions of t h e condition appear f o r

s p e c i a l cases, i n t h e Necessary Conditions.

It may not be

obvious t h a t t h i s i s an immediate g e n e r a l i s a t i o n of t h e necess a r y condition from t h e Theorem i n 7.2. t h e r e a r e only two rays, t h e semi-axes; either a l l

6F

6F

0, o r a l l

?:

2

There

0 ( c f . Exercise 1).

(Proof)

Consider f i r s t any winning p o s i t i o n

and l e t

u

E

x

W(0,8)\R,

b e a corresponding s t r a t e g y ; a l s o consider any

i n Q.

p a i r of quarry c o n t r o l values

q1,q2

pursuer responses

s t a t e responses

with i n i t i a l value i n (O,e].

m = 1, so

the assertion is that

p

i

= u[qil,

Then one has t h e x(-), y ( * )

x(O)= x = y(O), and termination times

F i n a l l y , both

x = x(t)

and

y = y(s)

t,s

are i n

R = {F = 01, so t h a t (Taylor's expansion, with i n t e g r a l re-

mainder term)

(3 1

0 = F(x)

-

%=1 F(Y) = [F(Y + 5(x-Y))l*=o

1 = J0DF(~+5(x-Y))d6

t + s > 0 and t a k e limits x ( t ) , y ( s ) have a common l i m i t

We w i l l subeequently divide by appropriately.

Note t h a t i f

(x-Y)

224

NONLINEAR GAMES

With t h i s prepared, assume xo sequence of points

xO

t

xk E

Then t h e r e i s a

R.

ek

such t h a t

w(o,ek)\R,

o < ek +

and times

xk

E ess

Therefore t h e r e e x i s t s t r a t e g i e s

0.

f o r each point

ak:

q

E Q,

interpreted as a constant quarry control, t h e r e a r e corresponding pursuer responses pk = ok[q1, s t a t e responses termination times tk. I n d e t a i l ,

t

5 + f(xk(s),pk(s),q)ds, 0 < tk 5 ek + 5, s ( t ) E

xk(t) = =

and

\(*)

O'

tk, t h e L i m i t Theorem f o r a t t a i n a b i l i t y s e t s (see 3.11) yields a subsequence x of

Since the

xk ( tk )

5

i s attained from xk

at

x . ( t )-x

++

2

E

j Herep of course, t h e subsequence

xk(*), t k ) . subsequence.

cvx f(xo,P,q).

xJ

w i l l depend on t h e

q E Q above ( t h i s then determines p k ( * ) , The usual t r i c k provides an almost universal

choice of t h e point

in

j

such t h a t one has convergence i n

Take a countable dense s e t

{q:

Q, and apply t h e foregoing argument t o

.

i = 1,2,. .)

ql,%,

... i n turn.

of sequences (Cantor's 'second diagonal of xk which method'), one obtains a single subsequence x By taking subsequences

works f o r a l l t h e

qi.

I n d e t a i l , f o r each

225

J

i

there exist

PURSUIT GAMES

sequences of times points

zi

t

such t h a t

ij

, points

o
8.

Treat e s s e n t i a l p o i n t s i n t h e game induced by t h e

second-order n-dimensional equation

..x = f(x,;,p,q)j

end: W ( X )= 0

with nonvoid compact c o n t r o l c o n s t r a i n t sets. each

x

with

q(x) = 0

and

s t a t e space, with a r b i t r a r y

Show t h a t , f o r

m ( x ) =/ 0, t h e point (x,;)

E

Rn,

i s essential.

in

(Hint:

ess R i s closed.)

4.

Obtain t h e version of t h e Necessary Conditions i n t h e

allonomous case (e.g.,

-

0)

5.

$

= f(x,t,p)

- g(x,t,q),

end: F ( t , x ) =

Using t h e n o t a t i o n from 1.4 o r Example 2 , apply t h e

Necessary Conditions t o Kelley's game.

6. Consider t h e game i n t h e Example of 3.4.

Prove t h a t ,

i f t h e c o n s t r a i n t sets a r e b a l l s about t h e o r i g i n i n

n

2

3, then t h e necessary condition

VcU

f o r isochronous

capture i s a l s o necessary f o r g e n e r a l capture.

233

and

Rn

(Hint:

In

PURSUIT GAMES

Vc U + L with L linear and two-dimensional, consider vectors perpendicular to L.) 7. Treat the (two-dimensional) homicidal chauffeur game with perfect capture; e.g., in the formulation of Example 2 from 7.2, with termination condition x = 0 = y. (Answer: nontrivial capture is impossible.) 8 . Treat the game of two cars with perfect capture, in the formulation of (4) of 1.6. (Answer: the Necessary Condition is inconclusive.) Remarks A fascinating sequence of necessary conditions has been

obtained in the linear case (see 4.2, isochronous capture.. capture.. second-order games..

.

.

.

5.1,

(12)):

dim MQ s dim MP dim MQ s 1 + dim MP dim MQ s 2 + dim MP

under appropriate assumptions. Here the first may be extraneous; nevertheless there remains a tempting conjecture to be investigated. Computationally feasible sufficient conditions for essential points were obtained for large targets in 7.2, and for linear games (and 'small' targets) in 5.2. It would be highly desirable to extend the latter to non-linear games, possibly first for dynamical equations with the players' effects separating. One obstacle in the direct approach is that Filippov's Lemma applied to the nonlinear analogue of the proof of Theorem 2 in 5.2 only pravides state-dependent strategies. PROBLEM In Example 2 with n = 3, Cockayne asserts 2 2 that sl > s2, sl/pl > s2/p2 is sufficient for capture from

all initial positions (see reference in 1.6); but then our necessary condition sl/pl 2 s2/p2 need not be satisfied, 234

NONLINEAR GAMES

e.g. w i t h El

3, s2 = 1,

=

p1 =

4,

p 2 = 1.

7.4 Isochronous Capture Sections 7.2 and 7.3 may be interpreted as the extension, t o t h e nonlinear situation, of t h e results established i n 5.1 and 5.2. It is natural t o ask whether there i s an analogous generalisation of t h e treatment of isochronous capture i n Chapter 4 . The game is indicated by

with the standard assumptions.

The termination condition 1

.

of class c THEOREM Assume t h a t , i n the game j u s t described, nont r i v i a l isochronous capture is possible (W(0) jd Par some 8 > 0). Then involves F: R" + R"

lWUPW(t)

(2)

*o+

and, for every point

x

49

i n t h e limeup, the s e t s

cvx DF(x)f(x,P,q)

(Q

E Q)

have a point i n common.

(Proof) This i s analogous to, but simpler than, t h a t of Theorem 1 i n 7.3. Corollary 2 i n 2.3 s t a t e s t h a t ( 2 ) holds; t h i s w i l l be used i n place of Proposition 2 from 7.1. Take any point

xo E limeup W(t):

*O+

(3 1

5*

Let

a k be an isochronous strategy, forcing

"0,

5 E W(tk)?

t i o n a t time tk. For each element {ql,%,

...I

dense i n

Q let

%

'tk %

t o termina-

of a countable s e t

pik = uk[(4] (thus pik: R 235

1

+

P,

PURSUIT GAMES

while t h e

qi

Also, l e t

are constants).

s t a t e response, i n i t i a t i n g a t p o s i t i o n Pikj

xik(-)

be t h e

xk, t o t h e c o n t r o l s

91:

set Yik = Xik(tk)' We have (omitting fndices f o r t h e moment) t h a t

+$

7= x

t

x o

=

;f(x(s),p(s),q)ds. 0

From t h e L i m i t Theorem f o r a t t a i n a b l e sets (3.11),

yitj'xj

(4)

I)

f o r esch index

zi

E cvx

f(xoyP,qi)

and s u i t a b l e subsequence

i

as j x

j

-+

m

5.

o f the

By Cantor's diagonal method, t h e r e i s a single subsequence xj, common f o r a l l Second, t h e

qi. a r e endpoints i n t h e t a r g e t s e t .

yij

f o r any i n d i c e s

Thus,

i,J,j

Since t h e p o i n t s

y

converge t o

(use (4) with ( 3 ) ) , t h e

xo

e n t i r e i n t e g r a l term converges t o DF(xO). By adding and subtracting

x

3

i n t h e second term, t h e l i m i t i s

according t o (4):

Take, say,

a

Since t h e

%

f o r any

= 1:

q E Q

COROLLARY = f(x,p)

for a l l

i = 1,2

form a set dense i n i n place of

q

i'

zi

-

z

a

,..., Q, we f i n d t h a t ( 5 ) holds

QED

I f t h e p l a y e r s ' dynamical e f f e c t s separate,

- g(x,q),

then t h e necessary condition becomes

236

NONLINEAR GAMES

If t h e i r c o n t r o l s appear l i n e a r l y , a s i n

;c

= a(x)+B(x)p+C(x)q,

with compact, convex, and symmetric c o n s t r a i n t s e t s , t h e n t h e necessary condition i s

(7)

DF(X)C(X)Q c DF(X)B(X)P. Example

L e t us revisit t h e homicidal chauffeur game,

with t h e n o t a t i o n of Example 2 i n 7.2 (including capture radius

e

> 0). The form (7) is not applicable, b u t (6) is,

with DF(x)g(x,Q) = {€s2sin(6+v): IvI

5 IT]

= [-Cs2,Cs21,

DF(x)f(x,P) = s € s i n 6 1 f o r t h e point

x E R w i t h p o l a r angle

8.

Condition (6) i s

thus

(8)

[-2€s2,2€s21 c {O] :

isochronous capture i s impossible. Exercises 1. I n t h e h o d c i d a l chauffeur game, common sense t e l l s

us t h a t once isochronous capture i s impossible f o r radius € > 0, it must a l s o be impossible f o r E = capture; on t h e o t h e r hand, (8) would be s a t i s f i e d S e t t l e t h e matter. ( P e r t i a l answer: g(x,Q) is a with radius 2. (Answer:

3. game

s2, f(x,P)

capture 0, p e r f e c t

f o r E = 0. circle

i s a single point.)

Treat t h e game of two c a r s f o r isochronous capture. our conditions a r e inconclusive. ) Obtain a concise necessary condition f o r t h e l i n e a r

$ = Ax

-p+q

with nonlinear termination condition 1 2 f o r F(x) = c ), t h e Jacobian

-

e. (Hint: i s x'M'M.) 4. The dolichobrachistochrone game has equations of IMxl

I;

motion

237

PURSUIT GAMES

(constant quarry's

U)

1.

> 0); t h e pursuer's c o n s t r a i n t i s

IvI < 1; t h e termination condition is

Ignoring t h e s t a t e c o n s t r a i n t y

2

s n, t h e

x = 0.

0, decide on t h e p o s s i b i l -

i t y of g e n e r a l o r isochronous capture.

(Answers:

i s necessarv f o r isochronous capture, and y > 0

y

2

m2/4

sufficient

f o r capture.) Remarks This s e c t i o n was motivated by t h e suggestion on p. 191 of A . Friedman : D i f f e r e n t i a l Games, Wiley-Interscience, New York, e t c . , lgll

t o study isochronoiis capture i n t h e homicidal chauffeur game. Notwithstanding t h e negative aspect of our s o l u t i o n i n t h e example, it would b e u s e f u l t o have companion s u f f i c i e n t conditions.

An o5vious candidate t o p a r a l l e l

(7) i s

.

DF(X)C(X)Q c I n t DF(X)B(X)P

238

CHAPTER VIII

STRATEGIES The f i r s t two sections a r e concerned with optimality:

in

reasonable circumstances, t h e r e e x i s t time-optimal s t r a t e g i e s . This i s 9 non-constructive existence theorem pure and simple, and thus i s a departure from our claim t h a t t h e meteri a l i s t o contribute t o t h e solution of games r a t h e r than t o

t h e i r study.

The r e s u l t i s interesing, and generalises

r e a d i l y t o a cost-optimal problem f o r a c l a s s of nonlinear games (see 8.2); however, t h i s alone would be an i r r e l e v a n t excuse.

More important i s t h a t t h e proof y i e l d s r e s u l t s which

a r e simple and important.

5.3 and 5.4:

Finally, it applies d i r e c t l y t o

once t h e r e i s assurance t h a t some optimal

s t r a t e g y e x i s t s , t h e methods developed t h e r e p r w i d e r e l a t i v e -

l y simple constructions of other optimal s t r a t e g i e s . The l a s t section t r e a t s games f o r which t h e known procedures a r e e i t h e r t o o complicated (e.g.,

t h e Second Re-

c i p r o c i t y Theorem does cover Example 2 ) o r inapplicable; and a solution may be highly desirable and not ruled out a p r i o r i by f a i l u r e of t h e necessary conditions.

If one i s then w i l l -

ing t o accept s t r a t e g i e s which, although forcing capture, a r e not necessarily optimal, one has t h e s i t u a t i o n t r e a t e d there.

8.1 Compactness Lemma I n l i n e a r control theory, t h e s h o r t e s t proof t h a t reachable s e t s a r e closed uses a weak compactness argument on t h e control functions.

Here we present an extension of t h i s

reasoning t o game theory:

t h e bare bones i n Lemma 1, t h e

239

PURSUIT GAMES

main i n t e r p r e t a t i o n as a compactness property of pursuer's s t r a t e g i e s , and f i n a l l y t h e development of t h e consequences. To each sequence of mappings

LXMMA 1

set

t o a compact metrisable space Y

X

mapping

f: X

+

f o r any countable subset

quence

fk

fk ( x ) j

Xo c X t h e r e i s a subse-

+

f ( x ) f o r every x

E x~.

By Tychonov's Theorem, t h e r e i s a subnet, say

(Proof)

fk(i), f o r

i n a directed s e t

i

converges pointwise t o a mapping

f:

I

x+

no subnet of t h i s needs be a sequence). able s e t of points f o r each

x1,x2,

r = 1,2,.

metrisable space Y x Y x

i w yi

t h e r e corresponds a

with

3

(2)

f(xr)

... from a

which i s t h e i r limit, i n t h e following

Y

sense:

i

fl,f2,

of indices, which Y.

(Unfortunately,

Now take any count-

... i n X. Then l i m f k ( i ) ( x r ) = i .. . I n t e r p r e t i n g t h i s i n t h e ... = n Y, we find t h a t t h e net m

I=l

converges t o y, where these points a r e i n t h e product

space and have t h e appropriate r - t h coordinates:

Now choose an index

i

1

so t h a t

from yj then take an index from y

each

y

i

+ j

has distance s 1/4 1 i2 so t h a t yi has distance

k . = k ( i .) J J

have

2 k(il)

< k(i2); e t c . % < k2 < . . .; t h e

and, i n addition, t h a t

The integer indices points

yi

y, so t h a t t h e i r coordinates s a t i s f y ( 2 ) f o r

.

x = x QE ,D r I n order t o apply t h i s r e s u l t t o a sequence of s t r a t e g i e s

i n a game, it i s necessary t o recognise o r adjust t h e needed

i s a strategy, and quarry controls q1 = q2 almost everywhere, then a l s o a[qll = a[q21 a.e.; thus t h e elements.

If

u

240

STRATEGIES

s t r a t e g i e s c a r r y Over t o mappings between t h e appropriate function spaces, with t h e almost-everywhere i d e n t i f i c a t i o n . N e x t , bounded pursuer c o n t r o l s (or r a t h e r , t h e i r equivalence

c l a s s e s ) belong t o t h e Hilbert space 1 ings p: R + Rn with

If t h e c o n s t r a i n t set

P

H

of measurable mapp-

i s compact and convex, t h e pursuer

c o n t r o l s form a closed, bounded, convex subset a celebrated theorem, Y

Y

of

By

H.

i s compact i n t h e weak topology ( H

i s a Hilbert space, hence it i s r e f l e x i v e , and t h e weak-star topolcgy coincides with t h e weak topology).

i s separable and Y

Finally, as

bounded, t h e weak topology f o r

Y

H

is

metrisable.

COMPACTNESS OF STRATEGIES

I f t h e pursuer's c o n s t r a i n t

s e t i s compact and convex, then t o each sequence of s t r a t e g i e s

,...

t h e r e corresponds a s t r a t e g y a which i s t h e i r 1 2 limit i n t h e following sense: f o r any countable s e t of quarry

u ,U

,... t h e r e i s a

controls

q ,q 1 2

u

subsequence uk

k [qr3 + o[qrl weakly a s j

3

+

Thus, f o r any fixed

8 2 0, bounded measurable

and

have

as

r = 1,2,

j

+

(Proof)

. . ., we

such t h a t

j 03.

F: R1

+

Rm,

m.

A s indicated above, we reduce t o t h e s i t u a t i o n of

Lemma 1by t a k i n g f o r

X

t h e set of a l l quarry controls

(without any s t r u c t u r e ) , and f o r

Y

t h e subset, of

s i s t i n g of t h e pursuer c o n t r o l s modulo e q u a l i t y a . e . , ed with t h e weak topology.

The r e s u l t i s a mapping 241

H, con-

equippf = 0

PURSUIT GAMES

such t h a t trol

u[q]: R

1

i s measurable f o r each quarry con-

P

4

q, and s a t i s f y i n g t h e limit r e l a t i o n a s a s s e r t e d .

It

u i s non-anticaptory (00 t h a t u Assume t h a t q1 = q2 a.e. (--,a) f o r

only remains t o show t h a t

i s indeed a s t r a t e g y ) . quarry c o n t r o l s 4.. Then u k [ q 1 1 = ak[q2] a.e. ( - m , a ) j a l s o , t h e r e i s a subsequence such t h a t , f o r b o t h r = 1,2, we have uk [4.] + u [ s ] weakly ( i n (--,a) a l s o ) . Hence t h e i r limits j satisfy

u[q

3

= a[q21 a . e .

1 Non-Example

I n t h e Compactness Lemma one cannot t a k e a

subsequence common t o sequence

(-=,a).

..,

ul,u2,.

all quarry

controls.

u,[ql(t) f o r measurable constant

q: R

q's).

1

+

=

COB

Consider t h e

kq(t),

[-1,11 (however, w e w i l l only need

Now, i f constant f u n c t i o n s converge weakly,

t h e n t h e i r l i m i t i s c o n s t a n t a.e.,

and i s t h e ordinary l i m i t

of t h e c o n s t a n t s . Thus, i f a s i n g l e subsequence cp

k = k

cos k q

t o convergence f o r a l l

3

leads

cp(q); here

q E [-1,11, t h e n 3 i s measurable, and, by t h e Bounded Convergence Theorem,

P

cp(q)dqt

a

sP

U

cos k.qdq = J

1 ( s i n k.P

J

3

-t

- s i n kJ. a ) + 0

or, con k . q -t 0 for a l .J most a l l q. For t h e s e q's, say q E %, we a l s o have 2 cos k.q/2 a ~ C O Sk.q 1+ -1. J J Thus t h e s e t [q/2: q E Qo] has measure 0, although &o has full measure: a c o n t r a d i c t i o n . The consequences w e wish t o draw concern, f i r s t , a l i n e a r f o r a l l r e a l a,P.

Thus cp = 0 a.e.,

-

game

( 31

= Ax

- p + q;

p(t)

E

P, q ( t )

I n 2.2 t h e r e was introduced t h e n o t a t i o n f o r ,winning sets.

E Q;

W(0,e)

end: x E R. and

Generalising s l i g h t l y , f o r a s u b s e t 242

W(@)

STRATEGIES

O c R1

let

denote t h e s e t of positions which can be

W(0)

forced t o t h e t a r g e t with a termination time i n t h e s e t 0 c R In t h e game (3) assume t h a t

LEMMA 2

convex, and

closed.

R

i s compact and

P

For any sequence of s e t s

Ok

within

a common compact i n t e r v a l we have

(41

l i m sup ~ ( 0c ~ W(lim ) sup x E limsup W(Ok)

(Proof) Consider a w point

ok).

thus, f o r a

j

subsequence of t h e indices ( t o be denoted by k ' s again), Xk

%

To each

E

within

R

by Coapactness of S t r a t e g i e s

Ok);

It i s plausible t h a t

t h e r e i s a ' l i m i t ' s t r a t e g y a. winning s t r a t e g y f o r forces

x

to

within t h e time s e t

R

f o r some sequence

k

(5)

1

5

A s each

u

is a

x, and we proceed t o v e r i f y t h a t indeed

proving (4)). Take any quarry control again).

*

t h e r e corresponds an appropriate s t r a t e g y ak

(forcing t o

u

5*

'('k),




i s absolutely continuous.

The e x e r c i s e i s t o f i n d t h e e r r o r . 2.

I n t h e a s s e r t i o n on Compactness of S t r a t e g i e s , prove

t h a t i f a l l uk

a r e b i l a t e r a l l y d e t e r m i n i s t i c , then s o i s

a; and s i m i l a r l y f o r almost stroboscopic, isochronous, i n -

different strategies.

What can be s a i d about

u

i f a l l uk

a r e stroboscopic, o r i f each has time-lag

3.

( P r i n c i p l e of Suboptimality)

show t h a t , whenever

0 g t

5

T(x), t h e r e e x i s t s a point

on some s t a t e response i n i t i a t i n g a t T(y) < T(x)

bk? For t h e l i n e a r game (3)

- t.

245

x, such t h a t

y,

PURSUIT GAMES

Remarks For a result analogous t o Compactness of S t r a t e g i e s , see E. Roxin:

On Varaiya's d e f i n i t i o n of a d i f f e r e n t i a l game, Proc. U.S. Japan Seminar on D i f f . and Funct. Eqs. (ed. Harris, Sibuya), Benjamin, New York, 1968.

-

The proof suggested t h e r e contains e r r o r s which have not succumbed t o repeated c o r r e c t i o n and adjustment of a s s e r t i o n . The result was intended t o g e n e r a l i s e a theorem due t o Varaiya on coapactness of s t a t e responses ( r a t h e r than s t r a t e g i e s ) . An exposition of t h e l a t t e r begins on p. 43 of A. Friedman:

D i f f e r e n t i a l Games, Wiley-Interscience, New York, e t c . , 1971

It may be noted t h a t t h i s properly concerns one-person games,

. ., c o n t r o l systems.

i e

It seems t h a t t h e a s s e r t i o n i n t h e t e x t i s t h e ' r i g h t '

one.

A r a t h e r clumsy proof was presented i n 0. G j e k :

Lectures on Linear h r s u i t Games (unpublished), Case Western Reserve Univers i t y , 1973.

The formulation of our compactness result i s somewhat cumbersome because of t h e i n s i s t e n c e on convergence of seThere i s a good reason

quences r a t h e r than of g e n e r a l n e t s . for this:

t h e s t r a t e g i e s appear under i n t e g r a l signs; and

i n t e g r a l s a r e continuous, a t b e s t , under convergence of sequences, b u t not f o r convergence of n e t s . E.g., i f cx i s 1 t h e mapping R + R1 with value 1 a t x and 0 elsewhere, then obviously

sup cx = 1 4 0 = sup X

8.2

X

sex.

Optimal Strategies

I n t h e preceding s e c t i o n , t h e existence of time-optimal s t r a t e g i e s appeared a s an easy corollary, i n t h e course of

246

STRATEGIES

e s t a b l i s h i n g some elementary p r o p e r t i e s of t h e minimum time Naturally, t h e e x i s t e n c e proof i t s e l f may be

function.

d i s s e c t e d f r o m t h i s ; an a n a l y s i s reveals t h a t it applies t o a considerably more general s i t u a t i o n . Consider a non-linear game i n

given t a r g e t set

R

Rn, with dynamical equation

c R ~ ,and cost f u n c t i o n a l C(t,x,p,q)

(where t h e arguments

x,p,q

A play of t h e

a r e flrnctions).

game proceeds froin an i n i t i a l p o s i t i o n , u n t i l t h e t a r g e t set

i s first reached, possibly not time-optimally; t h e c o s t i s then computed, with t h e pursuer attempting t o minimize, and t h e quarry, t o maximize.

The data include t h e elements

already mentioned, and also player c o n s t r a i n t sets

P,Q.

As i s u s u a l i n t h e s e complex s i t u a t i o n s , an unwieldy l i s t of assumptions and s p e c i f i c a t i o n s o b s t r u c t s t h e way t o t h e main r e s u l t .

Most of t h e assumptions a r e i n t h e category of

book-keeping;

t h e i n t e r e s t i n g ones a r e t h a t pursuer controls

appear l i n e a r l y i n t h e system dynamics (1);and t h a t t h e cost C(t,x,p,q)

t.

increase with

The d e f i n i t i o n s only serve t o

p i n down t h e obvious notion of cost-optimal pursuer s t r a t e g y . F i r s t , t h e conditions.

I n (l), f and

G

a r e mappings

f: Rn x Q + Rn, G: Rn x Q + Rm s a t i s e i n g appropriate versions of t h e standard assumptions The p l a y e r s ' c o n s t r a i n t sets

from 2.1. subsets of closed.

Rm, with

P

convex.

The cost f u n c t i o n a l

r e a l number whenever t 2 1 1 p: R + P and q: R + Q C(t,x,p,q)

The t a r g e t

C(t,x,p,q) 1 0, x: R 4 Rn

R c Rn

F'urthermore,

t, and continuous i n x,

uniformly with respect t o t h e o t h e r v a r i a b l e s . C(t,x,p,q)

is

i s a well-defined i s continuous, and

a r e measurable.

i s left-continuous i n

a r e compact

P,Q

Finally,

i s non-decreasing i n t, and quasi-convex i n p: 247

PURSUIT GAMES

ignoring t h e other variables,

( 31

C(up1+(l-u)p2) L max c(p1),c(p2) Given a winning s t r a t e g y

xo, set

position

C(u,q) = C(t,x,p,q)

P = u[ql,

X b ) =

su

5

1.

The notation from 2.3 w i l l

Second, t h e specifications. be needed again.

o

for

f o r an i n i t i a l

where

fl(s,xo,a[ql,q),

t = T(u,q),

and then c ( u ) = sup C(u,q), y = i n f c ( u ) , q(* ) supremum taken over a l l quarry controls q(- ), and infimum over winning pursuer s t r a t e g i e s .

i s then one which has

A cost-optimal s t r a t e g y

a

C(o) = Y.

Under t h e l i s t e d assumptions, t o each position

THEOREM

which can be forced t o termination within a bounded t i m e i n t e r v a l t h e r e corresponds a cost-optimal s t r a t e g y . (Proof) Assume t h a t t h e position

E Rn

‘(‘kjq)

= T(‘,q)

(c(e>xk,%,q)

+

(c(e>x,Pk#q)

xk,Pk#q))

= (‘(‘k>\,Pk,q)

+

8

- c(e,x,Pk#q))

- c(8,x,P,q))*

If necessary, t a k e subsequences i n such a fashion t h a t each

bracketed t e r m here has a l i m i t , t h a t and t h a t t h e

ek

converge,

pk

-+

+ e0 E [O,XI.

p weakly (cf. 8.1)

I n t h e following lemmas (with present n o t a t i o n and assumptions r e t a i n e d ) we w i l l show t h a t each of t h e brackets i n (6) has a non-negative limit, and complete t h e proof of (5). LEMMA 1 [ 0,h 1.

The s o l u t i o n s

xk

-+

x uniformly i n t h e i n t e r v a l

Furthermore,

e = T(a,q)

(7) (8)

lim (c(8,xk,Pk,q)

k

(Proof)

5

h,

- c(e,x#Pk,q))

=

O*

The s o l u t i o n s satisf‘y t h e appropriate i n t e g r a l ver-

s i o n of (1); we may then s u b s t r a c t and rearrange, omitting reference t o

q(

0 )

:

t

weakly, t h e l a s t i n t e g r a l term converges t o as

k

+

03,

The Lipschitz property (2)

uniformly i n [O,Xl.

t h e n y i e l d s t h e following estimate f o r 249

4, =

1%

- XI

:

0

PURSUIT GAMES

(9)

sk(t)

t const* fock(s)ds

5

constant involving t h e Lipschitz constant P 3 pk(s).

ck + 0,

It follows t h a t

+

0(1), and a bound on

p

%

and

Gronwall's i n e q u a l i t y ( 'Bellman's Leuuna' ).

sk

x, e.g. by

For a d i r e c t proof,

s ~ $ 5 , + Ek, multiply by t h e i n t e g r a t i n g rewrite ( 9 ) a s f a c t o r e-at and i n t e g r a t e :

Then x,,(ek)

E

yields

R

t h e first i n t e r s e c t i o n time

f i r s t r e l a t i o n i n (5).

x(e0) E R

i n t h e l i m i t j thus

e f o r x ( * ) s a t i s f i e s (7), t h e

F i n a l l y , (8) follaws from xk -+ x

a c o n t i n u i t y assumption on t h e c o s t f u n c t i o n a l .

-

lim(C(Bk,xk,Pk,q) k

l i m c(ek) 2 C(e).

>

and l a r g e

0

QE ,D

c(e#\'Pk,q))

Omitting repeated v a r i a b l e s , t h e a s s e r t i o n i s t h a t

(Proof) C

and

Since k.

Thus

ek

I)

eo, we have

l i m c(ek)

ek > e0

-

C

for

l i m c(eo-e) = c(eo)

2

CI)o+

by monotonicity and l e f t - c o n t i n u i t y ; t h e a s s e r t i o n now follows from (8).

&Fs

-

l i m (c(e,X,pk,q) c(e,x,P,q)) 2 0. k U s i n g abbreviated n o t a t i o n again, we wish t o show

3

(Proof)

+

i s the 'strong' limit of f i n i t e convex combinations of t h e pk. Also weakly, so t h e convex combinations need only pk+l -+ p involve pk with k 2 2 j e t c . Thus

that

C(p) s l i m C(%).

Since

pk

p weakly, p

S

k '

for suitable

04rr

quasi-convex i n

and p,

sk

2

k.

250

Since

C

k

i s continuous and

STRATEGIES

C(P) = l i m C( k

C ~ Q ? s~ )limsup k

mex

~(p,)

ksrrsk

= limsup ~ ( p , ) = l i m ~ ( p , ) .

&ED

k

The l i m i t r e l a t i o n s i n t h e

(Proof of Theorem, conclusion)

lemmas, applied t o (6) i n s u i t a b l e order, y i e l d l i m (C(ak,q) C(o,q))

2 Oj

of ( 5 ) .

&ED

-

t h i s , t o g e t h e r with (7),complete t h e v e r i f i c a t i o n

Exercises 1. Determine whether t h e following cost f u n c t i o n a l s s a t i s f y t h e requirements of t h e theorem:

l e a s t distance t o target, inf d i s t (x(t),n);

t2O

l e a s t sustained d i s t a n c e from t a r g e t i n f sup d i s t ( x ( s ) , n ) . t 2 0 srt 2.

Show t h a t Lemma 1 f a i l s If t h e requirement t h a t

p u r s u e r ' s c o n t r o l appear l i n e a r l y i s omitted. dimensional s t a t e space t i a l condition converges t o

x ( 0 ) = 0.

:=

Ipl

with Ip,(t)l

one-

-1 s p ( t ) < 1; i n i -

Construct a sequence

0 .weakly but has

(Example:

p k ( * ) which

= 1 8.e.)

Remarks I n t h e d e f i n i t i o n of cost-optimality it was not assumed t h a t t h e optimal s t r a t e g y a

f o r c e s t o t a r g e t within a com-

pact time i n t e r v a l , but only t h a t t h e termination times T(a,q)

251

PURSUIT GAMES

b e f i n i t e f o r each quarry control.

Neverthe l e s s , (5) shows

t h a t t h e s t r a t e g y studied i n t h e proof does have t h i s property. The requirement t h a t t h e cost be computed a t t h e f i r s t termination t i m e i s t r e a t e d , i n t h e theorem, by r e q u i r i n g t h a t C(t,x,p,q)

i n c r e a s e with

t j

a s seen i n Exercise 1, t h i s

precludes s e v e r a l n a t u r a l cases.

The monotonicity condition

could be dropped i f t h e pursuer were allowed t o choose, among t h e times a t which t h e s t a t e response meets t h e t a r g e t , t h a t a t which t h e c o s t i s t o be a s c e r t a i n e d .

8.3 Design of S t r a t e g i e s Given an i n t r a c t a b l e game with t a r g e t Rn,

and s t a t e space,

R

a p l a u s i b l e a t t i t u d e f o r f i n d i n g some s o l u t i o n s i s t o

decompose, t o select intermediate t a r g e t s (gambits) Ro,R1

(1)

,...,2,Ra = R,RO

= s t a t e space,

i n such a manner t h a t f i r s t , some o r a l l p o s i t i o n s i n each f$-l

can be forced t o

Rk, and second, t h a t t h e s e intermedi-

a t e end games b e considerably simpler t h a n t h e o r i g i n a l game. Obviously t h i s i s open-ended:

t o obtain an a c t u a l procedure

one must s p e c i f y t h e c l a s s e s of gambits and end games t o be used.

As one f a c e t of an example, suppose t h a t i n a l l t h e intermediate end games but t h e l a s t we r e q u i r e t h a t capture be isochronous.

Refer t o f i g . 1; and r e c a l l t h a t f o r c i n g t o

a t a r g e t set is not t h e same a s forcing t o a point within t h e t a r g e t (see t h e remarks i n 3 . 3 ) . a c t i o n of t h e quarry, a t t i m e end a t v a r i o a s p o i n t s

3

can be forced t o s2, etc. Since t2

4

ylgt2,

tl

Thus, depending on t h e t h e position

... w i t h i n

R2.

x1 E R1

will

Each of t h e s e

isochronously; e.g., Y1 a t t2, Y2 a t s i n general, t h e durations tl + t2

3:

2

4

tl + s2 f o r t h e composite motion of x1 t o forcing t o i s not isochronous. This negative result has p o s i t i v e

5

252

STRATEGIES

impact:

hopefully some games with g e n e r a l capture can be

assembled i n this fashion frm isochronous games.

Fig. 1 (schematic) Gambits t i o n of isochronous f o r c i n g t o R2

%

9.Composi-

with t h a t t o

r e s u l t s i n non-isochronous capture.

Some remarks a r e i n p l a c e here.

One should allow f o r t h e

p o s s i b i l i t y t h a t t h e f i n i t e sequence of gambits i n (1)i s replaced by a sequence i n f i n i t e i n one o r both d i r e c t i o n s j

253

PURSUIT GAMES

thus, i n Example 1, t h e appropriate s i t u a t i o n i s suggested by

...,

...,

Rf

R1, no = 0

If, i n (l), not a l l p o i n t s i n one gambit can be forced t o t h e

next, then one might a t l e a s t r e t r a c e t h e p a t h backward f r o n n R = QR, thus obtaining winning p o s i t i o n s i n R = R The

.

0

term end game may seem inappropriate f o r a l l b u t t h e l a s t stage.

Huwever, a t our level of vagueness, l o g i c a l l y nothing

prevents us from considering a game with

R

R -1

as l a s t target,

i s t h e end game. to R R4-2 4-1 F i n a l l y , t h e t r i v i a l case a = 1 of (1)amounts t o t h e enquiry whether t h e i n i t i a l l y given game i s some elementary end game.

f o r which t h e f o r c i n g from

Obviously t h e idea i s q u i t e f a m i l i a r .

Closer t o our

i n t e r e s t s , I am t o l d it i s considerably developed i n puzzlesolving i n t h e context of a r t i f i c i a l i n t e l l i g e n c e , and i n t h e study of l a r g e and heterogenous systems i n systems engineering. Even i n value-oriented treatment of game theory, some of t h e apparatus can be i n t e r p r e t e d , i n t h e present terms, a s t h e use of end-games of f i x e d s h o r t duration, a l t e r n a t e l y i n d i f f e r e n t and having a n t i c i p a t i o n allowed (and multitudinous ramificat i o n s of t h i s ) . Consider again t h e games of 5.3, with h a l f -

Example 1

spaces a s t a r g e t s ; r e t a i n t h e n o t a t i o n developed t h e r e r e c a p i t u l a t e t h e method, f o r given i n i t i a l p o s i t i o n

f i r s t solves

c'x = a

t

t

for

t

2

x

one

0, and then wes t h e i n d i f f e r -

ent s t r a t e g y u , u ( s ) = F ( t - s ) , t o f o r c e t o against all a c t i o n of quarry.

To

Taking

t

R

t

at time

minimal i n t h e f i r s t

s t e p amounts t o a pre-computation of t h e minimal time

T(x).

A simpler procedure may be applied i f t h e p l a y e r s ' con-

s t r a i n t s e t s a r e polytopes.

Then

u = f; -

needed f o r

a

t can be taken piecewise constant, with d i s c o n t i n u i t i e s having no f i n i t e accumulation points :

254

STRATEGIES

o = eo < el < ( s e e Remarks i n 5.3).

Let

R.

,1

...

r e a d i l y described by i t s boundary {x: c'x = a ] From t h e theorem i n 5.2 applied t o R

can be forced t o

R

t

t

t

for

6j

=ej-e

e

=

, each p o s i t i o n j -1

within a t i m e

R

W(ej),

be t h e hyperplane

x

3'

in

by

1-1

a constant s t r a t e g y

3 -1 p(t), t =

t o determine

Composing t h e s e end-games, we f i n d t h a t

j

T(x).

(ej +

t h e r e i s no need

ej+1)/2;

any winning p o s i t i o n i n one of t h e hyyerplanes

forced t o

R, within a time

elementary end games. by determination of

b1 +

... + 6 j

The computation of j

in

=

ej

T(x)

W(e ) can be j using j i s replaced

x E W(6Jj).

What l o s s does t h e s i m p l i f i c a t i o n e n t a i l ?

I n Example 1

of 5.3 (please r e f e r t o Fig. 2 ) t h e gambits a r e

I n t h e method j u s t described, some p o s i t i o n s a r e forced t o

W(n/4) (and subsequently t o R ) , although obviously those i n could have been forced t o R t h e curved p o r t i o n of W(O,rr/k) d i r e c t l y and f a s t e r ; thus t h e method i s not time-optimal. Second, although it seems p l a u s i b l e t h a t a l l winning p o s i t i o n s can be t r e a t e d thus, t h e e q u a l i t y needed f o r t h i s ,

has not been e s t a b l i s h e d . Section

5.4 suggests another candidate f o r gambits: t h e

l i n e a r subspaces i n v a r i a n t under t h e c o e f f i c i e n t matrix.

Be-

f o r presenting an i l l u s t r a t i o n of t h i s , l e t us dispose of one p o s s i b l e idea. If t h e c o e f f i c i e n t matrix

values, t h e r e a r e of matrix Rn

A'

n

i s simple with r e a l eigen-

A

independent r e a l eigenvectors cl,

( t h e ' l e f t eigenvectors' of+;

a s s t a t e space).

we a r e taking

The orthogonal conplement

i n v a r i a n t subspace of dimension 255

n

-

...,cn

I

cl

i s an

1; f o r c i n g t o it i s

PURSUIT GAMES

isochronous and stroboscopic, and p o s i t i o n s can be held i n it t h e r e a f t e r (Corollary 2 i n 5.3). process: etc.,

I

c1

while holding i n

Thus one may repeat t h e

I

R ~ ,cl,

c1

c1

I

n c2,

The sequence (1)i s

stepping down t h e dimensions. L

&

one can f o r c e t o

n c2, .. .,cl n I

I

... n cnA = 0;

and t h e suspicion t h a t t h e idea i s t o o b e a u t i f u l t o l a s t under s c r u t i n y i s confirmed by t h e l a s t t a r g e t : can one hope t o f o r c e t o o r i g i n .

Qc P

only i f

Speaking somewhat vaguely,

each instance of holding uses up one degree of freddom i n t h e dim P = n

pursuer's c o n t r o l c a p a b i l i t y ; unless

a t l e a s t one

of t h e holding manoeuvres i s impossible (see Exercises 2 and 3). Example 2

(Linear analogue of two-car game)

The game i s

induced by two n-dimensional second order equations,

with t h e p l a y e r c o n s t r a i n t sets

non-void,

U,V

compact, con-

Various assumptions w i l l be placed on t h e

vex, and symmetric.

data; t h e f i r s t i s t h e s u f f i c i e n t condition f o r isochronous capture, V c I n t U ( s e e 4.2,

and t h e Example i n 3 . 4 ) .

t h e c o n s t r a i n t sets a r e b a l l s and

n

2

I n case

3, V c U i s necessary

f o r g e n e r a l capture (Exercise 6 i n 7.3). There i s a standard d e s c r i p t i o n i n s t a t e space coordinates being

x,i,y,G

obvious reduction t o

2"

R4",

the

( c f . 3.4; we w i l l not need t h e via

x

- y,

i,;).

I n attempting t h e method described e a r i l e r , l e t us f i r s t

look f o r i n v a r i a n t subspaces.

These a r e associated with t h e

eigenvalues of t h e full c o e f f i c i e n t matrix (namely, 0 and those of A,B); but Example 1 tends t o discourage t h i s approach.

There a r e two obvious 3n-dimensional i n v a r i a n t

subspaces, with equations

k

- AX 256

=

0,

$

- By = 0.

STRATEGIES

Now, we a r e i n t e r e s t e d i n t h e u l t i m a t e behaviour of t h e v a r i able

x

- y;

i s o l a t i n g t h i s from t h e above, w e a r e l e d t o

consider

( 31

-1.

x + B

x - Y - A

-1. y = O

a s t h e equation of a subspace, L; it i s r e a d i l y v e r i f i e d t h a t L

i s indeed i n v a r i a n t under t h e complete c o e f f i c i e n t matrix.

With t h i s a s t h e gambit, t h e sequence (1)i s R

4n

, L,

a;

l e t u s study t h e fi = 2 end games involved. The f i r s t c o n s i s t s i n f o r c i n g t o t h e i n v a r i a n t t a r g e t It i s formally simpler t o use t h e reduction des-

L ( c f . 5.4).

cribed i n Exercise 5 of 6 . 3 (note t h a t t h i s i s not a reduction of t h e o r i g i n a l game:

6.3, here R + L is

4

i n connection with Proposition 1 i n

a ) . The new coordinate, according t o ( 3 ) ,

-1. -1. z = x - y - A x + B y,

by

k

(4 1

= A

-1 u

and t h e reduced game i s indicated

- B-1v;

The c o e f f i c i e n t matrix i s

end: z = 0; s t a t e space: Rn.

0; t h u s capture i s isochronous and

stroboscopic (Exercise 1 i n 5.4), B-% c A - 4 .

and n o n t r i v i a l i f f

The winning sets a r e W(0,t)

and R+(A%

=

W(t) = t(A-%J

* B-%)

i s t h e set of winning p o s i t i o n s . t u r e occurs i f f

B-%

c I n t A-%,

B-%),

= span ( A - 4 2

B-4)

I n p a r t i c u l a r , complete capwhich we s h a l l assume.

This t r a n s l a t e s e a s i l y i n t o t h e o r i g i n a l coordinates. summary of assumptions t o t h i s p o i n t i s t h a t

A,B

i n v e r t i b l e and

(5)

vc

Int

u, 257

B-4 c Int

A%.

are

A

PURSUIT GAMES

N e x t , consider t h e second end game, of f o r c i n g t h e out-

come x

-y

from an i n i t i a l p o s i t i o n i n L.

D i r e c t l y from

(21,

G(t) and s i m i l a r l y f o r

=

At e (xo

-As

By a second i n t e g r a t i o n and rear-

y(t).

rangement, we f i n d

-st ((;-'eArdr)u(s) 0

-

0

(sot-s

Br e dr)v(s))ds

(e.g. by Laplace transform methods, t h e i n t e g r a l of a convolution i s t h e convolution of one f a c t o r with t h e i n t e g r a l of t h e second).

Since t h e i n i t i a l values s a t i s f y (3), t h e

first bracket vanishes.

(This i s t h e purpose of t h e f i r s t

s t e p a s shown i n Exercise 5 , it i s not advisable t o hold withThe termination condition

L.)

(7 1

e

x = y

/

dr)u(s)ds =

0

A,B

as

0

([

may now be r e w r i t t e n a s

t-s 0

eBrdr)v(s)ds +

A t -1. A x0

+ If

t

(e

. .

- eB t B-1.yo).

a r e s t a b i l i t y matrices, t h e l a s t bracket approaches

t + +

m,

f o r any

x o, yo.

The s o l u t i o n procedure i s

then t o s p l i t t h e pursuer's c o n t r o l i n t o two components, appropriately compensating t h e r i g h t hand members i n ( 7 ) . A s our f i n a l assumption, l e t

(8)

A(I

- eAt)-l(I

-

eB)B-%

(note t h a t t h e s t a b i l i t y matrix eigenvalues, so I

-

A

c Int U

258

>0

has no pure imaginary

eAt i s n o n s i m a r for

To simplify notation, define

for a l l t

t >

01.

STRATEGIES

(9)

t

- eAt)-'(I - eBt)B-l

F ( t j = A(I

0

0 for

B rd r )

= (leArdr)'l(le

t > 0, and l e t F(0) = l i m F ( t ) = I, F(m) = l i m F ( t ) = AB'l t40 tA+m

From ( 5 ) and (8) we have

F(t)V c I n t U

for

0

5

t

L

t h a t , by c o n t i n u i t y and coxpactness, t h e r e i s a b a l l the origin i n

m,

so

G

about

with

Rn

for t

F(t)V + G c U

(10)

Consider (7) again.

Choose

- eB t

A t -1. e A xo

-1.

€I yo

( t h e left-hand member approaches neighbourhood of

t

>0

E

t 0

so t h a t

(eAs

- I)A-lG

ds

0, t h e r i g h t hand one i s a

t > 0).

increasing with

0

2 0

t > 0 and a l s o an i n t e g r a b l e mapping

This determines

tt+ g ( t )

E

G

with

yo = Jo(e A s - 1 ) A -1g ( s ) d s eA tA -1x. -e Bt B -1. 0

(11)

=

t 0

t-s Ar

(l

0

e

dr)g(t-s)ds.

Finally, define a stroboscopic s t r a t e g y

0,

U ( V , X )= F(t-S)V + g(t-S);

i t s values a r e i n U according t o (10). t h e pursuer c o n t r o l

s w U(S) = u ( v ( s ) , s )

Fro2 ( 9 ) and (ll),

w i l l satisfy the

termination condition (7). Conclusions: r e l a t i o n s ' (5),and

If

(8) holds, t o g e t h e r with t h e two 'limit

A,B

a r e s t a b i l i t y matrices, then complete

capture occurs f o r t h e game ( 2 ) :

any i n i t i a l p o s i t i o n s can be

be forced t o coincidence ( a c t u a l l y , by isochronous and stroboscopic s t r a t e g i e s ) , Example 3

(The two c a r game i n t h r e e dimension) 259

The

PURSUIT GAMES

game i s a s described i n 1.6; we s h a l l merely present t h e highl i g h t s of Rublein's s o l u t i o n ( c f . references t h e r e ) of t h e case

n

=

3 and S >l,(?) 1 S

a s i n 1.6, x

2

>p2 O 1

;

2

denotes t h e p o s i t i o n of t h e pursuer, and

y

t h a t of t h e quarry. The author shows t h a t (merely from t i o n can be forced t o one with and t h e n t o one w i t h y

- x.

Iy

ly

- x(

- XI

s1 > s 2), every posi-

a r b i t r a r i l y large,

l a r g e and

parallel t o

Then, from (12) he shows t h a t t h e following outcome

can b e forced: (y

(13)

- x)';

> 0; $ ,:

and y

-

x a r e coplanarj

x,y have equal components i n (y-x)

.A

.

Another formulation of t h e l a t t e r two conditions i n (13) i s t h a t t h e d i r e c t i o n of

y

-x

can be held constant, s o t h a t

t h e f i n a l s t a g e of t h e game is, e s s e n t i a l l y , one-dimensional. That t h i s procedure i s an instance of our gambits and end games i s obvious (e.g.,

a

=

4 i.n

( 1 ) ) j note t h a t holding

occurs, b u t not a t a l l s t a g e s . Exercises The f i r s t three of t h e s e a r e r e l a t e d t o Example 1.

.

1. Verify t h e following a s s e r t i o n s f o r a game i n

R

2

,

x = x + y + u , y = - y + v , w i t h [-1,1] a s common c o n s t r a i n t s e t . and

The eigenvalues a r e

-1, with corresponding eigenvectors a' = (2,1),

b' = (0,l).

The fundamental matrix s o l u t i o n i s

260

1

STRATEGIES

2.

I n t h e game above with t a r g e t set

winning set

show t h a t t h e i s t h e s t r i p described by -t O c 2 x + y s l - e