So-called differential games of kind (qualitative games) were considered involving two or more players each of whom possesses a target toward which he wished to steer the response of a dynamical system that was under the control of all players. Sufficient conditions were derived, which assure termination on a particular player's target. In general, these conditions were constructive in that they permited construction of a winning (terminating) strategy for a player. The theory is illustrated by a pursuit-evasion problem

Getz, W. M.

Leitmann, G.

English

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https://ntrs.nasa.gov/search.jsp?R=19780002882 2020-03-22T07:14:50+00:00Z
•NASA CR 152044
Qualitative Differential Game_ with Two Targets
W. M. Getz t and G. Leitmannit
(NASA-CH-152044)	 QUALITATIVF DIFFERENTIAL	 R78-10825
GAMES WITH TWO TARGETS (Nitional Rosearch
Iust. tot Mathematical)
	
19 p HC A02/MF .01
CSCI 12B	 Unclas
63/66 52033
August 1977
NSG - 2242
( National Research Institute for Mathematical Sciences, CSIR, Pretoria,
South Africa
h University of California, Berkeley, California, USA
kims—
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Abstract
We consider so-called differential games of kind (qualitative
games) involving two or more players each of whom possesses a target
toward which he wishes to steer the response of a dynamical system
that is under the control of all players. Sufficient conditions are
derived, which assure termination on a particular player's target.
In general, these conditions are constructive in that they permit
construction of a winning (terminating) strategy for a player. The
theory is illustrated by a pursuit-evasion problem.
i
Abstract
We consider so-called differential games of kind (qualitative
games) involving Cwo or more players each of whom possesses a target
toward which tie wishes to steer the response of a dynamical system
that is under the control of all players. Sufficient conditions are
derived, which assure termination on a particular player's target.
In general, these conditions are constructive in that they permit
construction of a winning (terminating) strategy for a player. The
theory is illustrated by a pursuit-evasion problem.
le
	 _	 ,^^
ii
1
i
r
i
1. INTRODUCTION
Differential games of kind, especially games of pursuit and evasion,
were introduced by Isaacs, Ref. 1. Much of the subsequent literature on this
subject deals with extensions and generalizations of problems discussed by
Isaacs; a particular favorite is the "homicidal chauffeur game", Refs. 2,3.
These games concern situations in which two players exert control over a
system; one player (pursuer) wishes to steer the state of the system to a
given set, while the other player (evader) desires to keep it out of that set.
Thus, such games involve a single set which is the pursuer's "target" and the
evader's "anti-target".
Differential games of kind (qualitative games) in which each of two
players has his own target toward which he wishes to steer the system's state
were considered by Dlaqui^re et al., Ref. 4; such multi-target games encompass
games with a single target, of course. A general discussion of two-target games
may be found in Ref. 4, where there are also presented conditions sufficient
to assure one player that a particular subset of the state space is one from
which his opponent cannot guarantee himself a win (that is, termination on
his target).
In this paper we consider qualitative two-target differential games.
For such games we give conditions sufficient to assure one player that a par-
ticular subset of the state space is one from which he is guaranteed a win;
that is, if he utilizes a strategy for which certain conditions are met and
play begins in an appropriate subset of the state space, then he is assured
of steering the state to his own target before his opponent can steer it to
his,
3
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z
AI
_.-Yom„._..	 ^^.....	 m-.	 ___	 1111•«.. ._zs;_ «. ..,w	 '
0	 1
-2-
2. PROBLEM STATEMENT
In order to admit a class of strategies sufficiently general to be
of interest in many applications we allow the system under discussion to be a
generalized dynamical system, Refs. 5, 6.
We suppose that n real numbers , x i , i = 1, 2, ..., n, fully
describe the system at a given instant of time, t E (-w, w) . Thus, the
system is described by a vector x = (x l , x 2 , ..., xn ) T E D , loosely called
the "state" of the system, where D is a domain or the closure of a domain
in Rn . The state evolves, that is, changes with time, as a trajectory of
a generalized dynamical system
x( t ) E F (x( t ), t)	 (1)
where F ( • ) is an appropriately defined vector-valued function from
DI
 R 	 into all nonempty subsets of R n .
Since the system is under the control of two agents (players),
we consider two prescribed sets, P ', i = 1, 2, of set-valued functions
d.
of x and t . Let U', i = 1, 2, be given subsets of R ' , the players'
control spaces.' The elements of P' are the i-th player's admissible
feedback controls (strategies) p'( • ) : D x R 1 + all nonempty subsets of U' .
Next let there be given a function f( • ) : D x U1 x U2 -r R 
and for given p'(•) a P' , i = 1, 2, define F( • ) by
F(x, t)	 (z e D I z = f(x, u l , u 2 ), u i E p'(x, t))
= f (x, 101 (x, t ),
 p2 ( x , t))•
For given (xo, to) E D x R1	 a solution of (1) is a function
One can allow for state and time-0ependent constraint sets U' 	 U'(x, t)
by prescribing set-valued functions U i ( • ) .
f.
3
O^YG A y V
_3_
x(') : [t0' tf) -} D , x(to) = x0 , that is absolutely continuous on all
compact subintervals of [t 0 , tf) and satisfies
x(t) G f(x(t),
 pl (x(t), t), p2 (x(t), t))	 (3)
a.e.	 [to , tf ) .
To assure existence of solutions of (3) we make the following assumption
Assumption 1. The sets of admissible strategies, P i , i = 1, 2 , are such
that for all p i ( • ) G P i , i = 1, 2, and all 	 (xo, to) G D x R l , there is
at least one solution of (3).
Of course, the choice of P i satisfying Assumption 1 depends on
function f( • ) . Conditions assuring existence of a solution at (xo, to)
are of the following kind (e.g., see Refs. 5,6,7):
i) F(xo, to) is compact and convex, and
ii) F( • ) is upper semicontinuous on a compact set containing (xo, to) .
Now let T l and T2 be two prescribed closed sets contained in
D and such that D ^ T1 u T2	 These are the "targets" of players 1 and 2,
respectively.
Before we can define a "winning strategy" and a "winning set" for
a player we need to introduce the concept of "play".
Definition 1. A play is a quadruple {(xo , to ) , p 1 (•), p 2 (•) , x(•)} such
that
1- i (xo, to) e [D \(TI U T2 )^ x R 
2.	 p i ( • ) a Pi , i = 1, 2,
tGiven two sets, A and B , A \ B -6 {a e A I a ^. B} .
-a-
3. x(-) : [to ' tf] .} D	 or x(-) : [to, t f) -' D if x(tf ) is not
defined, with x(to) - x D
 , is a solution of (3) generated by
p '(-)	 1 = 1, 2,
4. x(t)	 T  u T2 for t e [t D , tf)
5. either i) x(t f) e T1 u T2 for tf <	 or
ii) tf = - , or
iii) tf
 is a finite escape time for x(•)
that is, x(t) + x c- DD or 11x(t)II +
	
as
t + tf .
Now we can define the concepts of winning strategy at a point
and then of a winning set.
Definition 2. A strategy p l H E P1 is winning for player 1 at
( X D , to) E [D \(T1 u T2 )] x R l iff the set of all plays
{(XD, to) , p l (,) , p 2 (,) , 
x(-))
 
for all p 2 ( • ) E P2 is nonempty and
contains no members satisfying 5i) with x(t f) E T2 , or 5ii) or 5iii) of
Definition 1. A completely analogous defirltion, with 1 and 2 inter-
changed, holds for a winning strategy p 2 ( • ) E p2 at (xD , to ) .
Simply stated, a winning strategy for a player at (xD , to)
guarantees termination on his, and only his, target no matter what the
strategy of the other player.
Definition 3. A set W c D \(T 1 U T2 ) is 1-winning ( or 2-winning) iff
player 1 (or 2) Aas a strategy p l (•) (or p 2 (•)) that is winning for
him at all (x D , t o ) e W x Rl
3
An essential part of the solution of a game is the mapping of
D into its 1-winning and 2-winning subsets, along with the characterization
of the associated winning strategies, p l ( • ) and "p2 ( • )	 In the terminology
of Ref. 4, the union of all 1-winning (or 2-winning) sets together with T1
(or T2 ) is SE
 (or Sp ) .
In the next section we consider the problem of finding winning
strategies by giving a theorem of conditions sufficient for a set to be
1-winning (2-winning).
3. WINNING SETS
The game as defined in Section 2 is completely symmetrical with
respect to players 1 and 2. In this section we give conditions sufficient
for a set to be 1-winning. Of course, the theorem with 1 and 2 interchanged
holds equally for a 2-winning set.
The conditions in the theorem below depend on an a riori chosen
quadruple {V 1 ( • ) , V 2 (•) , k l , k 2 } where V
i
(-) : D + Rl , i = 1, 2, are
C l
 functions for which there exist constants C 1 and C2 such that
i) T1 D AI A {x E D I V 1 (x) s C 1 }
ii) T2 c A2 Q {x E D I V 2 (x) < C21
and
iii) k  > 0 , k 2 are scalar constants.
Given a quadruple (V 
l (-) 1 V 2 ( • ) , k l , k2 } , let
	
W(V 1 ,	 V 2 , k l , k 2 ) © {x E D \(T 1 u A2 )^ V1 xkl_ Cl > V 2 x k2 - C2 }	 (4)
t
r
-°d
ORIGINAL PAGB IS
Or POOR QUA13TY1
^p
-b-
Before stating a theorem, we introduce another assumption.
Assumption• n 2. If D = Rn , given p i (•) E P i
 , i = 1, 2, and a quadruple
(V l ( • ), V 2 (•), k it k 2} , no solution of (3) with (x O , to) a W(Vi) V 21 kit k 2 ) x R1
has a finite escape time.
This assumption is satisfied if
i) equation (3) satisfies a linear growth condition, or
ii) function V l (•) in Theorem 1 below is radially unbounded;	
^i
that is, V l (x)	 - as I1x11 -F w .
Now we are ready to state the main theorem.
THEOREM 1. If there exist a_ quadruple {V l (•), V 2 (•), kit k2 } , a strategy
P l (.) E P 1
 and a_ function p l ( )	 D + all nonempty subsets of U l , such that
i) P1 (X, t) = pl (x)	 V(x, t) e D x R1
ii) V x E W ( V 1 , V 2 , k i t k2 ) and V u l E 5 1 (X)
	sup	 VV (x) f(x, u l , U2 ) s - k
	 inf	 VV 2(x) f(x, u I , u 2 ) z - k
	
2EU	 U2	
1	 1	 2EU 2	
	 2
 
iii) Assumption 1 is met,
iv) D is an invariant set of (3) with p i ( ) = P1 ( • ) and all p 2 ( • ) c- P2
or else D = R n and Assump tion 2 is satisfied,
then W(V 1 , V 21 kit k 2 ) is 1-winning.
Proof. first we show that if a trajectory of (3), with p l (•) = plO ,
any p2 ( • ) E P 2 and (xo, to ) E W(V i ) V21 k i t k2) x R 1 , remains in
W(V I , V 2 , k it k 2 ) for t < tf , then it terminates on T 1 u T2
 ; that is,
5ii) of Definition 1 cannot occur. In view of condition iv), no such
trajectory has a finite escape time and so may be
11
-7-
extended over any interval [t	 t)	 including t =	 in case of0 f
	 f
non te rip ination. Consider a trajectory corresponding to solution
	
x( • )	 [to , tf) + D if t f = W , with
x(t0) = x0 G 41(V1' V2' k 
	 k 2 )	 In view of ii) and the supposition that
x(t) a W V 1 , V2 1 k l , k2) for t e [t0 , tf)
VV 1 (x(t)) x(t) 6 - kl
VV 2 (x(t)) X(t) > - k2
a.e. [to, t f )	 Upon integration, e.g., see Ref. 7, we obtain for
t e [t0 , tf)
V 1 (x0 ) > V 1 ( x ( t )) + kl ( t - to)	 (5)
V 2 (x0 ) 6 V 2 (x(t)) + k 2 (t - to )	 (6)
Let
o	 Vl(x0) - Cl
t = t0 +	 (7)k 
1
and suppose that 't < t f . Then it follows from (5) and (7) that
V 1 (x(t)) 6 C 1 .
By the definition of V 1 ( • ) , namely, Ti D 61 , this implies that
x(t) E T1 , contradicting the supposition of termination at t = t f or
else non-termination, tf = W . Thus, t > tf and termination must
occur on T1 U T2 at t = tf < m .
Next we demonstrate that termination takes place on T1 \ T2
Two cases must be considered, k 2 6 0 and k2 > 0 .
For k2 5 0 , since
ORIGINAL PAG1; 1S
OF POOR QUALM
V i (x) - C 1 > 0	 for	 x C- D \	 Ti (p)
V 2 (x) - C2 > 0
	 for	 x E D \ A2 (9)
it follows from (4) that
W(V i )	 V 21	 k i 9 	k2 )	 = D \(T i u p2 ) (10)
Hence,	 by condition iv),	 x(t) E W(V I ,	 V2, k i ,	 k 2 ) for	 t E [t o ,	 t f )	 .
In view of (9), for each	 x 0 a W(V i , V 2 1 k i ,	 k2 )	 there exists an	 e > 0
such that
V2 (x 0) = C 2 + e (11)
Since	 x(t) E W(V i , V2 ,	 k i ,	 k 2 )	 for all t c- [to , tf )	 ,	 it follows
from (6)	 that
V2 (x 0) < V 2 (x(t)) V	
t (=- [t o ,	 tf ) (12)
whence,	 by (11),
V 2 (x(t)) a C 2 + e V t E [tO, tf)
which implies that x(tf) I A2 :1 T2	Out x(tf ) E Ti U T2	so that
X(tf) E 
71 \
	 T2
	. ,
For	 k2 > 0 ,	 suppose there is a	 t < tf for which
X(t)	 W(V i ,	 V
21
	 k i , k 2 )	 .	 Then, by continuity of V i ( • ),	 V 2 ( • )	 and x(•),
there exists a
	 t < tf
	such that
x('t)	 E W(V i ,	 V 2 1 	k i , k 2 )	 V t e [to ,	 t)
and
k i	 2
V 1 (x( t ))-C i	V2(x(l))-C2
(13)
11
1
It
-9-
Now,	 M) and (9) hold for	 x(t) c W(V l , V21 k i t k 2 ) and so, by continuity
of	 V l (-)	 and	 V2 ( • )	 and by positivity of
	
k l and k2
V 1 (xo)- c l 	V l (x kt))-C l i	
(t -	 tC)
1	 1
•	 (14)
V 2 ( k ) - C2	 V2( xkt) ) -C 2
+	 t -	 t0 )	 .
2	 2
Since	 xo r_ W(V l ,	 V 2 ,
	
k it	 k 2 )	 ,	 it follows from (4) and (14)	 that
k l 	 k2> _
V l (x(t))-c l 	V2(x(t))-C2
(16)
This	 contradicts	 (13), and so
x(t) E W(V l , V2 9 	k it	 k 2 )	 V t c [to ,	 tf)
Thus, as shown earlier,
	 x(tf ) E Tl I T2 .
Novi suppose there is a	 t G t 	 such that x(t) E A 2 \	 T1 D T2 i T1
and	 x(t) 4- T1 a A2 a T1 u T2	 for all	 t e [to , t) Since	 x(t) e A2 \	 T l	 >
V 1 WO) > C-1
and
V 2 (x( t )) 6 C 2 (17)
Substitution of (16) and (17)	 in	 (14) yields
Vl 
^l -G^t ltp
I,
f	 a
-fu-
and
V 
x k2-C ^ 1
2 0
	
2	 t-t0
so that	 j
ki	
A
I
1 ( XOT 1 < 2x0 2V	 -C2
8
which contradicts x0 E W(V l ; V 2 , k l , k 2 )	 Hence, x(t)
	
A2 \ T1
	T2 \ T1
	 1
for t < tf
 so that x(tf) c Tl
 \A CT \ T2
Since these conclusions hold for all x
0
 E 
W(V 1 3 V2' k l' k 2 ) '
that set is a 1-winning set.
Remarks
1. Since a winning strategy "p l ( • ) with (x0 , t0 ) a W(V l , V 21 k l , k 2 ) x R1
guarantees that x(t) E W(V 1 , V2 , k l , k 2 ) for all t E [t 0 , t f ) , only
the restriction of the strategy to the 1-winning set need be determined.
2. There may exist 1-winning strategies at (x 0 , t0) ^'- W(V l , V 2 , k l , k2 ) x R1
In general, however, such strategies will not be 1-winning at other
initial points.
3. If k2 < 0 , region A2
 is an avoidance set in the sense of Ref. 8.
4. As k  + 0 , t given in (7), tends to - • Thus, if k  = 0 ,
termination cannot be guaranteed; in particular W(V 1 , V 2 , k l , k2)
need not be 1-winning. However, the inequality in (4) then requires
k2
 < 0 so that, by the second of conditions ii) of the Theorem,
W(V 1 , V21 kl , k2 ) cannot be 2-winning.
5. If T2 =	 , let V2 (x) = constant # C 2
 for all x E D \ Tl and
let k2 = 0	 The inequality in (4) is then satisfied trivially.
a
The set W(V l , V2) k l , k2 ) being 1-winning is then equivalent to
target Ti being capturable in finite time. Theorem 1 is thus
related to the target capture theorem of Ref. 9.
6. Let
S(R,) 6 (x a W(V l , V 2 , k l , k 2 ) I V l (x) 4 R} a Ti u T2
and suppose that it is required to accomplish termination on
Tl \ T2 for all x 0 a S(R) in a time interval not exceeding t* -
 to .
This can be assured by the conditions of Theorem 1 with
k 	
Q*Cl	
(18)
t -to
For, as shown in the proof of Theorem 1, for given x D c W(V l , V21 k i , k2)
t = tD+	 k	 a tf1
*
and x(tf ) e T  \ T2	 Thus we impose	 sup t 6 t	 whence
k-Cl
xoss(R.)
*
tf - t 0 6 -^6t - to
1
giving the condition (18).
;. If more than one set, say T 2 , T3 , ..., Tr , is to be avoided before
terminating on Ti \ (T2 u T3
 u ... u T r ), the conditions of Theorem 1
are augmented by introducing for i e (2, 3, ..., r}
Ai ^
_ {x e D I V i (x) 4 C i } :) Ti ,
replacing W(V l , V 29 k l , k 2 ) by
W(V l , V21 ...,
 
V r , k i , k 2 , ..., k r )	 (x e D \ (Tl u o2 u ... u Add
k l	 ki
V l x -C l	 V i x -Ci 	 i = 2, ...> r}
}
1
1.
i
and requiring V u l ® pl(x)
inf	 VV i (x) f(x, u l , u 2 ) > - k 
u"c-U2
4•	 EXAMPLE
Here we illustrate the use of Theorem 1 by means of a very simple
example, a pursuit-evasion game between inertialess objects P and E with
constant speeds vp and vE
	respectively. The equations of motion (see
Figure 1) are
R = v E cos (a + OE)
0 = v R sin	(a + OE)	 (19)
r = vp cos 0p - V E cos PIE
8 = r (vp sin O p - v E sin 0E)
The targets of E and P	 respectively, are
T
I
 = {(R, 0, r, 0) a R4	 R < PE = constant > 0}
(20)
T2 = ((R, 0, r, 0) E R 4	 r s pp = constant > 0} .
Thus, the evader, E , wishes to choose the values of O E so
that he reaches T1 (R = p E ) before he is intercepted (r = p p)	 no
matter how the ursuer
	 P	 choases the values of 0P	 , , p- .
To apply Theorem 1,	 let
vi
=R ,
	 V2	
t:
i
i	 d
A
so that Ai = Ti , i = 1, 2, and
VV l = ( 11 0 0 0, 0)
VV2 = (0 1 0, 1 1 0)
Conditions ii) of Theorem 1 become
VI: cos (a + OE ) < - k l < 0
-vp - ve cos O E > - k2
and W(V,, V21 k l , k2 ) is defined by
k
r - pp
	
k2 
(R - PE)
1
Condition (21) implies
k
-1 < cos (a + 0 E ) < - V1 < 0
E
1
(21)
(22)
(23)
(24)
so that
v
k l	 11
 
—
E
 for d e [0 5 m) .
i
For give., k l	region W is maximized by choosing the smallest
k2 such that (22) is met for all possible a = 0	 0	 This results in
VE
k 2 _ 
"vP + 11+6
whence
k2	 1 + l+d vP
71
	 (	 ) v E s
Hence, the largest W results from d = 0 	 namely,
j
v +v	
(25)r
-Pp> v P(R-pE)
P
i The c;:rresponding escape strategy, according.to (24), is given by
cos (a + O E ) = -1 ; that is, E moves radially inward and is assured
termination without interception by P provided the initial conditions
satisfy (25).
Strategy OE
 = n - (0	 is continuous on D = R4	Hence,
for sufficiently well-behaved OP
	system (19) possesses a solution at
every initial state. Furthermore, (19) satisfies a linear growth condition.
Thus Assumptions 1 and 2 are met and 'W given by (25) is indeed 1-winning,
that is, winning for the evader, E .
Since v 	 and vE
 are positive, (25) implies that
r - pp > R 
-PE no matter how much faster the evader is than the pursuer.
Thus, W(V l , V 2 , k l , k2 ) does'not include initial configurations for
which P lies between E and his target, Ti . This restriction is not
surprising since Theorem 1 relates to all initial	 g	 positions for :which the
i	
evader's strategy is winning (see Remark 2).
l
i
i
rf
References
1. R. Isaacs, "Differential Games", Wiley, N. "., 1965.
2. 0. Hajek, "Pursuit Games", Academic Press, N. Y., 1975.
3. A. Merz, The Homicidal Chauffeur - A Differential Game, Ph.D.
Dissertation, Stanford University, 1971.
4. A Blaquitre, F. Gerard and G. Leitmann, Quantitative and
Qualitative Games, Academic Press, N. Y., 1969.
5. E. Roxin, On Generalized Dynamical Systemis Defined by
Contingent Equations, Journal of Differential Equations 1
(1965), 188-205.
6. A. F. Filippov, Classical Solutions of Differential
Equations with Multi-Valued Right-Hand Side, SIAM Journal
on Control 5 (1967), 609-621.
7. Yu. I. Alimov, On the Application of Lyapunov's Direct
Method to Differential Equations with Ambiguous Right Sides,
Automation and Remote Control 22 (1961), 713-725.
8. G. Leitmann and J. Skowronski, Avoidance Control, Journal
of Optimization Theory and Applications, to appear.
9. D. J. Sticht, T. L. Vincent and D. G. Schultz, Sufficiency
Theorems for Target Capture, Journal of Optimization Theory
and Applications 17 (1965), 523-543.
Alll_
List of Symbols
x
	
lower case "eggs"
x
	
"multiplication" symbol
0 zero
T
	
script upper case "tee"
G
	
"belongs to" symbol
E
	
lower case epsilon
P
	
lower case rho
0
	
lower case theta
0
	
upper case theta
a
	
lower case alpha
a
	
lower case delta
A
	
upper case delta
7T
	
lower case pi
R
	
script lower case "ell"
"infinity" symbol
"empty" symbol
0
inertial
reference
Figure 1, Pursuit-Evasion
pt. Y AVt
o3 
A11^I
p1^1^^II` K
	
L
v^, yW


Qualitative differential games with two targets

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