Office of Naval Research

Contract Nont-1866 (16) NR- 372-012

DIFFERENTIAL GAMES

ANDOPTIMAL PURSUIT-EVASION STRATEGIES

F-COPY-. .. OF _.. _

E HARD-COPY $. ,.oo

LMICROFICHE $ 6,5-0

//PD. DC

By JA 0 1 0 oiYu-chi Ho, A.E. Bryson, Jr. &

DDCAIRA B

November 4, 1964

Technical Report No. 457

Cruft Laboratory

Division of Engineering and Applied Physics

Harvard University -Cambridge, Massachusetts

BEST

A n,-- . . ,A VA ILA LE 'y

Office of Naval Research

Contract Nonr-1866(16)

NR - 372 - 012

DIFFERENTIAL GAMES

AND OPTIMAL PURSUIT-EVASION STRATEGIES

by

*Yu-Chi Ho, A. E. Bryson, Jr., and S. Baron

November 4, 1964

The research reported in this document was made possible throughsupport extended to Cruft Laboratory, Harvard University, by theU. S. Army Research Office, the U. S. Air Force Office of ScientificResearch, and the U. S. Office of Naval Research under the JointServices Electronics Program by Contract Nonr-1866( 16). Repro-duction in whole or in part is permitted for any purpose of the UnitedStates Government.

NASA, Langley Field. On leave at Harvard University.

Technical Report No. 457

Cruft Laboratory

Division of Engineering and Applied Physics

Harvard University

Cambridge, Massachusetts

DIFFERENTIAL GAMES AND OPTIMAL PURSUIT-EVASION STRATEGIES

by

Yu-Chi Ho, A. E. Bryson, Jr., and S. Baron

Division of Engineering and Applied Physics

Harvard University, Cambridge, Massachusetts

ABSTRACT

In this report, we show that variational techniques can be applied to

solve games with differential constraints. Conditions for capture and for

optimality are derived for a class of optimal pursuit-evasion problems.

Results are used to demonstrate that the well known "proportional navigation

law" is actually an optimal intercept strategy.

-lili-

TR457 I. INTRODUCTION

The theory of differential games was initiated by Issacs in 1954 [1].

It was later studied in greater detail by Fleming and Berkovitz [2]. Recently,

because of the advances in the computational solution of variation problems,

renewed interest has developed in this subject. Stated simply, games are a

class of two-sided optimization problem. The simplest games are the type with

which we are most familiar. A function J of two discrete variables u and v

is given in tabular form. Each particular value of u or v is called a strategy.

The problem is the determination of the strategy for u and v such that J is

a saddle point (minimax). Because of the discrete nature of the available

strategies, it is generally not possible to realize a minimax with a simple

choice for u and v. Instead, a mixture of strategies must be employed to

realize a minimax on an average basis. The usual game theory devotes

considerable effort to the study of the properties of mixed strategies. On the

other hand, if u and v were to take on a continuous range of values, then a

saddle point could generally be realized by a pair of pure strategies for any

reasonably well behaved function J (u, v). The necessary and sufficient

conditions for a saddle point are simply,

JT 0 J -- 0()u v for all u , v

uu >0 v <0 (2)

This is in direct analogy to the one sided optimization problem of calculus.

A more involved version of the game problem can be stated as that of

It is the authors' understanding that Prof. Pontriagin lectured on the subjectduring the month of October 1964.

TR457 -2-

determining a saddle point for J(x, u, v) subject to the restriction that

i(x; u, v) = 0. In this case, u and v still have the usual meaning while

the variable x is an intermediate variable indirectly determined by the constraint

= 0. Of course, we can always eliminate x using the constraints and convert

the problem into a simple game by writing J = J (g- (u, v), u, v) where

x -1 (u, v) => { = 0. However, it is usually simpler both theoretically and

computationally to solve the problem by the introduction of a Lagrange

multiplier again in imitation of the one-sided calculus problem. From this

point, it is a straightforward conceptual generalization to consider the differential

game problem of determining a saddle point for

T

J = $(x(T), T) + L(x, u, v) dt (3)

0

subject to

x = f(x, u, v, t) ; x(t ) x0 (4)

where x plays the usual role of the state vector, and u, v the control vectors.

Extending our analogy with the one-sided optimizatj';n problem, we expect that

the techniques of variational calculus should be useful in this context. The

purpose of this report is to illustrate that this is indeed so by solving a class

of optimal pursuit-evasion problems and deriving conditions for optimality and

capture. As an interesting by-product, we shall show that the "proportional

navigation" law used in many missile guidance systems actually constitutes

a form of optimal pursuit strategy, under the usual simplifying approximations

to the equations of motion of the missile and the target.

TR457 -3-

II. A CLASS OF OPTIMAL PURSUIT-EVASION GAMES

We shall consider two dynamic systems

=F x +G u (5)

:e Fe xe + G v (6)

where the subscript p and e stand for pursuer and evader respectively.

The pursuing system desires to intercept or rendezvous with the evading

system at time T while the latter attempts to do the opposite. Both systems

have limited energy sources. Hence, a reasonable criterion would be

Tx()IA+I SHuI - ]dtll

SL11x(T)-x (T)11- + -LI [1ulIR 2 ] dt (7)p e

where A> 0 and R and Re > 0. The minus sign in front of the llvll 2 term

comes about due to the fact that we shall attempt to maximize J w. r. t. to the

variable v . Following the usual variational procedure, we introduce at this

point the multiplier functions X (t) and X (t) which are used to adjoin (5)p e

and (6) to (7). Now let us take a particular pair of strategies u(t) and

v(t) with the associated trajectories x (t) and x (t) and considerp e

variations 6u(t) and bv(t). The change in J to second order is

6J = [(X (T)-x (T).TAT_ % T(T)]6x (T)+[(X (T)-x (T))TAT- T(T)]6xe(T)p e p p e A e e

T+.- 115xp(T) (6xe(T) 112 + [ +H x 6X +HI'I 6Xe

0

T ] 0 6ul

+ H 6u + H 6v]dt + -1 5 [6 u 6 v]T dt (8a)0 0 - R 6v

e

TR457 -4-

where H is the Hamiltonian

H = 1 112- Ilv~il ) + X T (F x+G u)+%T (Fx + G V) (8b)Z(u R p p p e e e e

In order that (8) vanish to first order, we immediately derive the necessary

conditions

T X T F XpT(T) (x (T)- xe(T))T A T (9)p PP p p e

T- XT F X T (T) (xe(T) - Xp(T))TAT (10)

U 0 => u p p - GpTXp-1 T

Hv =0 => v =+ Re Ge Te (12)

Equations (5, 6, 9- 12 ) represent a linear two-point boundary value problem

which can be solved in terms of the fundamental matrix solution of the "p1 and

"e" system, P p(t, T ) and 4e(t, r). We have,

XpM) =1p (T, t) A (x p(T)-x e (T))

(13)

ke (t) = DeT (T, t) A(xe (T)-xp (T))

Substituting (13) into (5-6) and using (11-12), we obtain

x p(T) = (T, t)xp (t) - M pA (x p(T) - xe(T))p p p) p (14)

Xe( ) :@e(, t Xet). Me (p(W- e ()

TR457 -5-

where

T

M (Tt) j (T, 7) G R G T T(TIr)dT (15)p p p Pt

TM e(Tt) A= S e (T I T ) GeRe " G e T e T (T,.T)d (16)

t

Finally, (13) and (14) combine to yield the optimal pursuit and evasive

strategies

u(t) = -R-IG T PT(T,t)A[I+(M -M )A]- [o (T,t)x (t)- e(Tt)Xe (t)](17)

p p p p e P p

v(t) = -R-le Ge T ' (Tt)A[I+(Mp-M)A] -l[ (T, t)xp(t). e(Tt)xe(t)] (18)

Since 0 (T, t) x (t) has the interpretation of the predicted terminal state of a

dynamic system, the optimal pursuit-evasive strategies are simply linear

combinations of the predicted terminal miss - a very reasonable result.

Examination of the second-order terms in (8) shows that the analogous

Legendre-Clebsch condition for the saddle point is satisfied.

Huu Rp vv e= - R <0 (19)

Furthermore, the accessory minimax problem is

"determine a saddle point for T6J= 16xp(T)-6Xe(T) 112+ ' [11 b ll 2 1 -II vll- dt

0 P e(p)

subject to 6x = F 5x + G 6u; 6 x (0)= 0

6c =F 6x +G 6v ; 6 x (0) 0e e e e e

TR457 -6-

By exactly the same argument leading to the derivation of the conjugate point

conditions for the one sided variational problem [3], we find that

6J> 0 V 6u/0, 6v fixed,and 6J<0 V 6v /0, 6ufixed, if and only if

the matrix solution X (t) is nonsingular during the interval (0, T) where

X (t) obeys

-I TX F 0 -G CRG T 0 X X(T) = I

0 F 0 G R-IG Te e e ee T (20)

0 0 -F 0p T A -A')

A 0 0 0 -F e A A-

But from (20) ,

X (t) -p(t)T) 0e(tT K p-Me (21)

0 () L e A I-Me jThus, the nonsingularity of X (t) (i. e., nonexistence of conjugate point) is

equivalent to the condition

de +Mp- I = det[I+(M ^M )A]/0 (22)A Me Ae p e

which in view of (17) and (18) also guarantees the boundedness of the

strategies.

It is also interesting to consider the limiting case where A is

positive definite and infinite (e. g., infinite diagonal matrix). This is the

situation where u attempts to capture v with minimal energy. Condition

(22) becomes

TR457 -7-

det [I+ (Mp - M e)A]/O<:>det(A-I + (Mp -M e )) /0 (23)

<=> det (M -M ) /0p e

In terms of the usual definition of controllability, (23) simply says that the

pursuer must be "more controllable" than the evader - an eminently reasonable

conclusion. Since both M and M > 0 if the individual systems arep e

controllable, then a sufficient condition for capture is simply M - M > 0.p e

Furthermore, we have the following

Proposition: Let Rp and Re in (7) be scalar, and the optimal pursuit and~p e

T Tevasion energy for A= o be T jjujj 2 dr= cp jjvjjdt= ce) respectively,

0 C

then a necessary and sufficient condition for capture of an evader with energy

resources ce by a pursuer with energy resourses c is M -M > 0. Theep p e

proof of this proposition is a direct consequence of the utility interpretation

of Lagrange multipliers [4].

III. GUIDANCE LAW FOR TARGET INTERCEPTION

A special case of the class of problems treated in Section II can be

formulated as follows: the equations of motion (kinematic) in space for an

interceptor and a target are

v f + ap p p

r = VP P (24)v =f + a

e e e

r ve e

TR457 -8-

where

v = velocity of a body in three dimensions

r = position vector of a body in three dimensions

= external force per unit mass exerted on the body

a= control acceleration of a body

We assume that the altitutude difference between the pursuer and the evader

is small enough such that f = f . Hence, if we are only interested in thep e

difference Y- (t) - "- (t) , the effect of the external forces can be ignored.p e

Now consider the criterion

J = -l r(T) -- re(T))("r (T) - "r(T))

Zp e p e

T (25)

+ ) - c (a e'a)]dt2 [c ( e ep e

0

where c and c e represent the energy capacity of the pursuer and thepe

evader, respectively. Applying the result of the last section, it can be

directly verified that (17) and (18) become in this case

-e + () Va= p p e p e (26)

-- + -c ) (T - t) /3a (p e

- C (T-t)[% (t)-1" (t) + (e(t)-"Ve(t))(T-t)]a e 3 (27)

- + (c p-c e ) (T.-t) /3

TR457 -9-

We note immediately that if

(i) if c > ce (i. e., pursuer has more energy than the evader)p e

then the feedback control gain is always of one sign.

(ii) if c < ce (i. e., pursuer has less energy than the evader)p e

then the feedback gain will change sign at

1 + (c -c ) (T- t) 3/3 = 0 (28)a Cp e

for T sufficiently large.

But (28) is simply the conjugate point condition (22) specialized for this

problem. Hence, for case (ii), (26) is no longer optimal for large T. This

fact is, of course, obvious to startwith, particularly in the case when a = oo.

In that case, interception is not possible when c < ce (cf. M < M ). Assumingp e p e

(i) and letting a = co , the control strategy for the pursuer simplifies to

p 2 p - p (

(1- e) (T-t)Cp

Let the pursuer and the target be on a nominal collision course with range

R and closing velocity V = R/(T-t). Let x -xe represent the lateralc p e

deviation from the collision course as shown in Fig. 1

TR457 -10-

x ip _x-e Line of sight angle

R-Xe I (T Nominal line of sight

H(-"R = V(T-t) -)Pursuer Evader

Fig. 1 Geometry of Proportional Navigation.

Then the lateral control acceleration to be applied by the pursuer according

to (29) is

3a (lateral) V V c (30)C C

(1- -e)

Cp

which is simply proportional navigation with the effective navigation constant

K = 3 From experience it has been found that the "best" valuee c

Cp

for K ranges between 3 to 5 [5]. In view of (30), we see that the value of 3e

corresponds to the case when the target is not maneuverable[6] (ce = 0) ) whileC

the value of 5 corresponds to e 2c 5

p

TR457 -11-

REFERENCES

1. R. Issacs, "Differential Games I, II, III, IV, " RAND Corporation ResearchMemorandum RM-1391, 1399, 1411, 1468, 1954-56.

2. L. D. Berkovitz and W. H. Fleming, "On Differential Games with IntegralPayoff, " Annal. of Math. Study No. 39, pp. 413-435, PrincetonUniversity, 1957.

3. J. V. Breakwell and Y. C. Ho, " On the Conjugate Point Condition for theControl Problem," Int. Journal of Engineering Science (to appear) 1964.Also, Cruft Laboratory Technical Report No. 441, Harvard University,March 1964.

4. R. Bellman, Adaptive Control-A Guided Tour, Princeton University Press,Chapter VI, pp. 103-104, 1961.

5. A. Puckett and S. Ramo, Guided Missile Engineering, pp. 176-180,McGraw-Hill Book Company, New York, 1959.

6. A. E. Bryson, "Optimal Guidance Laws for Injection, Interception,Rendezvous, and Soft Landing, " AIAA Journal (to appear).

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