Rufus Isaacs - Differential Games: A Mathematical Theory with Applications to Warfare and Pursuit, Control and Optimization

(A1.1)

Both players have simple motion in a plane. The pursuer P is faster and could capture E readily were he not deterred by a barrage of gunfire assisting E . The firing is continuous and emanates from a battery located at a point O . The instantaneous hit probability is inversely proportional to the distance OP , and its time integral, as in Section A1, is to be the payoff.

Thus P seeks to capture with a minimum chance of being hit; he must choose a course which compromises between a direct chase of E and a wariness of being too close to the battery for too long. Similarly, E must inculcate into his flight from P maneuvering which lures P into lethal proximity to O .

We shall use the polar coordinates r , for P and the rectangular coordinates x , y for E , both systems having the same origin O with the lines = 0 and y = 0 coinciding. The speeds of P and E shall be 1 and w with w < 1, so that w is really the speed ratio of E to P . The control variables and are as in . The KE are thus

The payoff is integral with

a being a fixed positive constant. The terminal surface is the set where | PE | = l , a given positive number, and , the set with | PE | /.

The solution is obtained almost entirely by pure integration, this example being very exceptional in its freedom from singular phenomena. One singular surface there certainly is: when P , O , E lie on a straight line in this order, the two symmetrical ways that P may skirt O clearly imply a dispersal surface, with an instantaneous mixed strategy.

Our standard method then presents no difficulties (except possibly the common one of deciding the sign of the V i of the initial conditions). From the RPE, it will follow at once that E always travels a straight path. Any route of P , however, lies on a curve:

where c 1,... , c 4, K are constants; K may be of either sign, meaning that for some paths sin, cos, replace sinh, cosh.

is a scale plot of typical partie. Here w the dots indicate corresponding values of . Naturally any corresponding pair may be considered as a starting position.

Exercise A2.1 . Write the KE for this problem using three state variables instead of four.

Exercise A2.2 . Solve a version of the preceding problem which differs from it in that the battery, instead of being at a point, is distributed uniformly over a line . That is, G = a/d , where d = distance P to .

Research Problem A2.1 . Suppose in the problem of the text we had taken for G a function more rapidly decreasing than the inverse of r . Intuitively it might appear that, were this decrease rapid enough and had E time enough, the best strategy for E would be to get near O and to loiter there. Is this ever actually the case? If so, how is such a problem solved?