Interception in differential pursuit/evasion games

A qualitative criterion for a pursuer to intercept a target in a class of differential games is obtained in terms of \emph{future cones}: Topological cones that contain all attainable trajectories of target or interceptor originating from an initial position. An interception solution exists after some initial time iff the future cone of the target lies within the future cone of the interceptor. The solution may be regarded as a kind of Nash equillibrium. This result is applied to two examples: 1. The game of Two Cars: The future cone condition is shown to be equivalent to conditions for interception obtained by Cockayne. \cite{ref2} 2. Satellite warfare: The future cone for a spacecraft or direct-ascent antisatellite weapon (ASAT) maneuvering in a central gravitational field is obtained and is shown to equal that for a spacecraft which maneuvers solely by means of a single velocity change at the cone vertex. The latter result is illustrated with an analysis of the January 2007 interception of the FengYun-1C spacecraft.


I. INTRODUCTION
A variety of pursuit/evasion problems may be treated by the methods of differential game theory. 3,4 These range from idealized problems of an illustrative nature, such as the Homicidal Chauffeur or Two Cars games, 3,5,6 to detailed studies of optimal strategies in air-to-air combat. [7][8][9] In a previous paper 1 , a qualitative criterion for interception or capturability was devised for a class of differential games of kind suggested by Pontryagin. [10][11][12][13] The criterion is a simple one: Let the future cone K + be the set of all attainable trajectories available to a player subsequent to some initial time. Interception of the target by the interceptor is guaranteed to be possible if This latter problem is applicable to the study of a direct-ascent antisatellite weapon (ASAT) engaging a spacecraft in low Earth orbit. An account of the January 2007 interception of the FengYun-1C meteorological satellite by a Chinese ASAT appears in Section IV as a worked example.
Following a discussion of the results in Section V, a final Section VI presents conclusions.

II. CONDITIONS FOR GUARANTEED INTERCEPTION
This paper is concerned with differential games that describe the pursuit of a target with position y(t) at time t, by an interceptor whose position at time t is x(t). Both target and interceptor can maneuver freely and autonomously, subject to constraints. The evolution of x and y is of the form dx dt = F (x, u) and dy dt = G(y, v).
where F and G are assumed to be bounded analytic functions, and u = u(t) and v = v(t) are piecewise analytic controls.

A. The Future Cone
We recall the definition of the future cone of a maneuvering player as given in Ref. 1, and review its properties. The net effect of propulsive forces and of external forces such as gravity or aerodynamic drag cause the position r(t) of the interceptor (respectively, target) to evolve as it maneuvers. The evolution will be continuous, but not necessarily differentiable.
In addition, target and interceptor will be subject to physical limitations on their peak acceleration or total velocity change ∆v. Finally, target and interceptor may be expected to interact within a compact subset of R 4 .
At an initial time t 0 , call the position of the interceptor r(t 0 ). The set of all possible histories for r(t) originating at r(t 0 ) comprises a topological cone in R 4 with vertex r(t 0 ).
We call the set of points subsequently accessible to the interceptor in the time interval (t 1 , t 2 ), with t 0 ≤ t 1 < t 2 the future cone of r(t 0 ), written K + x (t 1 : t 2 ; r(t 0 )). 43 Following Ref. 1, we assume that the subset of either cone lying to the future of its vertex is a manifold with compact closure possessing a timelike foliation. We shall denote by K + x (t; r(t 0 )) a leaf of the foliation corresponding to a time t. In particular, there is no assumption that future cones of either target or interceptor, or their leaves, are necessarily convex.

B. Guaranteed Interception
The properties of optimal solutions to (2) and (3)  g. Refs. 5-9,23. These studies variously treat games of degree or of kind, but all restrict the form of the game to facilitate its analysis, by means such as limitation to a linear game, or to fixed velocity ratios for the players, or to piecewise constant radii of curvature.
We work here with (2) and (3) in fairly general form, but restrict attention to the more limited goal of finding qualitative conditions under which we may be confident that the interceptor can force an interception, optimally or no. The winning strategy for the interceptor is to choose a trajectory that intercepts the target at some time to the future of t 0 .
This choice amounts to a mapping Φ : K + x → K + x from the set of all trajectories available to the interceptor to its desired actual trajectory. A guaranteed intercept will be said to exist when an interception solution always exists, no matter how the target maneuvers within its future cone.
Then, a necessary and sufficient condition for the existence of a guaranteed intercept at t i ∈ [t 0 , t 1 ] is Proof : Sufficiency: Suppose that, of all the possible trajectories ⊂ K + y (t i ; y(t 0 )), the actual trajectory of the target is y * (t). The mapping from K + x (t α : t ω ; x(t α )) into K + x (t α : t ω ; x(t α )) ∩ K + y (t 0 : t 1 ; y(t 0 )) = K + y (t 0 : t 1 ; y(t 0 )) is given by the correspondance Φ(x) = {y * (t), t ∈ [t 0 , t 1 ]}, which is continuous. Its value is homeomorphic to a one-simplex in R 4 , and is thus acyclic. The cones K + x (t α : t ω ; x(t α )) and K + y (t 0 : t 1 ; y(t 0 )) are differentiable manifolds by construction, and thus triangulable. 17 Each may therefore be regarded as the polyhedron of a simply connected simplicial complex and is, in consequence, both an acyclic set 18 and an absolute neighborhood retract. 19 The conditions for the Eilenberg-Montgomery theorem are satisfied: A fixed point exists for t 0 < t i < t 1 . Combining this result with the tautological fixed point y * (t i ) ∈ K + y (t 0 : t 1 ; y(t 0 )) resulting from the target's ability to maneuver freely, we find   x * (t i ) The proof of the necessary condition is identical to that in Ref. 1 but is included here to make the treatment self-contained. We begin with a condition for the relation between leaves of the two cones at a single time. A necessary condition for the existence of a guaranteed intercept is that, at the time of intercept t i , for t 0 , t 1 < t i : Suppose a guaranteed intercept exists at time t i . Then, every point y in K + y (t i ; y(t 1 )) must coincide with some point x in K + x (t i ; x(t 0 )) in order that x − y = 0 for at least one pair of values of x and y. Were (7) false, there would be some portion of K + y (t i ) that lay outside the attainable set of interceptor positions at that time. Thus there would be a subset of K + y (t i ; y(t 1 )) for which x − y > 0, ∀x ∈ K + x (t i ; x(t 0 )). The necessary condition for the entire cone K + y (t 0 : t 1 ; y(t 1 )) is obtained by transfinite induction 16 . Let t α < t 0 < t β < t γ < t i < t 1 < t ω and take t i − t as an ordinal. We prove the result for K + y (t β : t 1 ; y(t 0 )) ⊂ K + y (t 0 : t 1 ; y(t 0 )) and extend to the full set K + y (t 0 : t 1 ; y(t 0 )) at the end.
We begin by showing the necessary condition holds at late times. Suppose that a guaranteed intercept exists at time t i for t γ , t i within any neighborhood of t 1 . As t γ → t 1 , K + y (t γ : t 1 ; y(t 0 )) → K + y (t 1 ; y(t 0 )).
By the condition for a single leaf, it follows that Next, suppose that at least one guaranteed intercept opportunity exists for time t i between t γ and t 1 . By the inductive hypothesis, We wish to examine the prospects at an earlier time t β . Consider the sets K + y (t β : t γ ; y(t 0 )) and K + x (t β : t γ ; x(t α )): K + y (t β : t 1 ; y(t 0 )) = K + y (t β : t γ ; y(t 0 )) ∪ K + y (t γ : t 1 ; y(t 0 )) (11) and similarly for K + x . But if a guaranteed intercept is to be possible ∀t ∈ (t β , t γ ), at no time t in (t β , t γ ) can it be that by the single-leaf condition. Letting t β → t 0 , we have (4).
As noted in Ref. 1, the result (6) may be interpreted as a Nash equillibrium. 21 The interceptor strategy in (6) is a strictly dominant one. 22 That is, while the strategy of the target allows it to move anywhere within its future cone, the interceptor always has available to it the strategy which places it at some future position of the target. On the other hand, no alternative to the target's strategy given in (6) will increase its chances of survival. We note that, having stipulated t 0 in K + y (t; y(t 0 )), any such alternative will necessarily constrict the volume available for maneuvering by the target, with (we may suppose) the effect of worsening its prospects. If we argue in this way, the strategy in (6) is preferable to any other available to the target, and thus strictly dominant. Both target and interceptor then have available to them a strictly dominant, hence optimal, strategy. We may conclude (6) is the unique Nash equillibrium for this problem.

A. Two Cars
In this game, introduced by Isaacs 3 , two cars maneuver in a plane. We have a target Car 2 with position (x 2 (t), y 2 (t)) at time t, pursued by an interceptor Car 1 whose position at time t is (x 1 (t), y 1 (t)). Players move with constant velocities v 1 and v 2 , respectively, and maneuver exclusively by steering. The motion of Car 1 in the plane is given by and and similarly for Car 2. The control law for θ 1 (t) is subject to the constrainṫ Cockayne 5 showed that sufficient and necessary conditions for Car 1 to intercept Car 2 and We show that the condition for guaranteed interception given by applying Theorem 1 to the game of two cars is equivalent to the paired conditions (16)- (17) found by Cockayne.
It is convenient to assume that the initial velocity vector of either car may point in any direction. This assumption ensures that the future cones are simply connected.
To see equivalence of the sufficient conditions, we adapt Isaacs' 3 construction of trajectories for Car 1 that result in interception of Car 2. Amongst the admissible trajectories ∈ K + 1 must be those produced by the Method of the Explicit Policy. The account of the method as applied to this problem given by Isaacs 3 can hardly be bettered (P is the pursuer, Car 1 and E, the evader, is Car 2): If P has the higher speed and at least as favorable a curvature restriction as E, capture can be attained. For P can first go to E's starting point and then follow his track.
It is clear that, if K + 2 ⊂ K + 1 , Car 1 can accomplish this for every trajectory of Car 2 ∈ K + 2 . Let K + XP be the cone comprised of all trajectories constructed by Explicit Policy: The resulting sufficient conditions required for the motion of Car 1 are two 3 : with and t f > t 2 large, but otherwise arbitrary. In (21), one may suppose that the difference between t 2 and t 1 includes an overestimate of the time for Car 1 to come about, should its initital velocity not be parallel to that of the target.
Within K + 1 (t 1 : t f ; r 1 (t 0 )) consider the future cone for Car 1 originating at r 2 (t 1 ). We wish to force an interception for t > t 2 . By hypothesis, Car 1's position at time t may be written formally as and The distance traveled from the origin at time t is The Schwartz inequality gives We thus find or By (29), the leaf K + 2 (t 2 ) of the target at a time t > t 2 is contained within a circle of radius R 2 , and the leaf K + 1 (t 2 ) within a circle of radius R 1 . From (22), or The Method of the Explicit Policy requireṡ From (13) and (14), and for Car 2, likewise, whence the acceleration by (15). Note that a · v = 0 at all times. Equations (32) and (37) immediately give The sufficient conditions (31) and (38) are those found by Cockayne.
We may see the equivalence of the necessary condition for guaranteed interception and (16)

B. Spacecraft maneuvering in a gravitational field
In this section, we calculate the future cone of a spacecraft maneuvering in a gravitational field by impulsive velocity changes. The result yields an account of a direct-ascent ASAT engagement with a target satellite. It is also applicable to the endgame of a co-orbital ASAT attack. 24 The motion of the spacecraft is given by where r is the spacecraft radius vector, v is its velocity, a(t) its acceleration, and µ = G N ewt. M ⊕ . (Because the spacecraft state descriptor includes the velocity, the condition for interception is modified in an obvious way.) We make use of an exact solution for the system (40) in the case of ballistic motion in the Earth's gravitational field. The motion of a spacecraft in a bound Keplerian orbit is obtained using Lagrange coefficients Φ 25 It is not possible to express the coefficients in (42) in closed form as functions of time.
However, it is possible to do so in terms of the corresponding true anomaly f . If f 0 is the and we have: In the foregoing, a is the semimajor axis, e the eccentricity, r 0 is the initial orbital radius at A complete description of the spacecraft motion also requires the eccentric and mean anomalies. These quantities are related by (Note that f 2 and E 2 always lie in the same quadrant.) The relation between E and the mean anomaly M given by Kepler's equation: By use of the mean motion (obtained from Kepler's third law) the time t corresponding to true anomaly f is obtained from M and the time of pericenter passage τ by The time interval between two points on a ballistic arc is thus We wish to calculate the future cone K + (t 1 : t 2 ; r(t 0 )) of a spacecraft maneuvering in the Earth's gravitational field. To that end, we model the motion of an idealized target or interceptor with a trajectory consisting of piecewise ballistic orbital segments punctuated by shocks in which The acceleration during impulsive maneuvers of actual spacecraft will be bounded and continuous, even when occurring in times much shorter than any other timescale characteristic of the motion. The description of impulsive velocity changes as instantaneous shocks is an idealization adopted for computational convenience. However, in the proof of Theorem 3 below, the acceleration history of the spacecraft is assumed continuous, however violent, in order to satisfy requirements of the Gronwall inequality. In order to include the limiting case of instantaneous shocks, we may treat the acceleration history as a distribution. The transition in (55) may then be regarded as the limit of a suitable sequence of good functions, and that limit may be taken at the end of other calculations.
We assume the spacecraft maneuvers by a number (possibly large) of small-impulse shocks, and that the total impulse available to it is limited by where x 2 = x · x. The spacecraft trajectory is thus a chain comprised of piecewise elliptical ballistic orbits that is continuous, but not necessarily differentiable at any point corresponding to a shock. For all values of the true anomaly between shocks t n < t < t n+1 the motion is given by the free-fall Lagrange coefficients, in expressions to be given, as a function of θ = f − f n 0 . In any ballistic interval of the spacecraft motion, f is the true anomaly of the osculating elliptical orbit that coincides with the spacecraft trajectory (in this connection, by "osculating" one should perhaps understand "liplocked" ).
We require a relation between the true anomaly and time at any point in the course of an engagement. A closed-from solution for the time (found by Kepler) may be expressed in terms of the eccentric and mean anomalies. Begin with the relations between true and eccentric anomalies within the n th ballistic arc for three points on the orbit: and In the foregoing, θ ≡ f − f n 0 (respectively, E − E 0 0 ) is the difference in true (respectively, eccentric) anomalies between the position r and the semimajor axis of the n th ballistic arc, while θ n ≡ f n − f n 0 (E n − E n 0 ) is the difference in true (eccentric) anomalies between the position r n and semimajor axis. The time interval between r and r n is given by (54) as At a time t n+1 , let a velocity change ∆v n+1 be imparted when the spacecraft is at position r n . The initial position and velocity are thus r n and v + ∆v n+1 . The semimajor axis of the new ballistic arc is obtained from the vis-viva equation The parameter (and hence the eccentricity) is given by and These quantities determine the Lagrange coefficient matrix Φ n+1 for the n + 1 st ballistic arc.
Call K + n (t 1 : t 2 ; r(t 0 )) the set of all trajectories originating from r(t 0 ) and subject to n shocks {∆v i , i = 1 . . . n, } respecting (56), with the understanding that for n = 1 the shock occurs at time t 0 . The future cone of a spacecraft maneuvering by ballistic motion between shocks is then We now prove a result relating the future cone K + ∞ (t 1 : t 2 ; r(t 0 )) of a spacecraft that maneuvers in a gravitational field by free fall punctuated with a countable sequence of shocks {∆v i }, subject to the overall upper limit on their sum (56), to the set K + 1 (t 1 : t 2 ; r(t 0 )) comprised of the union of single ballistic arcs with ∆v ≤ ∆v tot . If we stipulate that the spacecraft be capable of maneuvers for which ∆v = ∆v tot at the cone vertex, a particularly simple relation holds: By their respective definitions. K + ∞ ⊂ K + 1 : By induction. We first show that, corresponding to any piecewise ballistic trajectory with n shocks ∆v i , i = 0 · · · n − 1, an equivalent single ballistic trajectory exists connecting the cone vertex with any point on the piecewise trajectory subsequent to the final shock. We then show the velocity change ∆v n 0 at the vertex of the single ballistic arc obeys In the case n = 1 of a single shock at time t 0 , it is immediate that Assume that after n shocks the spacecraft location r n (t n ) is connected to the vertex by a single ballistic arc with n−1 i=0 ∆v i < ∆v tot . We show that after a further shock obeying n i=0 ∆v i ≤ ∆v tot , any point on the resulting ballistic orbit of the spacecraft r n+1 (t) with t > t n+1 is likewise connected to the cone vertex by a single ballistic arc ∈ K + 1 (t 1 : t 2 ; r(t 0 )). Take any point r n+1 (t) in (t n+1 , t 2 ). We seek a single ballistic arc connecting the cone vertex and r n+1 (t), v n+1 (t) of the form for Lagrangian coefficients Φ n+1 corresponding to some ballistic orbit ⊂ K + 1 (t 1 : t 2 ; r(t 0 )). To this relation we append the condition that determines the true anomaly corresponding to the final time of the single ballistic arc, Equations (66) and (68) The relation (69) between (θ, v n+1 0 ) and (t, r n+1 , v n+1 ) is, with stipulated r 0 , diffeomorphic.
As a result, a version of the implicit function theorem proved by Kumagai 28 supplies a (real) solution of (69) for (θ, v n+1 0 ). Thus, there exists a single ballistic arc corresponding to the n + 1 shock piecewise ballistic trajectory.
It remains to estimate the magnitude of ∆v n+1 The inductive hypothesis is that a single ballistic arc exists corresponding to a trajectory which has experienced n shocks, for which and Assume that (72) holds for n > 1. Now consider the full trajectory with n + 1 shocks, subject to the overall limit n i=0 ∆v i ≤ ∆v tot .
We may replace that portion of the trajectory corresponding to the first n shocks by the equivalent single ballistic arc. The entire trajectory may thus be replaced by one comprised of two ballistic arcs. Applying (65) and (72), we have ∆v n+1 We conclude Thus, a single ballistic arc ⊂ K + 1 (t 1 : t 2 ; r(t 0 )) connects each r n ∈ K + n (t 1 : t 2 ; r(t 0 )) with the cone vertex. Call the set of all such single ballistic arcs K + * 1 . Therefore, proving the theorem.
K + ∞ is a subset of B([t 0 , t 1 ]), the set of bounded functions on [t 0 , t 1 ], which is a complete metric space in the sup norm. K + ∞ is also closed : The mapping from in K + 1 ≡ K + ∞ is homeomorphic (diffeomorphic, in fact) and thus maps closed sets onto closed sets. The set {v n 0 } that generates K + 1 is delimited by ∆v n 0 ≤ ∆v tot ; it, and thus K + ∞ , is a closed set. The latter is therefore a complete metric space.
Theorem 2 gives us the future cone K + ∞ for a countable number of shocks in terms of the trajectories resulting from a single shock experienced at r(t 0 ) = r 0 . The generic multishock trajectory amounts to a Devil's staircase in ∆v proceeding from ∆v = 0 to ∆v = ∆v f inal ≤ ∆v tot . The theorem to be proved next extends this result to the case of a spacecraft maneuvering by continuous nongravitational acceleration as, for instance, in the case of the powered flight of a rocket: Theorem 3: Let r(t), t ∈ [t 0 , t 1 ] the trajectory of a continuously accelerating spacecraft, originating at r 0 with initial velocity v 0 , and subject to ∆v ≤ ∆ v tot . Then r(t) ∈ K + ∞ (t 1 : t 2 ; r(t 0 )) Proof : Let r(t) be the trajectory, originating at r(t 0 ) with velocity v 0 , that results from an imposed acceleration We assume that a is a integrable function of t. The motion of the spacecraft is given by The effect of continuous acceleration is approximated by a series of shocks ∆v i from which we construct a minimizing sequence of shock histories. 44 Let the trajectory resulting from the nth choice of shock history be given by with a n (t) a collection of impulsive velocity changes punctuating ballistic motion subject to gravitation alone. We may approximate a n (t) for t ∈ [t 0 , t 1 ] by a sequence of step functions as k n → ∞, where χ A (x) is the characteristic function for an a subset x ⊂ A and the support of a i n is supp(a i n ) ≡ τ i . The acceleration a n (t) integrates to ∆v(t) of the form Expanding to second order in δr r , We note that control laws for the shock history ∆v i exist that drive δr to small values It may thus be assumed without loss of generality that if in the sup norm on [t 0 , t 1 ] then ∃ α > 0 such that the second-order terms in (88) dominate the error resulting from linearization. Using (91) and the Schwartz inequality, we have on We also have Then, using the estimates (92) and (93), a reliable overestimate of the second-order error that is linear in |δr| is Upon substitution from (94) and replacement of − 3 r · δr r 2 r (95) by 3 r · |δr| r 2 r, (88) takes the form which, upon formal integration from t 0 to t becomes where elements of the matrix A(t) ≥ 0 on [t 0 , t 1 ] and we are at liberty to choose a n so that is positive on [t 0 , t 1 ] for sufficiently large n.
Using r n = r+δρ in place of r = r n +δr in (86), we find the expected changes to (88): The terms linear in δρ change sign compared to (88), while the form of F(t) remains unaltered.
Following the same development leading to (98), with the change that 3 r · |δρ| r 2 r (100) now replaces and setting δρ = −δr, we obtain with the identical form for the positive matrix function A(s). Both δr and −δr obey the same vector inequality. It must be the case that |δr| also obeys the inequality (98). The multivariate version of the Gronwall inequality 29-31 then gives where V(s, t) satisfies In particular, (103) gives us An elementary but ugly calculation using the Schwartz inequality and the mean value theorem for integrals converts (105) into the Lipschitz condition Examine now δw ∞ . We have, from (85), But, a being integrable, we may choose a shock history approximated by a sequence of step functions such that 32 Therefore, m exists such that for any δ > 0 such that there is an ǫ > 0 such that The set X = {δr, ∀ n > m} is a bounded subset of the set C( uniformly on [t 0 , t 1 ] for all δr ∈ X. X is therefore an equicontinuous set. The Arzelà-Ascoli Lemma then implies that a uniformly convergent subsequence δr n k exists on [t 0 , t 1 ] whose sup norm tends to zero. 33 Therefore, as n k → ∞, r n k (t) → r(t) uniformly on [t 0 , t 1 ]. Recall that K + ∞ is a complete metric space in the sup norm, and thus contains the limit of all its convergent sequences. We may now claim proving the theorem.
Clearly, the future cone of a maneuvering spacecraft is the union of piecewise ballistic trajectories and continuously accelerating ones, which gives us Theorem 4: K + (t 1 : t 2 ; r(t 0 )) = K + 1 (t 1 : t 2 ; r(t 0 )) Proof : K + 1 ⊂ K + : By their respective definitions. K + ⊂ K + 1 : By Theorems 2 and 3. The vertex for the ASAT cone was arbitrarily chosen as the point in its flyout at which the ASAT altitude exceeds 104 km, using the same launch azimuth and initial pitch assumed in Forden's analysis. At this point in the flight profile, the missile is late in its second stage burn. The total ∆v used to calculate K + ASAT is obtained from (113) using the partially expended second plus third stage mass for m i and the second stage tare plus third stage mass for m f . This procedure presumably overestimates slightly the ∆v likely available to the actual ASAT very near the cone vertex, but should be accurate enough for illustrative purposes.
The change in rocket velocity ∆v resulting from the exhaust of reaction mass is given by the rocket equation where I sp is the specific impulse in seconds, g 0 is the gravitational acceleration at sea level, and m i and m f are the initial and final masses of the rocket, respectively. The ideal specific impulse for a cold nitrogen gas jet is given as 76 seconds in Ref. 41 The superposition of the future cone K + ASAT of the ASAT and that part of the cone K + F Y 1C for FengYun-1C lying within K + ASAT is presented in Figures 1 and 2 as random-dot stereograms with approximately antipodal viewing geometries. The nominal encounter time 22:26:00 occurs at 450 seconds TALO. In the interval during which the cones intersect, and Theorem 1 guarantees the ASAT can intercept FengYun-1C.
That the ASAT could intercept FengYun-1C is hardly a novel conclusion-this much was demonstrated beyond dispute on 11 January 2007. Admittedly, FengYun-1C appears to have served as a passive target. 38 Consider, however, a hypothetical scenario in which FengYun-1C maneuvers during the engagement. The disparity in expansion between the cones K +

F Y −1C
and K + ASAT underscores the inability of FengYun-1C to evade the ASAT. If anything, this exercise understates the vulnerability of FengYun-1C on that date: The cone K + F Y 1C is computed assuming that the total ∆v remaining in the attitude control system would actually be available to the spacecraft for maneuver. It is unlikely this assumption holds in practice. At a minimum, the actual ∆v is limited by the total thrust the cold nitrogen jets are capable of producing if operated in a configuration that produces net impulse, even on the assumption that operators on the ground issue evasive commands.

V. DISCUSSION
It was noted in Ref. 1 that (6) in Theorem 1 amounts to establishing a (dominant strategy) Nash equillibrium: The target may move to any position available in its future, but so long as the interceptor's future contains that of the target, the interceptor can always maneuver to some future position r(t i ) of the target. Cockayne 5 uses the telling phrase "against all opposition" to describe the corresponding situation in the game of Two Cars (vide. Isaacs Ref. 3, p. 202, as well).
Theorem 1 was proved for simply connected future cones, but can be applied to certain multiply connected ones. If multiply connected future cones of target and interceptor are manifolds with a timelike foliation and are the finite union of polyhedra of simply connected simplicial complices, one may invoke Theorem 1 for individual polyhedra. Thus, if where each K + int.,i is a simply connected polyhedron and a subset K + targ. (t α : t ω ; r 0 t ) of K + targ.
is a simply connected polyhedron (or other acyclic absolute neighborhood retract) such that for some j ∈ 1, N then the existence of a guaranteed interception opportunity follows in any polyhedron of the interceptor cone that contains K + targ. (t α : t ω ; r 0 t ). This approach is presumably necessary for a treatment of the Game of Two Cars that relaxes the simplifying conditions regarding initial velocity and cone vertex times made in Section III A.
Theorems 2-4 need not give a good account of the future cone of a spacecraft for which the assumption that it may expend the entirety of its available ∆v in a single maneuver at the cone vertex is a poor approximation. In that event K + ⊂ K + 1 (∆v tot ) remains true, but it may happen that K + ⊂ K + 1 (∆v avail. ). A proper treatment of this case with the results of Section III B requires considering a sequence of future cones, for each of which the ∆v avail.
available in a given time interval of its motion is used in place of ∆v tot .

VI. CONCLUSION
The sample problems worked in Section III show that the future cone construction based on Theorem 1 offers a simple method of determining when a class of differential pursuit/evasion games of kind, in which both players maneuver freely, is guaranteed to have interception of the target as a possible outcome.
The method of analysis presented in Section III B is applicable to direct-ascent and coorbital ASAT engagements. The specific example of the FengYun-1C interception by a  0)) and K + ASAT (68 : 750; r(68)) for target and interceptor, respectively, in an Earth-centered inertial frame with times in seconds after interceptor launch.
The stereogram also shows the latitude and longitude lines of the assumed interceptor launch from 28 Figure 1. In this view, the typical interceptor cone trajectory points somewhat out of the page.