# American Institute of Mathematical Sciences

October  2016, 3(4): 335-354. doi: 10.3934/jdg.2016018

## Interception in differential pursuit/evasion games

 1 The Aerospace Corporation, P. O. Box 92957, Los Angeles, CA 90009, United States

Received  December 2013 Revised  March 2016 Published  October 2016

A qualitative criterion for a pursuer to intercept a target in a class of differential games is obtained in terms of 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 equilibrium. 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.
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.
Citation: John A. Morgan. Interception in differential pursuit/evasion games. Journal of Dynamics & Games, 2016, 3 (4) : 335-354. doi: 10.3934/jdg.2016018
##### References:
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Pontryagin, Lectures on Differential Games, Stanford University, CA (reprinted as DTIC AD724166), 1971. Google Scholar [37] L. S. Pontryagin, On the evasion process in differential games,, Appl. Math and Optimization, 1 (): 5.  doi: 10.1007/BF01449022.  Google Scholar [38] L. S. Pontryagin, Linear differential games of pursuit, Math. USSR Sbornik, 40 (1981), 285-303. Google Scholar [39] E. Roxin, Axiomatic approach in differential games, J. Opt. Theory and A, 3 (1969), 153-163. doi: 10.1007/BF00929440.  Google Scholar [40] E. Spanier, Algebraic Topology, Springer, New York, 1966.  Google Scholar [41] J. Shinar. V. Glizer, and V. Turetsky, A pursuit-evasion game with hybrid pursuer dynamics, European Journal of Control, 15 (2009), 665-684. doi: 10.3166/ejc.15.665-684.  Google Scholar [42] J. Shinar. V. Glizer and V. Turetsky, Robust pursuit of a hybrid evader, Applied Mathematics and Computation, 217 (2010), 1231-1245. doi: 10.1016/j.amc.2010.04.019.  Google Scholar [43] J. Shinar and S. Gutman, Three-Dimensional Optimal Pursuit and Evasion with Bounded Controls, IEEE Trans. Automat. Contr., AC-25 (1980), 492-496. doi: 10.1109/TAC.1980.1102372.  Google Scholar [44] J. Shinar and V. Turetsky, Three-dimensional validation of an integrated estimation/guidance algorithm against randomly maneuvering targets, J. Guidance, Control, and Dynamics, 32 (2009), 1034-1038. Google Scholar [45] , Space Trak,, , ().   Google Scholar [46] D. Y. Stodden and G. D. Galasso, Space system visualization and analysis using the Satellite Orbit Analysis Program (SOAP), IEEE Aerospace Applications Conference Proceedings, Aspen, CO, 2 (1995), 369-387. doi: 10.1109/AERO.1995.468892.  Google Scholar [47] G. Sutton, Rocket Propulsion Elements, John Wiley and Sons, New York, 1992. doi: 10.1063/1.3066790.  Google Scholar [48] L. 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show all references

##### References:
 [1] C. Aliprantis and O. Burkinshaw, Principles of Real Analysis, North Holland, New York, 1981.  Google Scholar [2] C. Aliprantis and O. Burkinshaw, Op. cit., North Holland, New York, 1981, 64-65. Google Scholar [3] R. Battin, An Introduction to the Mathematics and Methods of Astrodynamics, AIAA Educational Series, American Institute of Aeronautics and Astronautics, New York, 1987.  Google Scholar [4] R. Brooks, Game and information theory analysis of electronic countermeasures in pursuit-evasion games, IEEE Trans. on Syst., Man, Cybern., 38 (2008), 1281-1294. doi: 10.1109/TSMCA.2008.2003970.  Google Scholar [5] S. Cairns, The triangulation problem and its role in analysis, Bull. Amer. Math. Soc., 52 (1946), 545-571. doi: 10.1090/S0002-9904-1946-08610-3.  Google Scholar [6] A. Carter, Anti-satellite weapons, countermeasures, and arms control, U. S. Congress, Office of Technology Assessment OTA-ISC-281, U. S. Government Printing Office, Washington, DC, 1985. Google Scholar [7] J. Chandra and P. W. Davis, Linear generalizations of Gronwall's inequality, Proceedings of the American Mathematical Society, 60 (1976), 157-160.  Google Scholar [8] S. C. Chu and F. T. Metcalfe, On Gronwall's inequality, Proceedings of the American Mathematical Society, 18 (1967), 439-440.  Google Scholar [9] J. Dong, X. Zhang and X. Jia, Strategies of pursuit-evasion game based on improved potential field and differential game theory for mobile robots, 2012 Second International Conference on Instrumentation & Measurement, Computer, Communication, and Control, 15 (2012), 1452-1455. doi: 10.1109/IMCCC.2012.340.  Google Scholar [10] E. Cockayne, Plane pursuit with curvature constraints, SIAM J. Appl. Math., 15 (1967), 1511-1516. doi: 10.1137/0115133.  Google Scholar [11] S. Eilenberg and D. Montgomery, Fixed point theorems for multi-valued transformations, Amer. J. Math., 68 (1946), 214-222. doi: 10.2307/2371832.  Google Scholar [12] R. Feynman and A. Hibbs, Quantum Mechanics and Path Integrals, McGraw-Hill, New York, 1965. Google Scholar [13] G. Forden, GUI_Missile_Flyout: A general program for simulating ballistic missiles, Science and Global Security, 15 (2006), 133-146. Google Scholar [14] G. Forden, A Preliminary Analysis of the Chinese ASAT Test, unpublished, MIT, 2008. Google Scholar [15] A. Friedman, Differential Games, John Wiley and Sons, New York, 1971.  Google Scholar [16] W. Getz and M. Pachter, Capturability in a two-target "game of two cars", J. Guidance and Control, 4 (1981), 15-21. doi: 10.2514/3.19715.  Google Scholar [17] V. Glizer, Homicidal chauffeur game with target set in the shape of a circular angular sector: conditions for existence of a closed barrier, J. Optimization Theory and Applications, 101 (1999), 581-598. doi: 10.1023/A:1021738103941.  Google Scholar [18] V. Glizer, V. Turetsky and J. Shinar, Differential Game with Linear Dynamics and Multiple Information Delays, Proceedings of the $13^{th}$ WSEAS International Conference on Systems, World Scientific and Engineering Academy and Society (WSEAS), (2009), 179-184. Google Scholar [19] A. Granas and J. Djundji, Fixed Point Theory, Springer, New York, 2003. doi: 10.1007/978-0-387-21593-8.  Google Scholar [20] R. Isaacs, Differential Games, John Wiley and Sons, New York, 1965.  Google Scholar [21] N. Johnson, E. Stansbery, J.-C. Liou, M. Horstman, C. Stokely and D. Whitlock, The characteristics and consequences of the break-up of the FengYun-1C spacecraft, Acta Astronautica, 63 (2008), 128-135. doi: 10.1016/j.actaastro.2007.12.044.  Google Scholar [22] S. Kakutani, A generalization of Brouwer's fixed point theorem, Duke Math J., 8 (1941), 457-459. doi: 10.1215/S0012-7094-41-00838-4.  Google Scholar [23] J. L. Kelley, General Topology, Van Nostrand, Princeton, 1955.  Google Scholar [24] S. Kumagai, An implicit function theorem: Comment, J. Optimization Th. and A, 31 (1980), 285-288. doi: 10.1007/BF00934117.  Google Scholar [25] J. P. Marec, Optimal Space Trajectories, Elsevier, New York, 1979. Google Scholar [26] R. K. Maloy, K. Y. Lee and L. H. Sibul, A pursuit-evasion differential game in relative polar coordinates with state estimation, In American Control Conference, 1995. Proceedings of the IEEE, 3 (1995), 2463-2467. doi: 10.1109/ACC.1995.531418.  Google Scholar [27] C. R. F. Maunder, Algebraic Topology, Cambridge University Press, Cambridge-New York, 1980.  Google Scholar [28] T. Miloh, A note on three-dimensional pursuit-evasion game with bounded curvature, IEEE Trans. Automat. Contr., 27 (1982), 739-741. doi: 10.1109/TAC.1982.1102992.  Google Scholar [29] T. Miloh, M. Pachter and A. Segal, The effect of a finite roll rate on the miss-distance of a bank-to-turn missile, Computers Math. Applic., 26 (1993), 43-53. doi: 10.1016/0898-1221(93)90116-D.  Google Scholar [30] J. A. Morgan, Qualitative criterion for interception in a pursuit/evasion game, Proc. Roy. Soc. A., 466 (2010), 1365-1371. doi: 10.1098/rspa.2009.0552.  Google Scholar [31] J. F. Nash, Non-Cooperative Games, Ph.D. thesis, Princeton University in Princeton, 1950. Google Scholar [32] J. F. Nash, Equilibrium points in n-person games, Proc. Nat. Acad. Sci. USA, 36 (1950), 48-49. doi: 10.1073/pnas.36.1.48.  Google Scholar [33] S. P. Novikov, Topology of foliations, Trans. Amer. Math. Soc., 14 (1965), 248-278.  Google Scholar [34] C. Pardini and L. Anselmo, Evolution of the Debris Cloud Generated by the FengYun-1C Fragmentation event, Proceedings of the $20^{th}$ ISSFD, Annapolis, Maryland, 2007. Google Scholar [35] L. S. Pontryagin, On some differential games, J. SIAM Controls, 3 (1965), 49-52. doi: 10.1137/0303004.  Google Scholar [36] L. S. Pontryagin, Lectures on Differential Games, Stanford University, CA (reprinted as DTIC AD724166), 1971. Google Scholar [37] L. S. Pontryagin, On the evasion process in differential games,, Appl. Math and Optimization, 1 (): 5.  doi: 10.1007/BF01449022.  Google Scholar [38] L. S. Pontryagin, Linear differential games of pursuit, Math. USSR Sbornik, 40 (1981), 285-303. Google Scholar [39] E. Roxin, Axiomatic approach in differential games, J. Opt. Theory and A, 3 (1969), 153-163. doi: 10.1007/BF00929440.  Google Scholar [40] E. Spanier, Algebraic Topology, Springer, New York, 1966.  Google Scholar [41] J. Shinar. V. Glizer, and V. Turetsky, A pursuit-evasion game with hybrid pursuer dynamics, European Journal of Control, 15 (2009), 665-684. doi: 10.3166/ejc.15.665-684.  Google Scholar [42] J. Shinar. V. Glizer and V. Turetsky, Robust pursuit of a hybrid evader, Applied Mathematics and Computation, 217 (2010), 1231-1245. doi: 10.1016/j.amc.2010.04.019.  Google Scholar [43] J. Shinar and S. Gutman, Three-Dimensional Optimal Pursuit and Evasion with Bounded Controls, IEEE Trans. Automat. Contr., AC-25 (1980), 492-496. doi: 10.1109/TAC.1980.1102372.  Google Scholar [44] J. Shinar and V. Turetsky, Three-dimensional validation of an integrated estimation/guidance algorithm against randomly maneuvering targets, J. Guidance, Control, and Dynamics, 32 (2009), 1034-1038. Google Scholar [45] , Space Trak,, , ().   Google Scholar [46] D. Y. Stodden and G. D. Galasso, Space system visualization and analysis using the Satellite Orbit Analysis Program (SOAP), IEEE Aerospace Applications Conference Proceedings, Aspen, CO, 2 (1995), 369-387. doi: 10.1109/AERO.1995.468892.  Google Scholar [47] G. Sutton, Rocket Propulsion Elements, John Wiley and Sons, New York, 1992. doi: 10.1063/1.3066790.  Google Scholar [48] L. Tefatsion, Pure strategy nash equilibrium points and the Lefschetz fixed point theorem, International Journal of Game Theory, 12 (1983), 181-191. doi: 10.1007/BF01769884.  Google Scholar [49] W. Walter, Differential and Integral Inequalities, Erbebnisse der Mathematik und ihrer Grenzgebiete,Band 55 Springer-Verlag, New York-Berlin, 1970.  Google Scholar [50] X. Yuhang, 'FengYun 1 meteorological satellite detailed, reprinted in China Science and Technology, JPRS-CST-89-026, Foreign Broadcast Information Service, 1989, 1-6. Google Scholar
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