# 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:
 [1] C. Aliprantis and O. Burkinshaw, Principles of Real Analysis,, North Holland, (1981). Google Scholar [2] C. Aliprantis and O. Burkinshaw, Op. cit.,, North Holland, (1981), 64. Google Scholar [3] R. Battin, An Introduction to the Mathematics and Methods of Astrodynamics,, AIAA Educational Series, (1987). Google Scholar [4] R. Brooks, Game and information theory analysis of electronic countermeasures in pursuit-evasion games,, IEEE Trans. on Syst., 38 (2008), 1281. 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. doi: 10.1090/S0002-9904-1946-08610-3. Google Scholar [6] A. Carter, Anti-satellite weapons, countermeasures, and arms control,, U. S. Congress, (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. Google Scholar [8] S. C. Chu and F. T. Metcalfe, On Gronwall's inequality,, Proceedings of the American Mathematical Society, 18 (1967), 439. 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, 15 (2012), 1452. doi: 10.1109/IMCCC.2012.340. Google Scholar [10] E. Cockayne, Plane pursuit with curvature constraints,, SIAM J. Appl. Math., 15 (1967), 1511. 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. doi: 10.2307/2371832. Google Scholar [12] R. Feynman and A. Hibbs, Quantum Mechanics and Path Integrals,, McGraw-Hill, (1965). Google Scholar [13] G. Forden, GUI_Missile_Flyout: A general program for simulating ballistic missiles,, Science and Global Security, 15 (2006), 133. Google Scholar [14] G. Forden, A Preliminary Analysis of the Chinese ASAT Test,, unpublished, (2008). Google Scholar [15] A. Friedman, Differential Games,, John Wiley and Sons, (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. 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. 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, (2009), 179. Google Scholar [19] A. Granas and J. Djundji, Fixed Point Theory,, Springer, (2003). doi: 10.1007/978-0-387-21593-8. Google Scholar [20] R. Isaacs, Differential Games,, John Wiley and Sons, (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. 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. doi: 10.1215/S0012-7094-41-00838-4. Google Scholar [23] J. L. Kelley, General Topology,, Van Nostrand, (1955). Google Scholar [24] S. Kumagai, An implicit function theorem: Comment,, J. Optimization Th. and A, 31 (1980), 285. doi: 10.1007/BF00934117. Google Scholar [25] J. P. Marec, Optimal Space Trajectories,, Elsevier, (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, 3 (1995), 2463. doi: 10.1109/ACC.1995.531418. Google Scholar [27] C. R. F. Maunder, Algebraic Topology,, Cambridge University Press, (1980). Google Scholar [28] T. Miloh, A note on three-dimensional pursuit-evasion game with bounded curvature,, IEEE Trans. Automat. Contr., 27 (1982), 739. 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. 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. doi: 10.1098/rspa.2009.0552. Google Scholar [31] J. F. Nash, Non-Cooperative Games,, Ph.D. thesis, (1950). Google Scholar [32] J. F. Nash, Equilibrium points in n-person games,, Proc. Nat. Acad. Sci. USA, 36 (1950), 48. doi: 10.1073/pnas.36.1.48. Google Scholar [33] S. P. Novikov, Topology of foliations,, Trans. Amer. Math. Soc., 14 (1965), 248. 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, (2007). Google Scholar [35] L. S. Pontryagin, On some differential games,, J. SIAM Controls, 3 (1965), 49. doi: 10.1137/0303004. Google Scholar [36] L. S. Pontryagin, Lectures on Differential Games,, Stanford University, (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. Google Scholar [39] E. Roxin, Axiomatic approach in differential games,, J. Opt. Theory and A, 3 (1969), 153. doi: 10.1007/BF00929440. Google Scholar [40] E. Spanier, Algebraic Topology,, Springer, (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. 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. 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. 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, 32 (2009), 1034. 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, 2 (1995), 369. doi: 10.1109/AERO.1995.468892. Google Scholar [47] G. Sutton, Rocket Propulsion Elements,, John Wiley and Sons, (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. doi: 10.1007/BF01769884. Google Scholar [49] W. Walter, Differential and Integral Inequalities,, Erbebnisse der Mathematik und ihrer Grenzgebiete, (1970). Google Scholar [50] X. Yuhang, 'FengYun 1 meteorological satellite detailed,, reprinted in China Science and Technology, (1989), 89. Google Scholar

show all references

##### References:
 [1] C. Aliprantis and O. Burkinshaw, Principles of Real Analysis,, North Holland, (1981). Google Scholar [2] C. Aliprantis and O. Burkinshaw, Op. cit.,, North Holland, (1981), 64. Google Scholar [3] R. Battin, An Introduction to the Mathematics and Methods of Astrodynamics,, AIAA Educational Series, (1987). Google Scholar [4] R. Brooks, Game and information theory analysis of electronic countermeasures in pursuit-evasion games,, IEEE Trans. on Syst., 38 (2008), 1281. 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. doi: 10.1090/S0002-9904-1946-08610-3. Google Scholar [6] A. Carter, Anti-satellite weapons, countermeasures, and arms control,, U. S. Congress, (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. Google Scholar [8] S. C. Chu and F. T. Metcalfe, On Gronwall's inequality,, Proceedings of the American Mathematical Society, 18 (1967), 439. 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, 15 (2012), 1452. doi: 10.1109/IMCCC.2012.340. Google Scholar [10] E. Cockayne, Plane pursuit with curvature constraints,, SIAM J. Appl. Math., 15 (1967), 1511. 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. doi: 10.2307/2371832. Google Scholar [12] R. Feynman and A. Hibbs, Quantum Mechanics and Path Integrals,, McGraw-Hill, (1965). Google Scholar [13] G. Forden, GUI_Missile_Flyout: A general program for simulating ballistic missiles,, Science and Global Security, 15 (2006), 133. Google Scholar [14] G. Forden, A Preliminary Analysis of the Chinese ASAT Test,, unpublished, (2008). Google Scholar [15] A. Friedman, Differential Games,, John Wiley and Sons, (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. 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. 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, (2009), 179. Google Scholar [19] A. Granas and J. Djundji, Fixed Point Theory,, Springer, (2003). doi: 10.1007/978-0-387-21593-8. Google Scholar [20] R. Isaacs, Differential Games,, John Wiley and Sons, (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. 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. doi: 10.1215/S0012-7094-41-00838-4. Google Scholar [23] J. L. Kelley, General Topology,, Van Nostrand, (1955). Google Scholar [24] S. Kumagai, An implicit function theorem: Comment,, J. Optimization Th. and A, 31 (1980), 285. doi: 10.1007/BF00934117. Google Scholar [25] J. P. Marec, Optimal Space Trajectories,, Elsevier, (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, 3 (1995), 2463. doi: 10.1109/ACC.1995.531418. Google Scholar [27] C. R. F. Maunder, Algebraic Topology,, Cambridge University Press, (1980). Google Scholar [28] T. Miloh, A note on three-dimensional pursuit-evasion game with bounded curvature,, IEEE Trans. Automat. Contr., 27 (1982), 739. 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. 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. doi: 10.1098/rspa.2009.0552. Google Scholar [31] J. F. Nash, Non-Cooperative Games,, Ph.D. thesis, (1950). Google Scholar [32] J. F. Nash, Equilibrium points in n-person games,, Proc. Nat. Acad. Sci. USA, 36 (1950), 48. doi: 10.1073/pnas.36.1.48. Google Scholar [33] S. P. Novikov, Topology of foliations,, Trans. Amer. Math. Soc., 14 (1965), 248. 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, (2007). Google Scholar [35] L. S. Pontryagin, On some differential games,, J. SIAM Controls, 3 (1965), 49. doi: 10.1137/0303004. Google Scholar [36] L. S. Pontryagin, Lectures on Differential Games,, Stanford University, (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. Google Scholar [39] E. Roxin, Axiomatic approach in differential games,, J. Opt. Theory and A, 3 (1969), 153. doi: 10.1007/BF00929440. Google Scholar [40] E. Spanier, Algebraic Topology,, Springer, (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. 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. 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. 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, 32 (2009), 1034. 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, 2 (1995), 369. doi: 10.1109/AERO.1995.468892. Google Scholar [47] G. Sutton, Rocket Propulsion Elements,, John Wiley and Sons, (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. doi: 10.1007/BF01769884. Google Scholar [49] W. Walter, Differential and Integral Inequalities,, Erbebnisse der Mathematik und ihrer Grenzgebiete, (1970). Google Scholar [50] X. Yuhang, 'FengYun 1 meteorological satellite detailed,, reprinted in China Science and Technology, (1989), 89. Google Scholar
 [1] Martino Bardi, Shigeaki Koike, Pierpaolo Soravia. Pursuit-evasion games with state constraints: dynamic programming and discrete-time approximations. Discrete & Continuous Dynamical Systems - A, 2000, 6 (2) : 361-380. doi: 10.3934/dcds.2000.6.361 [2] Lesia V. Baranovska. Pursuit differential-difference games with pure time-lag. Discrete & Continuous Dynamical Systems - B, 2019, 24 (3) : 1021-1031. doi: 10.3934/dcdsb.2019004 [3] Songtao Sun, Qiuhua Zhang, Ryan Loxton, Bin Li. Numerical solution of a pursuit-evasion differential game involving two spacecraft in low earth orbit. Journal of Industrial & Management Optimization, 2015, 11 (4) : 1127-1147. doi: 10.3934/jimo.2015.11.1127 [4] Jingzhen Liu, Ka-Fai Cedric Yiu. Optimal stochastic differential games with VaR constraints. Discrete & Continuous Dynamical Systems - B, 2013, 18 (7) : 1889-1907. doi: 10.3934/dcdsb.2013.18.1889 [5] Alain Bensoussan, Jens Frehse, Christine Grün. Stochastic differential games with a varying number of players. Communications on Pure & Applied Analysis, 2014, 13 (5) : 1719-1736. doi: 10.3934/cpaa.2014.13.1719 [6] Ellina Grigorieva, Evgenii Khailov. Hierarchical differential games between manufacturer and retailer. Conference Publications, 2009, 2009 (Special) : 300-314. doi: 10.3934/proc.2009.2009.300 [7] Piernicola Bettiol. State constrained $L^\infty$ optimal control problems interpreted as differential games. Discrete & Continuous Dynamical Systems - A, 2015, 35 (9) : 3989-4017. doi: 10.3934/dcds.2015.35.3989 [8] Beatris Adriana Escobedo-Trujillo, José Daniel López-Barrientos. Nonzero-sum stochastic differential games with additive structure and average payoffs. Journal of Dynamics & Games, 2014, 1 (4) : 555-578. doi: 10.3934/jdg.2014.1.555 [9] Beatris Adriana Escobedo-Trujillo, Alejandro Alaffita-Hernández, Raquiel López-Martínez. Constrained stochastic differential games with additive structure: Average and discount payoffs. Journal of Dynamics & Games, 2018, 5 (2) : 109-141. doi: 10.3934/jdg.2018008 [10] Alain Bensoussan, Shaokuan Chen, Suresh P. Sethi. Linear quadratic differential games with mixed leadership: The open-loop solution. Numerical Algebra, Control & Optimization, 2013, 3 (1) : 95-108. doi: 10.3934/naco.2013.3.95 [11] Alain Bensoussan, Jens Frehse, Jens Vogelgesang. Systems of Bellman equations to stochastic differential games with non-compact coupling. Discrete & Continuous Dynamical Systems - A, 2010, 27 (4) : 1375-1389. doi: 10.3934/dcds.2010.27.1375 [12] Beatris A. Escobedo-Trujillo. Discount-sensitive equilibria in zero-sum stochastic differential games. Journal of Dynamics & Games, 2016, 3 (1) : 25-50. doi: 10.3934/jdg.2016002 [13] Qingmeng Wei, Zhiyong Yu. Time-inconsistent recursive zero-sum stochastic differential games. Mathematical Control & Related Fields, 2018, 8 (3&4) : 1051-1079. doi: 10.3934/mcrf.2018045 [14] Alan Beggs. Learning in monotone bayesian games. Journal of Dynamics & Games, 2015, 2 (2) : 117-140. doi: 10.3934/jdg.2015.2.117 [15] Konstantin Avrachenkov, Giovanni Neglia, Vikas Vikram Singh. Network formation games with teams. Journal of Dynamics & Games, 2016, 3 (4) : 303-318. doi: 10.3934/jdg.2016016 [16] Hassan Najafi Alishah, Pedro Duarte. Hamiltonian evolutionary games. Journal of Dynamics & Games, 2015, 2 (1) : 33-49. doi: 10.3934/jdg.2015.2.33 [17] Yonghui Zhou, Jian Yu, Long Wang. Topological essentiality in infinite games. Journal of Industrial & Management Optimization, 2012, 8 (1) : 179-187. doi: 10.3934/jimo.2012.8.179 [18] Cyril Imbert, Sylvia Serfaty. Repeated games for non-linear parabolic integro-differential equations and integral curvature flows. Discrete & Continuous Dynamical Systems - A, 2011, 29 (4) : 1517-1552. doi: 10.3934/dcds.2011.29.1517 [19] Rui Mu, Zhen Wu. Nash equilibrium points of recursive nonzero-sum stochastic differential games with unbounded coefficients and related multiple\\ dimensional BSDEs. Mathematical Control & Related Fields, 2017, 7 (2) : 289-304. doi: 10.3934/mcrf.2017010 [20] Tyrone E. Duncan. Some linear-quadratic stochastic differential games for equations in Hilbert spaces with fractional Brownian motions. Discrete & Continuous Dynamical Systems - A, 2015, 35 (11) : 5435-5445. doi: 10.3934/dcds.2015.35.5435

Impact Factor: