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).

[2]

C. Aliprantis and O. Burkinshaw, Op. cit.,, North Holland, (1981), 64.

[3]

R. Battin, An Introduction to the Mathematics and Methods of Astrodynamics,, AIAA Educational Series, (1987).

[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.

[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.

[6]

A. Carter, Anti-satellite weapons, countermeasures, and arms control,, U. S. Congress, (1985).

[7]

J. Chandra and P. W. Davis, Linear generalizations of Gronwall's inequality,, Proceedings of the American Mathematical Society, 60 (1976), 157.

[8]

S. C. Chu and F. T. Metcalfe, On Gronwall's inequality,, Proceedings of the American Mathematical Society, 18 (1967), 439.

[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.

[10]

E. Cockayne, Plane pursuit with curvature constraints,, SIAM J. Appl. Math., 15 (1967), 1511. doi: 10.1137/0115133.

[11]

S. Eilenberg and D. Montgomery, Fixed point theorems for multi-valued transformations,, Amer. J. Math., 68 (1946), 214. doi: 10.2307/2371832.

[12]

R. Feynman and A. Hibbs, Quantum Mechanics and Path Integrals,, McGraw-Hill, (1965).

[13]

G. Forden, GUI_Missile_Flyout: A general program for simulating ballistic missiles,, Science and Global Security, 15 (2006), 133.

[14]

G. Forden, A Preliminary Analysis of the Chinese ASAT Test,, unpublished, (2008).

[15]

A. Friedman, Differential Games,, John Wiley and Sons, (1971).

[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.

[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.

[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.

[19]

A. Granas and J. Djundji, Fixed Point Theory,, Springer, (2003). doi: 10.1007/978-0-387-21593-8.

[20]

R. Isaacs, Differential Games,, John Wiley and Sons, (1965).

[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.

[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.

[23]

J. L. Kelley, General Topology,, Van Nostrand, (1955).

[24]

S. Kumagai, An implicit function theorem: Comment,, J. Optimization Th. and A, 31 (1980), 285. doi: 10.1007/BF00934117.

[25]

J. P. Marec, Optimal Space Trajectories,, Elsevier, (1979).

[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.

[27]

C. R. F. Maunder, Algebraic Topology,, Cambridge University Press, (1980).

[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.

[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.

[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.

[31]

J. F. Nash, Non-Cooperative Games,, Ph.D. thesis, (1950).

[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.

[33]

S. P. Novikov, Topology of foliations,, Trans. Amer. Math. Soc., 14 (1965), 248.

[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).

[35]

L. S. Pontryagin, On some differential games,, J. SIAM Controls, 3 (1965), 49. doi: 10.1137/0303004.

[36]

L. S. Pontryagin, Lectures on Differential Games,, Stanford University, (1971).

[37]

L. S. Pontryagin, On the evasion process in differential games,, Appl. Math and Optimization, 1 (): 5. doi: 10.1007/BF01449022.

[38]

L. S. Pontryagin, Linear differential games of pursuit,, Math. USSR Sbornik, 40 (1981), 285.

[39]

E. Roxin, Axiomatic approach in differential games,, J. Opt. Theory and A, 3 (1969), 153. doi: 10.1007/BF00929440.

[40]

E. Spanier, Algebraic Topology,, Springer, (1966).

[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.

[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.

[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.

[44]

J. Shinar and V. Turetsky, Three-dimensional validation of an integrated estimation/guidance algorithm against randomly maneuvering targets,, J. Guidance, 32 (2009), 1034.

[45]

, Space Trak,, , ().

[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.

[47]

G. Sutton, Rocket Propulsion Elements,, John Wiley and Sons, (1992). doi: 10.1063/1.3066790.

[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.

[49]

W. Walter, Differential and Integral Inequalities,, Erbebnisse der Mathematik und ihrer Grenzgebiete, (1970).

[50]

X. Yuhang, 'FengYun 1 meteorological satellite detailed,, reprinted in China Science and Technology, (1989), 89.

show all references

References:
[1]

C. Aliprantis and O. Burkinshaw, Principles of Real Analysis,, North Holland, (1981).

[2]

C. Aliprantis and O. Burkinshaw, Op. cit.,, North Holland, (1981), 64.

[3]

R. Battin, An Introduction to the Mathematics and Methods of Astrodynamics,, AIAA Educational Series, (1987).

[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.

[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.

[6]

A. Carter, Anti-satellite weapons, countermeasures, and arms control,, U. S. Congress, (1985).

[7]

J. Chandra and P. W. Davis, Linear generalizations of Gronwall's inequality,, Proceedings of the American Mathematical Society, 60 (1976), 157.

[8]

S. C. Chu and F. T. Metcalfe, On Gronwall's inequality,, Proceedings of the American Mathematical Society, 18 (1967), 439.

[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.

[10]

E. Cockayne, Plane pursuit with curvature constraints,, SIAM J. Appl. Math., 15 (1967), 1511. doi: 10.1137/0115133.

[11]

S. Eilenberg and D. Montgomery, Fixed point theorems for multi-valued transformations,, Amer. J. Math., 68 (1946), 214. doi: 10.2307/2371832.

[12]

R. Feynman and A. Hibbs, Quantum Mechanics and Path Integrals,, McGraw-Hill, (1965).

[13]

G. Forden, GUI_Missile_Flyout: A general program for simulating ballistic missiles,, Science and Global Security, 15 (2006), 133.

[14]

G. Forden, A Preliminary Analysis of the Chinese ASAT Test,, unpublished, (2008).

[15]

A. Friedman, Differential Games,, John Wiley and Sons, (1971).

[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.

[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.

[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.

[19]

A. Granas and J. Djundji, Fixed Point Theory,, Springer, (2003). doi: 10.1007/978-0-387-21593-8.

[20]

R. Isaacs, Differential Games,, John Wiley and Sons, (1965).

[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.

[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.

[23]

J. L. Kelley, General Topology,, Van Nostrand, (1955).

[24]

S. Kumagai, An implicit function theorem: Comment,, J. Optimization Th. and A, 31 (1980), 285. doi: 10.1007/BF00934117.

[25]

J. P. Marec, Optimal Space Trajectories,, Elsevier, (1979).

[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.

[27]

C. R. F. Maunder, Algebraic Topology,, Cambridge University Press, (1980).

[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.

[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.

[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.

[31]

J. F. Nash, Non-Cooperative Games,, Ph.D. thesis, (1950).

[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.

[33]

S. P. Novikov, Topology of foliations,, Trans. Amer. Math. Soc., 14 (1965), 248.

[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).

[35]

L. S. Pontryagin, On some differential games,, J. SIAM Controls, 3 (1965), 49. doi: 10.1137/0303004.

[36]

L. S. Pontryagin, Lectures on Differential Games,, Stanford University, (1971).

[37]

L. S. Pontryagin, On the evasion process in differential games,, Appl. Math and Optimization, 1 (): 5. doi: 10.1007/BF01449022.

[38]

L. S. Pontryagin, Linear differential games of pursuit,, Math. USSR Sbornik, 40 (1981), 285.

[39]

E. Roxin, Axiomatic approach in differential games,, J. Opt. Theory and A, 3 (1969), 153. doi: 10.1007/BF00929440.

[40]

E. Spanier, Algebraic Topology,, Springer, (1966).

[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.

[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.

[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.

[44]

J. Shinar and V. Turetsky, Three-dimensional validation of an integrated estimation/guidance algorithm against randomly maneuvering targets,, J. Guidance, 32 (2009), 1034.

[45]

, Space Trak,, , ().

[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.

[47]

G. Sutton, Rocket Propulsion Elements,, John Wiley and Sons, (1992). doi: 10.1063/1.3066790.

[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.

[49]

W. Walter, Differential and Integral Inequalities,, Erbebnisse der Mathematik und ihrer Grenzgebiete, (1970).

[50]

X. Yuhang, 'FengYun 1 meteorological satellite detailed,, reprinted in China Science and Technology, (1989), 89.

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