March  2019, 24(3): 1021-1031. doi: 10.3934/dcdsb.2019004

Pursuit differential-difference games with pure time-lag

National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute", 27 Enthuziastiv st., apt.16, 02154, Kyiv, Ukraine

* Corresponding author

Received  October 2017 Revised  April 2018 Published  March 2019 Early access  January 2019

Fund Project: The author was partially supported by the National Academy of Sciences of Ukraine, project 2275-f.

The analytical approach for solution of pursuit differential-difference games with pure time-lag is considered. For the pursuit local problem with the fixed time the scheme of the method of resolving functions and Pontryagin's first direct method are developed. The integral presentation of game solution based on the time-delay exponential is proposed at first time. The guaranteed times of the game termination are found, and corresponding control laws are constructed. Comparison of the times of approach by the method of resolving functions and Pontryagin's first direct method for the initial problem are made.

Citation: Lesia V. Baranovska. Pursuit differential-difference games with pure time-lag. Discrete and Continuous Dynamical Systems - B, 2019, 24 (3) : 1021-1031. doi: 10.3934/dcdsb.2019004
References:
[1]

J. AlbusA. MeystelA. A. ChikriiA. A. Belousov and A. J. Kozlov, Analitical method for solution of the game problem of softlanding for moving objects, Cybernetics and Systems Analysis, 37 (2001), 75-91.  doi: 10.1023/A:1016620201241.

[2]

J-P. Aubin and H. Frankovska, Set-Valued Analysis, Birkhäuser, Boston, 1990.

[3]

L. V. BaranovskayaA. A. Chikrii and Al. A. Chikrii, Inverse Minkowski functional in a nonstationary problem of group pursuit, Izvestia Academii Nauk. Teoria i Sistemy Upravlenia, 36 (1997), 101-106. 

[4]

Al. A. Chikrii, On nonstationary game problem of motion control, Journal of Automation and Information Sciences, 47 (2015), 74-83.  doi: 10.1615/JAutomatInfScien.v47.i11.60.

[5]

A. A. Chikrii, Conflict-Controlled Processes, Kluwer Academic Publishers Group, Dordrecht, 1997. doi: 10.1007/978-94-017-1135-7.

[6]

A. A. Chikrii, Analitical method in dynamic pursuit games, Proceedings of the Steklov Institute of Mathematics, 271 (2010), 69-85.  doi: 10.1134/S0081543810040073.

[7]

A. A. Chikrii and S. D. Eidelman, Generalized Mittag-Leffler matrix functions in game problems for evolutionary equations of fractional order, Cybernetics and Systems Analysis, 36 (2000), 315-338.  doi: 10.1007/BF02732983.

[8]

A. A. ChikriyL. V. Baranovskaya and A. A. Chikriy, An approach game problem under the failure of controlling devices, Journal of Automation and Information Sciences, 32 (2000), 1-8.  doi: 10.1615/JAutomatInfScien.v32.i5.10.

[9]

L. V. Baranovska, On quasilinear differential-difference games of approach, Journal of Automation and Information Sciences, 49 (2017), 53-67.  doi: 10.1615/JAutomatInfScien.v49.i8.40.

[10]

L. V. Baranovska, The modification of the method of resolving functions for the difference-differential pursuit's games, Naukovi Visti NTUU KPII, 4 (2012), 14-20. 

[11]

L. V. Baranovska, Method of resolving functions for the differential-difference pursuit game for different-inertia objects, in Advances in Dynamical Systems and Control. Studies in Systems, Decision and Control (eds. V. Sadovnichiy and M. Zgurovsky), Springer, 69 (2016), 159-176. doi: 10.1007/978-3-319-40673-2.

[12]

L. V. Baranovska, A method of resolving functions for one class of pursuit problems, Eastern-European Journal of Enterprise Technologies, 74 (2015), 4-18. 

[13]

L. V. Baranovska, The group pursuit differential-difference games of approach with non-fixed time, Naukovi Visti NTUU KPI, 4 (2011), 18-22. 

[14]

G. G. Baranovskaya and L. V. Baranovskaya, Group pursuit in quasilinear differential-difference games, Journal of Automation and Information Sciences, 29 (1997), 55-62.  doi: 10.1615/JAutomatInfScien.v29.i1.70.

[15]

G. G. Baranovskaya and L. V. Baranovska, On differential-difference group pursuit game., Dopov. Akad. Nauk Ukr, 3 (1997), 12-15. 

[16]

L. V. Baranovskaya and Al. A. Chikrii, Game Problems for a Class of Hereditaly Systems, Journal of Automation and Information Sciences, 29 (1997), 87-97.  doi: 10.1615/JAutomatInfScien.v29.i2-3.120.

[17]

L. V. Baranovskaya, About one class of difference games of group rapprochement with unfixed time, Science and World, 1 (2015), 10-12. 

[18] R. Bellman and K. L. Cook, Differential-Difference Equations, Academic Press, New York, 1963. 
[19]

Ya. I. Bigun, I. Iu. Kryvonos, Al. A. Chikrii and K. A. Chikrii, Group approach under phase constraints, Journal of Automation and Information Sciences. 46 (2014), 1-8. doi: 10.1615/JAutomatInfScien.v46.i4.10.

[20]

A. A. Chikrii and G. Ts. Chikrii, Matrix resolving functions in game problems of dynamics, Proceedings of the Steklov Institute of Mathematics, 291 (2015), 56-65.  doi: 10.1134/s0081543815090047.

[21]

A. A. Chikrii and A. A. Belousov, On linear differential games with integral constrains, Trudy Instituta Matematiki i Mekhaniki UrO RAN, 15 (2009), 290-301. 

[22]

A. A. ChikriiL. V. Baranovskaya and Al. A. Chikrii, The game problem of approach under the condition of failure of controlling devices, Problemy Upravleniya I Informatiki (Avtomatika), 4 (1997), 3-13. 

[23]

A. A. ChikriiI. S. Rappoport and K. A. Chikrii, Multivalued mappings and their selectors in the theory of conflict-controlled processes, Cybernetics and Systems Analysis, 43 (2007), 719-730.  doi: 10.1007/s10559-007-0097-8.

[24]

A. A. Chikrii and P. V. Prokopovich, Simple pursuit of one evader by a group, Cybernetics and Systems Analysis, 28 (1992), 438-444.  doi: 10.1007/BF01125424.

[25]

J. Diblk and D. Y. Khusainov, Representation of solutions of linear discrete systems with constant coefficients and pure delay, Advances in Difference Equations, 2006 (2006), Art. ID 80825, 13 pp. doi: 10.1155/ADE/2006/80825.

[26]

O. Hajek, Pursuit Games, Acad. New York, 1975.

[27]

A. D. Joffe and V. M. Tikhomirov, Theory of Extremal Problems, North Holland, Amsterdam, 1979.

[28]

P. O. Kas'yanov and V. S. Mel'nik, On properties of subdifferential mappings in Frėchet spaces, Ukr Math J., 57 (2005), 1621-1634.  doi: 10.1007/s11253-006-0017-5.

[29]

D. Y. Khusainov, J. Diblk and M. Ruzhichkova, Linear Dynamical Systems with Aftereffect. Representation of Decisions, Stability, Control, Stabilization, GP Inform-Analyt. Agency, Kiev, 2015.

[30]

D. Y. KhusainovD. D. Benditkis and J. Diblk, Weak Delay in Systems with an Aftereffect, Functional Differential Equations, 9 (2002), 385-404. 

[31]

N. N. Krasovskii and A. J. Subbotin, Positional Differential Games, Nauka, Moscow, 1974 (in Russian).

[32]

I. Iu. KryvonosAl. A. Chikrii and K. A. Chikrii, On an approach scheme in nonstationary game problems, Journal of Automation and Information Sciences, 45 (2013), 32-40.  doi: 10.1615/JAutomatInfScien.v45.i8.40.

[33]

N. F. KyrychenkoL. V. Baranovska and Al. A. Chikrii, On the class of linear differential-difference games of pursuit, Dopov. Akad. Nauk Ukr, 6 (1997), 24-26. 

[34]

M. S. Nikolskii, L.S. Pontryagins First Direct Method in Differential Games, Izdat. Lomonosov Moscow State University (Izdat. Gos. Univ.), Moscow, 1984.

[35]

V. A. Pepelyaev and Al. A. Chikrii, On the game dynamics problems for nonstationary controlled processes, Journal of Automation and Information Sciences, 49 (2007), 13-23.  doi: 10.1615/JAutomatInfScien.v49.i3.30.

[36]

P. V. Prokopovich and A. A. Chikrii, Quasilinear conflict-controlled processes with non-fixed time, Journal of Applied Mathematics and Mechanics, 55 (1991), 48-55.  doi: 10.1016/0021-8928(91)90061-X.

[37]

B. N. Pshenitchnyi, Convex Analysis and Extremal Problems, Nauka), Moscow, 1980.

[38]

M. Z. Zgurovskii and V. S. Mel'nik, Penalty Method for Variational Inequalities with Multivalued Mappings. Ⅲ, Cybernetics and Systems Analysis, 37 (2001), 203-213.  doi: 10.1023/A:1016794802307.

show all references

References:
[1]

J. AlbusA. MeystelA. A. ChikriiA. A. Belousov and A. J. Kozlov, Analitical method for solution of the game problem of softlanding for moving objects, Cybernetics and Systems Analysis, 37 (2001), 75-91.  doi: 10.1023/A:1016620201241.

[2]

J-P. Aubin and H. Frankovska, Set-Valued Analysis, Birkhäuser, Boston, 1990.

[3]

L. V. BaranovskayaA. A. Chikrii and Al. A. Chikrii, Inverse Minkowski functional in a nonstationary problem of group pursuit, Izvestia Academii Nauk. Teoria i Sistemy Upravlenia, 36 (1997), 101-106. 

[4]

Al. A. Chikrii, On nonstationary game problem of motion control, Journal of Automation and Information Sciences, 47 (2015), 74-83.  doi: 10.1615/JAutomatInfScien.v47.i11.60.

[5]

A. A. Chikrii, Conflict-Controlled Processes, Kluwer Academic Publishers Group, Dordrecht, 1997. doi: 10.1007/978-94-017-1135-7.

[6]

A. A. Chikrii, Analitical method in dynamic pursuit games, Proceedings of the Steklov Institute of Mathematics, 271 (2010), 69-85.  doi: 10.1134/S0081543810040073.

[7]

A. A. Chikrii and S. D. Eidelman, Generalized Mittag-Leffler matrix functions in game problems for evolutionary equations of fractional order, Cybernetics and Systems Analysis, 36 (2000), 315-338.  doi: 10.1007/BF02732983.

[8]

A. A. ChikriyL. V. Baranovskaya and A. A. Chikriy, An approach game problem under the failure of controlling devices, Journal of Automation and Information Sciences, 32 (2000), 1-8.  doi: 10.1615/JAutomatInfScien.v32.i5.10.

[9]

L. V. Baranovska, On quasilinear differential-difference games of approach, Journal of Automation and Information Sciences, 49 (2017), 53-67.  doi: 10.1615/JAutomatInfScien.v49.i8.40.

[10]

L. V. Baranovska, The modification of the method of resolving functions for the difference-differential pursuit's games, Naukovi Visti NTUU KPII, 4 (2012), 14-20. 

[11]

L. V. Baranovska, Method of resolving functions for the differential-difference pursuit game for different-inertia objects, in Advances in Dynamical Systems and Control. Studies in Systems, Decision and Control (eds. V. Sadovnichiy and M. Zgurovsky), Springer, 69 (2016), 159-176. doi: 10.1007/978-3-319-40673-2.

[12]

L. V. Baranovska, A method of resolving functions for one class of pursuit problems, Eastern-European Journal of Enterprise Technologies, 74 (2015), 4-18. 

[13]

L. V. Baranovska, The group pursuit differential-difference games of approach with non-fixed time, Naukovi Visti NTUU KPI, 4 (2011), 18-22. 

[14]

G. G. Baranovskaya and L. V. Baranovskaya, Group pursuit in quasilinear differential-difference games, Journal of Automation and Information Sciences, 29 (1997), 55-62.  doi: 10.1615/JAutomatInfScien.v29.i1.70.

[15]

G. G. Baranovskaya and L. V. Baranovska, On differential-difference group pursuit game., Dopov. Akad. Nauk Ukr, 3 (1997), 12-15. 

[16]

L. V. Baranovskaya and Al. A. Chikrii, Game Problems for a Class of Hereditaly Systems, Journal of Automation and Information Sciences, 29 (1997), 87-97.  doi: 10.1615/JAutomatInfScien.v29.i2-3.120.

[17]

L. V. Baranovskaya, About one class of difference games of group rapprochement with unfixed time, Science and World, 1 (2015), 10-12. 

[18] R. Bellman and K. L. Cook, Differential-Difference Equations, Academic Press, New York, 1963. 
[19]

Ya. I. Bigun, I. Iu. Kryvonos, Al. A. Chikrii and K. A. Chikrii, Group approach under phase constraints, Journal of Automation and Information Sciences. 46 (2014), 1-8. doi: 10.1615/JAutomatInfScien.v46.i4.10.

[20]

A. A. Chikrii and G. Ts. Chikrii, Matrix resolving functions in game problems of dynamics, Proceedings of the Steklov Institute of Mathematics, 291 (2015), 56-65.  doi: 10.1134/s0081543815090047.

[21]

A. A. Chikrii and A. A. Belousov, On linear differential games with integral constrains, Trudy Instituta Matematiki i Mekhaniki UrO RAN, 15 (2009), 290-301. 

[22]

A. A. ChikriiL. V. Baranovskaya and Al. A. Chikrii, The game problem of approach under the condition of failure of controlling devices, Problemy Upravleniya I Informatiki (Avtomatika), 4 (1997), 3-13. 

[23]

A. A. ChikriiI. S. Rappoport and K. A. Chikrii, Multivalued mappings and their selectors in the theory of conflict-controlled processes, Cybernetics and Systems Analysis, 43 (2007), 719-730.  doi: 10.1007/s10559-007-0097-8.

[24]

A. A. Chikrii and P. V. Prokopovich, Simple pursuit of one evader by a group, Cybernetics and Systems Analysis, 28 (1992), 438-444.  doi: 10.1007/BF01125424.

[25]

J. Diblk and D. Y. Khusainov, Representation of solutions of linear discrete systems with constant coefficients and pure delay, Advances in Difference Equations, 2006 (2006), Art. ID 80825, 13 pp. doi: 10.1155/ADE/2006/80825.

[26]

O. Hajek, Pursuit Games, Acad. New York, 1975.

[27]

A. D. Joffe and V. M. Tikhomirov, Theory of Extremal Problems, North Holland, Amsterdam, 1979.

[28]

P. O. Kas'yanov and V. S. Mel'nik, On properties of subdifferential mappings in Frėchet spaces, Ukr Math J., 57 (2005), 1621-1634.  doi: 10.1007/s11253-006-0017-5.

[29]

D. Y. Khusainov, J. Diblk and M. Ruzhichkova, Linear Dynamical Systems with Aftereffect. Representation of Decisions, Stability, Control, Stabilization, GP Inform-Analyt. Agency, Kiev, 2015.

[30]

D. Y. KhusainovD. D. Benditkis and J. Diblk, Weak Delay in Systems with an Aftereffect, Functional Differential Equations, 9 (2002), 385-404. 

[31]

N. N. Krasovskii and A. J. Subbotin, Positional Differential Games, Nauka, Moscow, 1974 (in Russian).

[32]

I. Iu. KryvonosAl. A. Chikrii and K. A. Chikrii, On an approach scheme in nonstationary game problems, Journal of Automation and Information Sciences, 45 (2013), 32-40.  doi: 10.1615/JAutomatInfScien.v45.i8.40.

[33]

N. F. KyrychenkoL. V. Baranovska and Al. A. Chikrii, On the class of linear differential-difference games of pursuit, Dopov. Akad. Nauk Ukr, 6 (1997), 24-26. 

[34]

M. S. Nikolskii, L.S. Pontryagins First Direct Method in Differential Games, Izdat. Lomonosov Moscow State University (Izdat. Gos. Univ.), Moscow, 1984.

[35]

V. A. Pepelyaev and Al. A. Chikrii, On the game dynamics problems for nonstationary controlled processes, Journal of Automation and Information Sciences, 49 (2007), 13-23.  doi: 10.1615/JAutomatInfScien.v49.i3.30.

[36]

P. V. Prokopovich and A. A. Chikrii, Quasilinear conflict-controlled processes with non-fixed time, Journal of Applied Mathematics and Mechanics, 55 (1991), 48-55.  doi: 10.1016/0021-8928(91)90061-X.

[37]

B. N. Pshenitchnyi, Convex Analysis and Extremal Problems, Nauka), Moscow, 1980.

[38]

M. Z. Zgurovskii and V. S. Mel'nik, Penalty Method for Variational Inequalities with Multivalued Mappings. Ⅲ, Cybernetics and Systems Analysis, 37 (2001), 203-213.  doi: 10.1023/A:1016794802307.

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