We consider the pursuit problem for 2-person conflict-controlled process with single pursuer and single evader. The problem is given by the system of the linear functional-differential equations of neutral type. The players pursue their own goals and choose controls in the form of functions of a certain kind. The goal of the pursuer is to catch the evader in the shortest possible time. The goal of the evader is to avoid the meeting of the players’ trajectories on a whole semiinfinite interval of time or if it is impossible to maximally postpone the moment of meeting. For such a conflict-controlled process, we derive conditions on its parameters and initial state, which are sufficient for the trajectories of the players to meet at a certain moment of time for any counteractions of the evader.
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Altmann E, Pourtallier O (2001) Advances in dynamic games and applications. Birkhäuser, Boston. https://doi.org/10.1007/978-1-4612-0155-7
Aubin J, Frankowska H (1990) Set-valued analysis. Birkhäuser, Boston
Aumann R (1965) Integrals of set valued functions. J. Math. Anal. Appl. 12:1–12
Bellman R, Cooke K (1963) Differential-difference equations. Academic Press, New York
Chikrii A (1995) Multivalued mappings and their selections in game control problems. J Autom Inf Sci 27(1):27–38
Chikrii A (1996) Quasilinear controlled processes under conflict. J Math Sci 80(1):1489–1518
Chikrii A (2008) Optimization of game interaction of fractional-order controlled systems. Optim Methods Softw 23(1):39–72
Chikrii A (2008) Game dynamic problems for systems with fractional derivatives. In: Chinchuluun A, Pardalos PM, Migdalas A, Pitsoulis L (eds) Pareto optimality, game theory and equilibria. Springer optimization and its applications, vol 17. Springer, Berlin, pp 349–387
Chikrii A (2010) An analytical method in dynamic pursuit games. Proc Steklov Inst Math 271:69–85
Chikrii A (2013) Conflict-controlled processes. Springer, Berlin
Eidel’man S, Chikrii A (2001) Game problems of control for quasilinear systems with fractional Riemann–Liouville derivatives. Kibern Sist Anal 6:66–99
Hajek O (1975) Pursuit games. Mathematics in science and engineering. Academic, New York
Halanay A (1966) Differential equations: stability, oscillations, time lags. Academic Press, New York
Hale J (1977) Theory of functional differential equations. Springer, New York
Isaacs R (1965) Differential games: a mathematical theory with applications to warfare and pursuit, control and optimization. Wiley, New York
Krasovskii N (1968) Theory of Motion control, linear systems. Nauka, Moscow (in Russian)
Kumkov S, Le Ménec S, Patsko V (2017) Zero-sum pursuit-evasion differential games with many objects: survey of publications. Dynamic games and applications 7:609. https://doi.org/10.1007/s13235-016-0209-z
Nikolskii M (1982) Stroboscopic strategies and the first direct Pontryagin’s method in quasilinear nonstationary differential pursuit-evasion games. Problems of control and information theory/Problemy Upravlen. Teor Inform 11(5):373–377
Pilipenko Y, Chikrii A (1993) Oscilatory conflict-control processes. Prikl Mat Mech 57(3):3–14
Pittsyk M, Chikrii A (1982) On a group pursuit problem. J Appl Math Mech 46(5):584–589
Pontryagin L (1966) On the theory of differential games. Russ Math Surv 21:4
Pontryagin L, Boltyanskii V, Gamkrelidze R, Mischenko E (1962) The mathematical theory of optimal processes. Wiley (Interscience), New York
Pshenichnyi B, Chikrii A (1977) The differential game of evasion. Izv AN SSSR Tekh Kibern 1:3–10 (in Russian)
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Liubarshchuk, I., Bihun, Y. & Cherevko, I. Non-stationary Differential-Difference Games of Neutral Type. Dyn Games Appl 9, 771–779 (2019). https://doi.org/10.1007/s13235-019-00298-z
- Differential-difference games
- Dynamic games
- Pursuit problem
- The Method of Resolving Functions