# A.A. Chikrii, G.Ts. Chikrii. Game problems of approach for quasilinear systems of general form ... P. 273-287

We study a conflict-controlled process of the approach of a trajectory to a cylindrical terminal set. The problem statement encompasses a wide range of quasilinear functional–differential systems.We use the technique of set-valued mappings and their selections to derive sufficient conditions for the game termination in a finite time. The methodology used is close to the scheme that involves the time of the first absorption. By way of illustration, quasilinear integro-differential games are examined. For this purpose, their solutions are presented in the form of an analog of the Cauchy formula. The calculations are performed for the case of a system with a simple matrix; the control sets of the players are balls centered at the origin and the terminal set is a linear subspace. Depending on the relations between the initial state of the system and the parameters of the process, sufficient conditions for the game termination are derived. An explicit form of the guaranteed time is found in one specific case.

Keywords: conflict-controlled process, selection of a set-valued mapping, Aumann’s integral, support function, integro-differential equation.

The paper was received by the Editorial Office on October 4, 2017.

Arkadii Alekseevich Chikrii, Dr. Phys.-Math. Sci., Prof., Corresponding Member of National Academy
of Sciences of Ukraine, Glushkov Cybernetics Institute of NASU, Kiev, 03187 Ukraine,
e-mail:chik@d165.icyb.kiev.ua .

Greta Tsolakovna Chikrii, Dr. Phys.-Math. Sci., Senior Scientific Researcher, Glushkov Cybernetics
Institute of NASU, Kiev, 03187 Ukraine, e-mail:g.chikrii@gmail.com.

REFERENCES

1. Krasovskii N.N. Igrovye zadachi o vstreche dvizhenii. [Game problems on the encounter of motions]. Moscow, Nauka Publ., 1970, 420 p.

2. Krasovskii N.N., Subbotin A.I. Game-theoretical control problems. N Y: Springer, 1987, 517 p. This book is substantially revised version of the monograph Pozitsionnye differentsial’nye igry, Moscow, Nauka Publ., 1974, 456 p.

3. Pshenichnyi B.N. Linear differential games. Automat. Remote Control, 1968, no. 1, pp. 55–67.

4. Subbotin A.I., Chentsov A.G. Optimizatsiya garantii v zadachakh upravleniya. [Optimization of a guarantee in control problems]. Moscow, Nauka Publ., 1981, 288 p.

5. Osipov Yu.S., Kryazhimskii A.V. Inverse problems for ordinary differential equations: dynamical solutions. Basel: Gordon and Breach, 1995, 625 p.

6. Kurzhanskii A.B. Upravlenie i nablyudenie v usloviyakh neopredelennosti. [Control and Observation Under the Conditions of Uncertainty]. Moscow, Nauka Publ., 1977, 456 p.

7. Chentsov A.G.,Morina S.J. Extensions and relaxations. Boston, London, Dordrecht: Kluwer Acad. Publ., 2002, 408 p. doi: 10.1007/978-94-017-1527-0 .

8. Pontryagin L.S. Izbrannye nauchnye trudy. T. 2. [Selected Scientific Works, Vol. 2]. Moscow, Nauka Publ., 1988, 576 p.

9. Pshenichnyi B.N. The structure of differential games. Dokl. AN USSR, 1969, vol. 184, no. 2, pp. 285–287 (in Russian).

10. Hajek O. Pursuit games. N Y, Acad. Press, 1975, 266 p. ISBN: 0123172608 .

11. Chikrii A.A. Conflict-controlled processes. Boston, London, Dordrecht, Springer Science and Busines Media, 2013, 424 p. ISBN: 9401711364 .

12. Rockafellar R. Convex analysis. Princeton, Princeton University Press, 1970, 451 p. ISBN: 0691015864 . Translated to Russian under the title Vypuklyi analiz, Moscow, Mir Publ., 1973, 470 p.

13. Aubin J.-P., Frankowska H. Set-valued analysis. Boston, Basel: Berlin, Birkh$\ddot{\mathrm{a}}$user, 1990, 461 p. ISBN: 0817634789 .

14. Aubin J.P., Ekeland I. Applied nonlinear analysis. N Y, Wiley-Interscience, 1984, 518 p. ISBN: 0-471-05998-6 . Translated to Russian under the title Prikladnoi nelineinyi analiz. Moscow, Mir Publ., 1988, 512 p.

15. Natanson I.P. Theory of functions of a real variable. N Y, Frederick Ungar Publishing Co., 1955, 277 p. (Translation of chapters I to IX of the 1st edition). Original Russian text (3rd ed.) published in Teoriya funktsii veshchestvennoi peremennoi, Moscow, Nauka Publ., 1974, 480 p.

16. Mikhlin S.G. Linear Integral Equations. Delhi, Hindustan Publ. Corp., 1960, 223 p. Original Russian text published in Lektsii po lineinym integral’nym uravneniyam. Moscow, Fizmatgiz Publ., 1959, 304 p.

17. Smirnov V.I. A course of higher mathematics. Vol. IV . [Integral equations and partial differential equations]. Oxford: N Y, Pergamon Press 1964, 811 p. ISBN: 9781483194714 . Original Russian text (6th ed.) published in Kurs vysshei matematiki. Moscow, Nauka Publ., 1974, vol. 4, Ch. 1, 236 p.