Functional differential games with nonatomic difference operator

  • L. A. Vlasenko Kharkiv National University of Radio Electronics
  • A. G. Rutkas Kharkiv National University of Radio Electronics
  • A. O. Chikrii Institute of Cybernetics of the National Academy of Sciences of Ukraine, Kyiv
Keywords: differential game, functional differential equation, Hilbert space, partial differential equation


UDC 517.9

We study a differential game of approach in a system whose dynamics is described
by a linear functional differential equation. The coefficients of the equation are closed linear operators on Hilbert spaces. The operator multiplying the state derivative at the current time is generally non-invertible. The main assumption is a restriction imposed on the characteristic operator pencil of the equation on a ray of real the positive semi-axis. Solutions of the equation are represented with the help of a formula of variation of constants where the delay effect is taken into account by summing shift type operators. To obtain conditions for the approach of the system dynamic vector to a cylindrical terminal set, we use constraints on support functionals of two sets defined by the behavior of pursuer and evader.
The paper contains an example to illustrate the differential game in a pseudoparabolic system described by a partial functional differential equation.


R. Isaacs, Differential games, John Wiley and Sons, New York ect. (1965).

A. Friedman, Differential games of pursuit in Banach spaces, Math. Anal. and Appl., 25, 93 – 113 (1969); DOI:

A. A. Chikrii, Conflict-controlled processes, Springer Sci. and Business Media, Dordrecht (2013); DOI:

J. Yong, Differential games: a concise introduction, World Scientific Publishing: New Jersey ect. (2015); DOI:

N. N. Krasovskyi, Yu. S. Osypov, Lyneinye dyfferentsyalno-raznostnye yhry, Dokl. AN SSSR, 197, 777 – 780 (1971).

E. N. Chukwu, Capture in linear functional differential games of pursuit, J. Math. Anal. and Appl., 70, 326 – 336 (1979); DOI:

A. A. Chykryi, H. Ts. Chykryi, Hruppovoe presledovanye v dyfferentsyalno-raznostnykh yhrakh, Dyfferents. uravnenyia, 20, 802 – 810 (1984).

P. V. Reddy, J. C. Engwerda, Feedback properties of descriptor systems using matrix projectors and

applications to descriptor differential games, SIAM J. Matrix Anal. and Appl., 34, 686 – 708 (2013); DOI:

J. H. Lightbourne, S. M. Rankin, A partial functional differential equation of Sobolev type, J. Math. Anal. and Appl., 93, 328 – 337 (1983); DOI:

J. K. Hale, S. Verduyn M. Lunel, Introduction to functional differential equations, Springer-Verlag, New York (1993), DOI:

A. G. Rutkas, L. A. Vlasenko, Time-domain descriptor models for circuits with multiconductor transmission lines and lumped elements, Proc. 5th IEEE Int. Conf. Ultrawideband and Ultrashort Impulse Signals (Sevastopol, Crimea), 102 – 104 (2010); DOI:

E. Hille, R. S. Phillips, Functional analysis and semi-groups, Providence, Phode Island (1957).

K. Yosida, Functional analysis, Springer-Verlag, Berlin etc. (1980).

L. A. Vlasenko, A. G. Rutkas, Optimal control of undamped Sobolev-type retarded systems, Math. Notes, 102, 297 – 309 (2017); DOI:

A. Pazy, Semigroups of linear operators and applications to partial differential equations, Springer-Verlag, New York ect. (1983), DOI:

J. L. Lions, Optimal control of systems governed by partial differential equations, Springer-Verlag, New York ect. (1971). DOI:

O. A. Boichuk, V. L. Makarov, V. A. Feruk, A criterion of solvability of resonant equations and construction of their solutions, Ukr. Math. J., 71, № 11, 1510 – 1521 (2020); DOI:

A. A. Chikrii, An analytical method in dynamic pursuit games, Proc. Steklov Inst. Math., 271, 69 – 85 (2010); DOI:

L. A. Vlasenko, A. G. Rutkas, On a differential game in a system described by an implicit differential-operator equation, Different. Equat., 51, 798 – 807 (2015); DOI:

L. A. Vlasenko, A. A. Chikrii, On a differential game in a system with distributed parameters, Proc. Steklov Inst. Math., 292, Issue 1 Supplement, 276 – 285 (2016); DOI:

A. V. Balakrishnan, introduction to optimization theory in a hilbert space, Springer-Verlag, Berlin ect. (1971). DOI:

R. E. Showalter, T. W. Ting, Pseudoparabolic partial differential equations, SIAM J. Math. Anal., 1, 1 – 26 (1970); DOI:

A. Rutkas, L. Vlasenko, Implicit operator differential equations and applications to electrodynamics, Math. Methods Appl. Sci., 23, 1 – 15 (2000);<1::AID-MMA100>3.0.CO;2-5 DOI:<1::AID-MMA100>3.0.CO;2-5

V. L. Makarov, N. V. Maiko, Vahovi otsinky tochnosti metodu peretvorennia Keli dlia abstraktnykh kraiovykh zadach u banakhovomu prostori, Dop. NAN Ukrainy, № 5, 3—9 (2020); DOI:

V. L. Makarov, N. V. Mayko, Weighted estimates of the cayley transform method for boundary value problems in a banach space, Numer. Funct. Anal. and Optim., 42, 211 – 233 (2021); DOI:

How to Cite
Vlasenko, L. A., A. G. Rutkas, and A. O. Chikrii. “Functional Differential Games With Nonatomic Difference Operator”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 74, no. 2, Feb. 2022, pp. 164 -77, doi:10.37863/umzh.v74i2.6895.
Research articles