Geodesic
Finite-dimensional approximations of neutral-type conflict-controlled systems
Izvestiya Instituta Matematiki i Informatiki Udmurtskogo Gosudarstvennogo Universiteta, Tome 49 (2017), pp. 111-122.

Voir la notice de l'article provenant de la source Math-Net.Ru

The paper deals with a conflict-controlled dynamical system which motion is described by neutral-type functional-differential equations in J. Hale's form. Approximations of this system by controlled high-dimensional systems of ordinary differential equations are investigated. A mutual aiming procedure between the initial system and its finite-dimensional approximation that guarantees proximity between their motions is elaborated. Stability properties of the procedure with respect to measurement errors are established, an illustrative example is considered. An application of the procedure is given for solving a guarantee optimization problem in which a motion of the dynamical system is described by linear functional-differential equations of neutral type in J. Hale's form and the quality index evaluates a motion history and realizations of control and disturbance actions. For this purpose an auxiliary control problem for the approximating system is formulated and its solution is constructed by the upper convex hulls method. It is shown that the optimal guaranteed result in the auxiliary problem approximates the optimal guaranteed result in the initial problem, and with the use of optimal in the auxiliary problem motions of the approximating system as guides an optimal control law is constructed. An illustrative example is considered, numerical simulation results are shown.
Keywords: control theory, differential games, neutral-type systems. 
@article{IIMI_2017_49_a4,
     author = {M. I. Gomoyunov and A. R. Plaksin},
     title = {Finite-dimensional approximations of neutral-type conflict-controlled systems},
     journal = {Izvestiya Instituta Matematiki i Informatiki Udmurtskogo Gosudarstvennogo Universiteta},
     pages = {111--122},
     publisher = {mathdoc},
     volume = {49},
     year = {2017},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/IIMI_2017_49_a4/}
}
TY  - JOUR
AU  - M. I. Gomoyunov
AU  - A. R. Plaksin
TI  - Finite-dimensional approximations of neutral-type conflict-controlled systems
JO  - Izvestiya Instituta Matematiki i Informatiki Udmurtskogo Gosudarstvennogo Universiteta
PY  - 2017
SP  - 111
EP  - 122
VL  - 49
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/IIMI_2017_49_a4/
LA  - ru
ID  - IIMI_2017_49_a4
ER  - 
%0 Journal Article
%A M. I. Gomoyunov
%A A. R. Plaksin
%T Finite-dimensional approximations of neutral-type conflict-controlled systems
%J Izvestiya Instituta Matematiki i Informatiki Udmurtskogo Gosudarstvennogo Universiteta
%D 2017
%P 111-122
%V 49
%I mathdoc
%U http://geodesic.mathdoc.fr/item/IIMI_2017_49_a4/
%G ru
%F IIMI_2017_49_a4
M. I. Gomoyunov; A. R. Plaksin. Finite-dimensional approximations of neutral-type conflict-controlled systems. Izvestiya Instituta Matematiki i Informatiki Udmurtskogo Gosudarstvennogo Universiteta, Tome 49 (2017), pp. 111-122. http://geodesic.mathdoc.fr/item/IIMI_2017_49_a4/

[1] Krasovskii N.N., “The approximation of a problem of analytic design of controls in a system with time-lag”, J. Appl. Math. Mech., 28:4 (1964), 876–885 | DOI | MR | Zbl

[2] Repin Yu.M., “On the approximate replacement of systems with lag by ordinary dynamical systems”, J. Appl. Math. Mech., 29:2 (1965), 254–264 | DOI | MR | Zbl

[3] Kurzhanskii A.B., “On the approximation of linear differential equations with delay”, Differents. Uravn., 3:12 (1967), 2094–2107 (in Russian) | Zbl

[4] Kryazhimskii A.V., Maksimov V.I., “Approximation in linear difference-differential games”, J. Appl. Math. Mech., 42:2 (1978), 212–219 | DOI | MR

[5] Banks H.T., Burns J.A., “Hereditary control problems: numerical methods based on averaging approximations”, SIAM J. Control Optim., 16:2 (1978), 169–208 | DOI | MR | Zbl

[6] Banks H.T., Kappel F., “Spline approximations for functional differential equations”, J. Differential Equations, 34:3 (1979), 496–522 | DOI | MR | Zbl

[7] Kunisch K., “Approximation schemes for nonlinear neutral optimal control systems”, J. Math. Anal. Appl., 82:1 (1981), 112–143 | DOI | MR | Zbl

[8] Fabiano R., “A semidiscrete approximation scheme for neutral delay-differential equations”, Int. J. Numer. Anal. Mod., 10:3 (2013), 712–726 | MR | Zbl

[9] Krasovskii N.N., Kotel'nikova A.N., “Stochastic guide for a time-delay object in a positional differential game”, Proc. Steklov Inst. Math., 277, suppl. 1 (2012), S145–S151 | DOI | MR | Zbl

[10] Lukoyanov N., Plaksin A., “On approximations of time-delay control systems”, IFAC-PapersOnLine, 48:25 (2015), 178–182 | DOI | MR

[11] Plaksin A.R., “Finite-dimensional guides for conflict-controlled linear systems of neutral type”, Differential Equations, 51:3 (2015), 406–416 | DOI | DOI | MR | Zbl

[12] Hale J.K., Cruz M.A., “Existence, uniqueness and continuous dependence for hereditary systems”, Ann. Mat. Pura Appl., 85:1 (1970), 63–81 | DOI | MR | Zbl

[13] Lukoyanov N.Yu., Plaksin A.R., “On the approximation of nonlinear conflict-controlled systems of neutral type”, Proc. Steklov Inst. Math., 292, suppl. 1 (2016), S182–S196 | DOI | MR | Zbl

[14] Krasovskii N.N., Control of a dynamic system, Nauka, M., 1985, 516 pp.

[15] Krasovskii N.N., Subbotin A.I., Game-theoretical control problems, Springer, New York, 1988, 517 pp. | MR | Zbl

[16] Krasovskii A.N., Krasovskii N.N., Control under lack of information, Birkhäuser, Berlin etc., 1995, 322 pp. | MR

[17] Isaacs R., Differential games, John Wiley and Sons, Inc., New York, 1965, 384 pp. | MR | Zbl

[18] Gomoyunov M., Plaksin A., “On a problem of guarantee optimization in time-delay systems”, IFAC-PapersOnLine, 48:25 (2015), 172–177 | DOI

[19] Lukoyanov N.Yu. A differential game with integral performance criterion, Differential Equations, 30:11 (1994), 1759–1766 | MR | Zbl

[20] Gomoyunov M.I., Lukoyanov N.Yu., Plaksin A.R., “Existence of the value and saddle point in positional differential games for neutral-type systems”, Trudy Inst. Mat. Mekh. Ural Otd. Ross. Akad. Nauk, 22:2 (2016), 101–112 (in Russian) | DOI | Zbl