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Relative controllability of stochastic nonlinear systems with delay in control. (English) Zbl 1254.93029

Summary: The existence of solutions to systems is a natural premise to carry our study about controllability. Under the basic and readily verified conditions to guarantee the existence of the solutions to a system, in this paper, we prove the relative controllability (approximate controllability ) of the stochastic differential systems with delay in control. Sufficient conditions are given firstly for the relative controllability and relative approximate controllability in finite dimensional spaces, and these results are then generalized to infinite-dimensional Hilbert spaces. Finally, examples are given to illustrate the effectiveness of the proposed methods.

MSC:

93B05 Controllability
34H10 Chaos control for problems involving ordinary differential equations
47N10 Applications of operator theory in optimization, convex analysis, mathematical programming, economics
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[1] Bensoussan, A.; Da Prato, G.; Delfour, M. C.; Mitter, S. K., (Representation and Control of Infinite Dimension Systems. Representation and Control of Infinite Dimension Systems, Systems and Control: Foundations and Applications, vol. 2 (1993), Birkhauser) · Zbl 0790.93016
[2] Bashirov, A. E.; Mahmudov, N. I., On concepts of controllability for deterministic and stochastic systems, SIAM J. Control Optim., 37, 1808-1821 (1999) · Zbl 0940.93013
[3] Liu, B., Controllability of impulsive neutral functional differential inclusions with infinite delay, Nonlinear Anal., 60, 1533-1552 (2005) · Zbl 1079.93008
[4] Wang, Z. D.; Shu, H. S.; Fang, J. A., Robust stability for stochastic Hopfield neural networks with time delays, Nonlinear Anal. Real World Appl., 7, 1119-1128 (2006) · Zbl 1122.34065
[5] Shen, L. J.; Shi, J. P.; Sun, J. T., Complete controllability of impulsive stochastic integro-differential systems, Automatica, 46, 1068-1073 (2010) · Zbl 1192.93021
[6] Zhang, J.; Shi, P.; Qiu, J., Non-fragile guaranteed cost control for uncertain stochastic nonlinear time-delay systems, J. Franklin Institute, 346, 676-690 (2009) · Zbl 1298.93364
[7] Yang, R.; Shi, P.; Gao, H., A new delay-dependent stability criterion for stochastic systems with time delays, IET Control Theory Appl., 2, 966-973 (2008)
[8] Zhang, J.; Shi, P.; Qiu, J.; Yang, H., A new criterion for exponential stability of uncertain stochastic neural networks with mixed delays, Math. Comput. Modelling, 47, 1042-1051 (2008) · Zbl 1144.34388
[9] Gong, C.; Su, B., Delay-dependent robust stabilization for uncertain stochastic fuzzy system with time-varying delays, Int. J. Innov. Comput., Inf. Control, 5, 1429-1440 (2009)
[10] Alcorta-Garcia, M. A.; Basin, M.; Sanchez, Y., Risk-sensitive approach to optimal filtering and control for linear stochastic systems, Int. J. Innov. Comput., Inf. Control, 5, 1599-1614 (2009)
[11] Liu, Y. S.; Wang, W., Fuzzy \(H\)-infinity filtering for nonlinear stochastic systems with missing measurements, ICIC Express Lett., 3, 739-744 (2009)
[12] Li, H. Y.; Chen, B.; Zhou, Q.; Fang, S. L., Robust exponential stability for uncertain stochastic neural networks with discrete and distributed time-varying delays, Phys. Lett. A, 372, 19, 3385-3394 (2008) · Zbl 1220.82085
[13] Li, H. Y.; Chen, B.; Zhou, Q.; Lin, C., Delay-dependent robust stability for stochastic time-delay systems with polytopic uncertainties, Int. J. Robust Nonlinear Control, 18, 15, 1482-1492 (2008) · Zbl 1232.93092
[14] Mahmudov, N. I., Controllability of linear stochastic systems, IEEE Trans. Autom. Contr., 46, 724-731 (2001) · Zbl 1031.93034
[15] Mahmudov, N. I., Controllability of linear stochastic systems in Hilbert spaces, J. Math. Anal. Appl., 259, 64-82 (2001) · Zbl 1031.93032
[16] Goreac, D., Approximate controllability for linear stochastic differential equations in infinite dimensions, Appl. Math. Optim., 60, 105-132 (2009) · Zbl 1211.93020
[17] Sirbu, M.; Tessitore, G., Null controllability of an infinite dimensional SDE with state- and control-dependent noise, Systems & Control Lett., 44, 385-394 (2001) · Zbl 0987.93073
[18] Klamka, J., Stochastic controllability of systems with variable delay in control, Bull. Polon. A: Tech., 56, 279-284 (2008)
[19] Klamka, J., On the controllability of linear systems with delays in control, Internat. J. Control, 25, 875-883 (1977) · Zbl 0361.93012
[20] Arapostathis, A.; George, R. K.; Ghosh, M. K., On the controllability of a class of nonlinear stochastic systems, Systems & Control Lett., 44, 25-34 (2001) · Zbl 0986.93007
[21] Mahmudov, N. I.; Zorlu, S., Controllability of non-linear stochastic systems, Internat. J. Control, 76, 95-104 (2003) · Zbl 1111.93301
[22] Mahmudov, N. I., Approximate controllability of semilinear deterministic and stochastic evolution equations in abstract spaces, SIAM J. Control Optim., 42, 1604-1622 (2003) · Zbl 1084.93006
[23] Klamka, J., Relative controllability of non-linear systems with distributed delays in control, Internat. J. Control, 28, 811-819 (1978) · Zbl 0462.93009
[24] Balachandran, K.; Karthikeyan, S.; Park, J. Y., Controllability of stochastic systems with distributed delays in control, Internat. J. Control, 82, 1288-1296 (2009) · Zbl 1168.93004
[25] Balasubramaniam, P.; Park, J. Y.; Kumar, A. V.A., Existence of solutions for semilinear neutral stochastic functional differential equations with nonlocal conditions, Nonlinear Anal., 71, 1049-1058 (2009) · Zbl 1171.34054
[26] Mahmudov, N. I.; Denker, A., On controllability of linear stochastic systems, Internat. J. Control, 73, 144-151 (2001) · Zbl 1031.93033
[27] Hernandez, E.; O’Regan, D., Controllability of Volterra-Fredholm type systems in Banach spaces, J. Franklin Inst., 346, 95-101 (2009) · Zbl 1160.93005
[28] Triggiani, R., Addendum: a note on the lack of exact controllability for mild solutions in Banach spaces, SIAM J. Control Optim., 18, 98-99 (1980) · Zbl 0426.93013
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