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Sufficient conditions for the local controllability of systems with random parameters for an arbitrary number of system states. (English. Russian original) Zbl 1157.93322
Russ. Math. 52, No. 3, 34-44 (2008); translation from Izv. Vyssh. Uchebn. Zaved., Mat. 2008, No. 3, 38-49 (2008).
Summary: We obtain sufficient conditions for the existence of a nonanticipating control for linear systems with stationary random parameters. We consider the case of a bounded control and an arbitrary number of system states. We estimate the probability that the system is nonanticipatingly locally controllable on a fixed time interval. We formulate the main assertions in terms of Lyapunov functions, choosing the latter in the class of piecewise continuously differentiable functions.

MSC:
93B05 Controllability
93E03 Stochastic systems in control theory (general)
93D30 Lyapunov and storage functions
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[1] R. Z. Khasminsky, ”Limit Theorem for a Solution of the Differential Equation with a Random Right Part,” Prob. Theor. Appl. 11(3), 444–462 (1966).
[2] O. V. Baranova, ”The Uniform Global Controllability of Linear Systems with Stationary Random Parameters,” Differents. Uravneniya 27(11), 1843–1850 (1991).
[3] F. Colonius and R. Jonson, ”Local and Global Null Controllability of Time Varying Linear Control Systems,” Control, Optimization and Calculus of Variations 2, 329–341 (1997). · Zbl 0899.93004
[4] D. P. De Farias, J. C. Geromel, J. B. R. Do Val, and O. L. V. Costa, ”Output Feedback Control of Markov Jump Linear Systems in Continuous-Time,” IEEE Trans. Autom. Contr. 45(5), 944–949 (2000). · Zbl 0972.93074
[5] Ye. Tsarkov, ”Asymptotic Methods for Stability Analysis of Markov Impulse Dynamical Systems,” Nonlin. Dynam. Syst. Theor. 2(1), 103–115 (2002). · Zbl 1023.34050
[6] S. Ibrir and E. K. Boukas, ”A Constant-Gain Nonlinear Estimator for Linear Switching Systems,” Nonlin. Dynam. Syst. Theor. 5(1), 49–59 (2005). · Zbl 1080.93010
[7] A. I. Subbotin and A. G. Chentsov, Guarantee Optimization in Control Problems (Nauka, Moscow, 1981) [in Russian]. · Zbl 0542.90106
[8] N. N. Krasovskii, Control of a Dynamic System (Nauka, Moscow, 1985) [in Russian].
[9] S. F. NIkilaev and E. L. Tonkov, ”Differentiability of Speed Vector and Positional Control of a Linear Subcritical System,” Differents. Uravneniya 36(1), 76–84 (2000).
[10] S. F. Nikilaev and E. L. Tonkov, ”Some Problems Connected with the Existence and the Construction of Nonanticipating Control for Nonstationary Control Systems,” Vestn. Udmurtsk. Univ., No. 1, 11–32 (2000).
[11] Yu. V. Masterkov and L. I. Rodina, ”Controllability of a Linear Dynamic System with Random Parameters,” Differents. Uravneniya 43(4), 457–464 (2007). · Zbl 1131.93008
[12] Yu. V. Masterkov and L. I. Rodina, ”Construction of a Nonanticipating Control for Systems with Random Parameters,” Vestn. Udmurtsk. Univ., No. 1, 101–114 (2005).
[13] Yu. V. Masterkov and L. I. Rodina, ”Conditions of Local Controllability for Systems with Random Parameters,” Vestn. Udmurtsk. Univ., No. 1, 81–94 (2006).
[14] Yu. V. Masterkov and L. I. Rodina, ”The Sufficient Conditions of Local Controllability for Linear Systems with Random Parameters,” Nonlin. Dynam. Syst. Theor. 7(3), 303–314 (2007). · Zbl 1133.93008
[15] A. N. Shiryaev, Probability (Nauka, Moscow, 1989) [in Russian].
[16] V. S. Korolyuk et al., Reference-Book on the Theory of Probability and Mathematical Statistics (Nauka, Moscow, 1985) [in Russian].
[17] I. P. Kornfel’d, Ya. G. Sinai, and S. V. Fomin, Ergodic Theory (Nauka, Moscow, 1980) [in Russian].
[18] Yu. A. Rozanov, Stationary Random Processes (Nauka, Moscow, 1990) [in Russian]. · Zbl 0721.60040
[19] N. N. Krasovskii, Theory of Motion Control (Nauka, Moscow, 1968) [in Russian].
[20] L. I. Rodina and E. L. Tonkov, ”Conditions of Complete Controllability of a Nonstationary Linear System in a Critical Case,” Kibern. i Sist. Analiz 40(3), 87–100 (2004). · Zbl 1068.93013
[21] E. A. Galperin and N. N. Krasovskii, ”Stabilization of Stationary Motions in Nonlinear Control Systems,” Prikl. Matem. i Mekh. 27, 1–24 (1963).
[22] E. Roxin, ”Stability in General Control Systems,” J. Different. Equat. 1(2), 115–150 (1965). · Zbl 0143.32302
[23] A. F. Filippov, Differential Equations with a Discontinuous Right-Hand Side (Nauka, Moscow, 1985) [in Russian]. · Zbl 0571.34001
[24] E. A. Galperin, ”Some Generalization of Lyapunov’s Approach to Stability and Control,” Nonlin. Dynam. Syst. Theor. 2(1), 1–24 (2002). · Zbl 1034.34058
[25] E. A. Barbashin, Lyapunov Functions (Nauka, Moscow, 1970) [in Russian].
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