Abstract
The paper deals with qualitative methods of investigation of dynamic systems with control. Classification of singular sets of two-dimensional bilinear control systems is proposed. Constructive criteria of the presence of different kinds of singular sets in the phase portrait are obtained. The dependence of dynamic properties of systems such as oscillation, stability, instability, and controllability on the types of singular sets is investigated. Analytical conditions under which analysis of the above properties is relatively simple are obtained.
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A. V. Babichev, A. G. Butkovskiy, and N. L. Lepe, “Singular sets on phase portraits of dynamic systems with control. I, II,” Automat. Remote Control, 47, May, Jul., 563–571, 878–884 (1986).
A. G. Butkovskiy, Phase Portraits of Controlled Dynamic Systems [in Russian], Nauka, Moscow (1985).
R. Gabasov and F. M. Kirillova, Singular Optimal Control [in Russian], Nauka, Moscow (1973).
A. P. Molchanov and Ye. S. Pjatnitsky, “Lyapunov functions that specify necessary and sufficient conditions of absolute stability of non-linear, non-stationary systems. II,” Automat. Remote Control, 47, Apr., 443–451 (1986).
A. F. Philippov, “Stability conditions in homogeneous systems with arbitrary regime switchings,” Automat. Remote Control, 41, Aug., 1078–1085 (1980).
V. N. Zhermolenko, “Oscillativity of two-dimensional bilinear systems,” Automat. Remote Control (2005).
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Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 11, No. 8, pp. 105–117, 2005.
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Zhermolenko, V.N. Singular sets and dynamic properties of bilinear control systems. J Math Sci 147, 6623–6630 (2007). https://doi.org/10.1007/s10958-007-0498-2
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DOI: https://doi.org/10.1007/s10958-007-0498-2