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STABILITY OF THE DIFFERENTIAL EQUATIONS WITH VARIABLE STRUCTURE AND NON FIXED IMPULSIVE MOMENTS USING SEQUENCES OF LYAPUNOV'S FUNCTIONS

R.B. Chukleva, A.B. Dishliev, K.G. Dishlieva

Abstract


A specific class of non-linear non-autonomous systems ordinary differential equations with variable structure and impulses are studied in the paper. The change of the system right side and impulsive effects of the solution are realized at the moments, at which the so-called switching functions, defined in the system phase space, are canceled. Sufficient conditions for stability, uniform stability and uniform asymptotically stability of zero solution for the systems investigated are obtained. The results are received using a modification of the Lyapunov's Second Method. For this purpose, there are introduced sequences of the scalar piecewise - continuous functions of the Lyapunov class. The consecutive change (activation) of the Lyapunov's functions from the sequence are synchronized with the changing of the structure of the system investigated. Note that, it is allowed each one of the Lyapunov's functions of the sequence to be piecewise - continuous. The points of discontinuity coincide with the points of the set of switching of the corresponding right side of the system.

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