Discontinuity, Nonlinearity, and Complexity
Asymptotic Stability of Nonzero Solutions of Discontinuous Systems of Impulsive Differential Equations
Discontinuity, Nonlinearity, and Complexity 6(2) (2017) 201218  DOI:10.5890/DNC.2017.06.008
K. G. Dishlieva
Department of Differential Equations, Faculty of Applied Mathematics and Informatics,Technical University of Sofia, Sofia, 1000, Bulgaria
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Abstract
Discontinuous systems of nonlinear nonautonomous differential equations with impulsive effects are the main object of investigation in the paper. These systems consist of two basic parts: (i) A set of nonlinear nonautonomous systems of ordinary differential equations that define the continuous parts of the solutions. The righthand sides of the systems are elements of the set of functions f = { f1, f2, ...} ; (ii) The conditions which consistently determine “the switching moments”. The structural change (discontinuity) of the righthand side and impulsive perturbations take place at the moments of switching. In these moments, the trajectory meets the “switching sets”. They are parts of the hyperplanes, situated in the phase space of the system considered. Sufficient conditions are found so that the nonzero solutions of the studied discontinuous system with impulsive effects are asymptotically stable.
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