Stability analysis for different classes of impulsive coupled systems of differential equations

We study the equilibrium stability of different classes of impulsive systems. For some class of a non-linear impulsive system consisting of two subsystems we use a piecewise time-differentiable Lyapunov functions as components of the Lyapunov vector function. Further, we implement two possibilities of construction of the Lyapunov function for the coupled system. The first of them involves the use of the weighted maximum of components of a Lyapunov vector function, and the second one involves the use of a Lyapunov vector function combined with the comparison method. For the class of linear coupled impulsive systems consisting of two time-invariant subsystems a matrix-valued Lyapunov function was constructed. Unlike many versions of small-gain stability conditions for coupled systems, we do not assume the property of asymptotic stability of the independent subsystems in this case. This makes it possible to study the asymptotic stability of a coupled system, in the case when the well known small-gain techniques are not applicable. We use the direct Lyapunov method in combination with the discretization method. The periodic case and the case when the Floquet theory is not applicable are considered separately. We also consider the problem of the asymptotic stability of linear large-scale impulsive systems with periodic coefficients in this case, the moments of the impulsive action are not assumed to be periodic, thus the Floquet theory is not applicable here. To construct convenient Lyapunov function we use the discretization method and classical identities of the commutator calculus.