Necessary conditions of partial asymptotic stability of impulsive systems (Q2906194)

A system of differential equations with impulse effect of the form NEWLINE\[NEWLINE \frac{dx}{dt} = f(t,x), \quad t \neq \tau_i(x), \;t\in {\mathbb R}_+ , NEWLINE\]NEWLINE NEWLINE\[NEWLINE \Delta x = I_i (x) , \quad t = \tau_i(x), \;i\in {\mathbb N} , NEWLINE\]NEWLINE is considered. The partial stability of its zero solution is studied by means of Lyapunov functions. A converse theorem on uniform partial asymptotic stability is stated and proved.