On the General Problem of Stability for Impulsive Differential Equations

Criteria for stability, asymptotical stability and instability of the nontrivial solutions of the impulsive system dx dt = f( t,x ), t� θi (x), ∆x|t =θi (x) = Ii (x), i ∈ N ={ 1, 2 ,... }, where ∆x|t =θ := x(θ+) − x(θ), x(θ+) = limt →θ + x(t) are obtained by Lyapunov’s second method. The construction of a reduced system in the neighbourhood of a nontrivial solution is a central auxiliary result of the paper.