On the general problem of stability for impulsive differential equations

Abstract

Criteria for stability, asymptotical stability and instability of the nontrivial solutions of the impulsive system dx dt =f(t,x), t≠θ i(x),Δx| t=θ i(x) =I i(x), i∈ N={1,2,…}, where Δx| t= θ := x( θ+)− x( θ), x( θ+)=lim t→ 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.