On the asymptotic stability of systems with impulses by the direct method of Lyapunov (Q1823371)

The paper deals with motions (x(t),y(t)) which satisfy a system of differential equations of the type (1) \(\dot x=f(t,x)+g(t,y),\) \(\dot y=h(t,x,y)\) for all \(t\neq t_ i(x,y)\) \((i=1,2,...)\), but perform certain jumps \((2)\quad \Delta x|_{t=t_ i(x,y)}=A_ i(x)+B_ i(y),\) \(\Delta y|_{t=t_ i(x,y)}=C_ i(x,y)\) when passing the hypersurfaces \(t=t_ i(x,y)\). In six theorems sufficient conditions for (local or global) asymptotic stability are given making use of the concepts of Lyapunov's direct method for the trajectories of (1). A crucial assumption is that the distance of a motion from the origin (and also the value of its Lyapunov function) is non-increasing at each jump (2).