Stability and attractivity of solutions of differential equations with impulses at fixed times
The authors consider two impulsive systems of the form \[ x'=F(t, x),\;t\neq t_k, \quad x(t^+_k)=x(t_k)+ I_k\bigl(x(t_k) \bigr),\;t=t_k,\quad x(t^+_0)= x_0,\tag{1} \] and \[ y'=f(t,y),\;t\neq t_k,\quad y(t^+_k)= \psi_k\bigl( y(t_k)\bigr),\;t=t_k, \quad y(t^+_0)= y_0,\tag{2} \] with solutions \(x(t)\) to (1) and \(y(t)\) to (2) and suppose that (1) and (2) are connected by a function \(v(t, x)\) which is locally Lipschitzian. Sufficient conditions for the inequality \[ v\bigl(t,x(t) \bigr)\leq \overline y(t),\;t\in\text{dom} x\cap\text{dom} \overline y, \] where \(\overline y(t)\) is an upper solution to (2) are given. Some generalized notions of well-known dynamic properties are introduced. The authors find sufficient conditions for the transfer of this properties of (2) to (1) by help of \(v(t,x)\).
scientific article; zbMATH DE number 1452805