Justification of partially-multiplicative averaging for a class of functional-differential equations with variable structure and impulses (Q1066346)

The averaging method is justified for the first time for a class of functional-differential equations of neutral type with variable structure and impulses. The system under consideration is of the form \(\dot x(t)=\mu A(t,x_ t,\dot x_ t) X(t,x(t)),\) \(t\neq t_ i(x)\), where \(x_ t=x(h(t,x(t)))\) and \(\mu >0\) is a small parameter. For \(t=t_ i(x)\) the trajectory is subject to instant jumps (of order \(\mu)\) and the matrix A(.) is also changed. The averaged system however is of constant structure and without impulses. This clearly simplifies the qualitative investigation and the integration of this class of functional- differential equations.