Justification of partially-multiplicative averaging for a class of functional-differential equations with impulses (Q5895469)

Let \(\sigma_ i:\) \(t=t_ i(x)\), \(i=1,2,..\). be a family of hypersurfaces such that for \(x\in D\subset R^ n\) they lie in the half- space \(t>0\) and \(t_ i(x)<t_{i+1}(x)\), \(i=1,2,... \).Let the point \(P_ t=(t,x(t))\) move along the trajectory of the system \[ (1)\quad x'(t)=\epsilon A(t,x(t),x(\Delta (t,x(t)),x'(\Delta (t,x(t)))X(t,x(t)),\quad t>0 \] and \(t\neq t_ i(x)\); \(x(t)=\phi (t,\epsilon)\), \(t\in <\)-\(\delta\),0\(>\), \(x'(t)=\phi '(t,\epsilon)\), \(t\in <\)-\(\delta\),0\(>\), where \(\epsilon\) is a small parameter, \(A(t,x,y,z)=(a_{ij}(t,x,y,z))_{nm}^ a \)matrix, \(\delta >0\) and t- \(\delta\leq \Delta (t,x)\leq t\) for \(t\geq\)-\(\delta\), \(x\in D\). Let \(I_ i(x)\), \(i=1,2,..\). be a family of vector functions defined on D. The problem is to find the solution x(t) of (1) which makes the jump \(x^+_ i=x_ i^{-1/2}+\epsilon I_ i(x_ i^{-1/2})\) when meeting the hypersurface \(\sigma_ i\). Let \(\lim(1/T)\int^{t+T}_{t}A(\tau,x,x,0)d\tau =A_ 0(x)\) exist and \(\lim(1/T)\sum_{t<t_ i<t+T}I_ i(x)=I_ 0(x)\) as \(T\to \infty\). The averaged system belonging to (1) is (2) \(\bar x'=\epsilon [A_ 0(\bar x(t))X(t,\bar x(t))+I_ 0(\bar x(x))],\) \(\bar x(0)=x_ 0\). The authors give an estimate of \(| x(t)-\bar x(t)|\) on an interval with length \(O(\epsilon^{-1})\).