A reduction principle for systems of differential equations that have impulses
The author considers the system with impulse action \[ (*)\quad x'=Ax+f(x,y,t),\quad y'=By+g(x,y,t)\quad (t\neq t_ i),\quad \Delta x|_{t=t_ i}=Cx+I_ i^{(1)}(x,y), \] \[ \Delta y|_{t=t_ i}=I_ i^{(2)}(x,y), \] where \(f,g,I_ i^{(1)}\), \(I_ i^{(2)}\) are continuous in t and Lipschitzian in x and y with sufficiently small constant and \(f(0,0,t)=I_ i^{(1)}(0,0)=0\), \(g(0,0,t)=I_ i^{(2)}(0,0)=0\). The eigenvalues of A have negative real parts and the eigenvalues of B have zero real parts. Under some additional assumptions she proves the existence of a function h such that the trivial solution of the equation \(y'=By+g(h(y,t),y,t)\) \((t\neq t_ i)\), \(\Delta y|_{t=t_ i}=I_ i^{(2)}(h(y,t_ i\)-0),y) has the same type of Lyapunov stability as the trivial solution of (*).
scientific article; zbMATH DE number 3881083
scientific article