Toroidal integral sets of systems with impulses (Q1821929)

Consider the system \(x'=Ax+P(x,\phi,t)\), \(\phi '=\omega +Q(x,\phi,t)\), \((t\neq t_ i)\), with \(\Delta x|_{t_ i}=Bx+H_ i(x,\phi),\Delta \phi |_{t_ i}=G_ i(x,\phi),\) where x is an n-dimensional vector and \(\phi\) is an m-dimensional vector, and where A and B are \(n\times n\) constant matrices with A in real canonical form. Assume that \(P,Q,H_ i\), and \(G_ i\) are bounded and piecewise continuous, periodic in \(\phi\) and satisfy a Lipschitz condition in x and \(\phi\) uniformly in t. Let \(t_{i+1}-t_ i\geq \theta\) for all i, \(\gamma =\max_{j} R\{\lambda_ j(A)\},\alpha^ 2=\max_{j} \lambda_ j\{(I+B)^ T(I+B)\}\) and suppose \(\gamma +\log \alpha /\theta <0.\) Then the system has a toroidal integral set \(x=u(\phi,t)\) with u(\(\phi\),t) periodic in \(\phi\), and this integral set is exponentially stable.