Splitting of linear systems with impulses
Let \(J=[t_0,\infty)\), \(\{t_k\}\subset (t_0,\infty)\) be an increasing sequence, \(\lim_{k\to \infty}t_k= \infty\) and \(J_k= (t_{k-1},t_k]\). The authors consider the impulsive system \[ y'(t)=\bigl( A(t)+B(t) \bigr)y(t), \quad \Delta y(t_k)= (C_k+D_k) y(t_k), \quad k\in\mathbb{N},\;t\neq t_k,\tag{1} \] where \(A(t)\), \(B(t)\) are \(n\times n\)-matrix functions on \(J\) which are uniformly continuous on each interval \(J_k\), \(C_k\), \(D_k\) are \(n\times n\)-matrices and \(I+C_k\) are invertible. It is proved that if the nonperturbed system \(x'=A(t)x\), \(\Delta x(t_k) =C_kx(t_k)\), has an exponential dichotomy with projection \(P\) and \(PA(t)= A(t)P\), \(PC_k= C_kP\), for \(t\in J\) and \(k\in\mathbb{N}\), then there exists a linear change of variables \(y(t)= S(t)z(t)\) which transforms (1) to the form \[ z'\bigl(A(t) +\widetilde B(t) \bigr)z,\;\Delta z(t_k)= (C_k+\widetilde D_k) z(t_k), \] with \(P\widetilde B(t)= \widetilde B(t)P\) and \(P\widetilde D_k= \widetilde D_kP\). By this result, a theorem of roughness for exponential dichotomies of (1) is proved.
scientific article; zbMATH DE number 1439647
scientific article