Stability criteria for linear Hamiltonian systems under impulsive perturbations

where p(t) and q(t) are real-valued functions and p(t) 6= 0 for any t ∈ R. In what follows we assume that a(t), b(t), and c(t) satisfy the periodicity conditions a(t+ T ) = a(t), b(t+ T ) = b(t), c(t+ T ) = c(t), t ∈ R. The system (2) (or (3)) is said to be stable if all solutions are bounded on R, unstable if all nontrivial solutions are unbounded on R, and conditionally stable if there exits a nontrivial solution bounded on R. The following well known stability theorem was given by M. Krein in [5]. Theorem 1.1. If b(t) ≥ 0, c(t) ≥ 0, b(t)c(t)− a(t) ≥ 0;