On Ψ-boundedness and Ψ-stability of matrix Lyapunov systems

In this paper we obtain a necessary and sufficient condition for the existence of at least one Ψ-bounded solution and also obtained sufficient conditions for Ψ-(uniform) stability of the Kronecker product system associated with the matrix Lyapunov system X′(t)=A(t)X(t)+X(t)B(t)+F(t).