On Ψ-stability of nonlinear Lyapunov matrix differential equations

On The Ψ – Asymptotic Stability Of Nonlinear Lyapunov Matrix Differential Equations

“…Remark. The Lemma 7, [6], is a similar result for Ψ− (uniform) stability of the trivial solution of the equation (1).…”

“…Murty, G. Suresh Kumar and A. Diamandescu in [5], [13], [14]. In our paper [6] have been proved (necessary and) sufficient conditions for Ψ− (uniform) stability of the trivial solutions of (nonlinear) Lyapunov matrix differential equations (1), (2) and (3). The purpose of this paper is to prove (necessary and) sufficient conditions for Ψ− asymptotic stability of trivial solution of the nonlinear Lyapunov matrix differential equation Z = A(t)Z + ZB(t) + F(t, Z) (1) and the linear equation…”

“…Lemma 5. ( [6]) The matrix function Z(t) is a solution of (1) on the interval J ⊂ R + if and only if the vector valued function z(t) = VecZ(t) is a solution of the differential system…”

On the Ψ-Strong Stability of Nonlinear Lyapunov Matrix Differential Equations

“…Kronecker product of A and B, written A ⊗ B, is defined to be the partitioned matrix We recall that the vectorization operator Vec has the following properties as concerns the calculations (see in [8], [9]): In the equation (1.1) we assume that A and B are continuous n × n matrices on R + = [0, ∞) and F : R + × M n×n → M n×n is a continuous n × n matrix such that F (t, O n ) = O n (null matrix of order n × n).…”

“…In our papers [6], [7], [8] are proved (necessary and) sufficient conditions for Ψ-(uniform) stability, Ψ-asymptotic stability and Ψ-instability of the trivial solutions of (nonlinear) Lyapunov matrix differential equations (1.1), (1.2) and (1.3).…”