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

Abstract

In this paper we obtain a necessary and sufficient condition for the existence of at least one -bounded solution and also obtained sufficient conditions for -(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).

Ψ -asymptotic stability of non-linear matrix Lyapunov systems

“…From (i) of Theorem 3 [7], it follows that the trivial solution of (2.2) is Ψ-stable on R + . For anyX(t 0 ) ∈ R n 2 , we have…”

“…Further, these concepts are extended to non-linear volterra integro-differential equations by Diamandescu [[3], [4]]. Recently, Murty and Suresh Kumar [ [7], [8]] extended the concepts of Ψ-boundedness, Ψ-stability and Ψ-instability to matrix Lyapunov systems.…”

“…Here, we extend the concept of Ψ-stability in [7] to Ψ-asymptotic stability for matrix Lyapunov systems.…”

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).…”

“…Recent results for Ψ-boundedness, Ψ-stability, dichotomy and conditioning for Lyapunov matrix differential equations have been given in [5], [9], [10], [11].…”