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

Abstract

The paper provides (necessary and) sufficient conditions for Ψ-strong stability of the trivial solution of a linear Lyapunov matrix differential equations. Further, sufficient condition are obtained for Ψ-strong stability of the trivial solution of a nonlinear Lyapunov matrix differential equation.

On the Ψ - Exponential Asymptotic Stability of Nonlinear Lyapunov Matrix Differential Equations

“…It follows that the system (10) is I n ⊗ Ψ− exponentially asymptotically stable on R + and I n ⊗ Ψ− strongly stable on R + . Now, from Lemma 2.1 and Lemma 2.7, [6], it results that the equation (3) is Ψ− exponentially asymptotically stable on R + and Ψ− strongly stable on R + .…”

“…Remark 2.1. The definitions of various types of Ψ− stability on R + for a vector differential equation (6) was given in [5], [6], [7].…”

“…Recent results for Ψ− boundedness, Ψ− stability, Ψ− instability, dichotomy and conditioning for Lyapunov matrix differential equations have been given in many papers. See, for example, [4], [5], [6], [8], [9], [11].…”

On the Ψ-Conditional Asymptotic Stability of Nonlinear Lyapunov Matrix Differential Equations

“…Sufficient conditions for Ψ− conditional asymptotic stability of equations (4) and (9) are given in the following theorems.…”

“…If the linear equation 4is only Ψ− conditionally asymptotically stable on R + , then the perturbed equation 9can't be Ψ− conditionally asymptotically stable on R + . This is shown by the Example 3.8, [5], adapted to linear matrix differential equation 4and (9).…”