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

Abstract

It is proved (necessary and) sufficient conditions for Ψ − conditional asymptotic stability of the trivial solution of linear or nonlinear Lyapunov matrix differential equations.

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

“…Def inition 2.1. ( [7], [12], [13]). The solution z(t) of the differential equation z = f(t,z) (where z ∈ R d and f is a continuous d vector function) is said to be Ψ− stable on R + if for every ε > 0 and every t 0 ∈ R + , there exists a δ = δ(ε, t 0 ) > 0 such that, any solution z(t) of the equation which Remark 2.1.…”

“…Def inition 2.5. ( [11], [12]) A matrix function M : R + −→ M d×d is said to be Ψ− bounded on R + if the matrix function Ψ(t)M (t) is bounded on R + (i.e. there exists m > 0 such that | Ψ(t)M (t) | ≤ m, for all t ∈ R + ).…”

“…Def inition 2.6. ( [11], [12]). The solution Z(t) of the matrix differential equation Z = F (t, Z) is said to be Ψ− conditionally stable on R + if it is not Ψ− stable on R + but there exists a sequence (Z n (t)) of solutions of the equation defined on R + such that…”

“…where b ∈ R, b = 0, then, a fundamental matrix for the perturbed corresponding Kronecker product system associated with equation 2is As in Example 4.1, [12], we have that the all columns of (I 2 ⊗ Ψ) Z 0 (t) are unbounded on R + . From this and Theorem 3.1 (variant for systems), the corresponding Kronecker product system associated with equation 2is not I 2 ⊗ Ψ− conditionally exponentially asymptotically stable on R + .…”

“…Let X(t) be a Ψ− bounded solution of (11). Let Y (t) be defined by (12); this function is a Ψ− bounded solution of (4). Let Z(t) = Y (t)−U (t)P 1 X(0), for t ≥ 0.…”