In this article, the authors investigate the stability theory for generalized linear differential equation (GLDE) given by \[ \dfrac{dx}{d\tau} = D[A(t)x + g(t)], \] where \(A \colon [0, +\infty) \to \mathcal{L} (\mathbb R^n)\) and \(g \colon [0, +\infty) \to \mathbb R^n\) are functions of locally bounded variation. The authors characterize the uniform stability in terms of the boundedness of the transition matrix \(U(t, s)\) associated to the homogeneous GLDE \[ \dfrac{dx}{d\tau} = D[A(t)x]. \] Also, they provide a characterization of the uniform asymptotic stability with a uniform exponential decay of the transition matrix of the form \[ \| U(t, s)\| \le K e^{-\alpha (t - s)}, \ \ \textrm{for} \ K, \alpha > 0 \ \textrm{and any} \ t \geq s \geq 0. \] Further, they introduce definitions of global asymptotic stability and global uniform exponential stability for the trivial solution of the homogeneous GLDE \[ \dfrac{dx}{d\tau} = D[A(t)x] \] and provide a characterization of these both concepts of stability in terms of the transition matrix \(U(t, s)\).
Applying the Floquet theory for periodic systems given by \( \frac{dx}{d\tau} = D[A(t)x]\), the authors provide a necessary and sufficient condition ensuring global uniform exponential stability for the equation. They apply the found results to the linear impulsive differential equations, using the known correspondence between these two types of equations, presenting analogues results for this class of equations. They also provide an equivalence between the variational asymptotic stability of the trivial solution for the homogeneous GLDE \( \frac{dx}{d\tau} = D[A(t)x]\) to the uniform asymptotic stability for the linear case.
This interesting article brings relevant results for stability for the theory of generalized linear differential equations, through different concepts of stabilities and applications.