Lyapunov stability of systems of lineargeneralized ordinary differential equations

Effective necessary and sufficient conditions are established for the stability in theLyapunov sense of solutions of the linear system of generalized ordinary differential equations dx(t)=dA(t)⋅x(t)+df(t), where A : ℝ+ → ℝn×n and ƒ: ℝ+ → ℝn (ℝ+ = [0,+∞[) are, respectively, matrix- and vector-functions with bounded total variation components on every closed interval from ℝ+, having properties analogous to the case of systems of ordinary differential equations with constant coefficients. The obtained results are realized for linear systems of both impulsive equations and difference equations.