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Lyapunov stability of systems of linear generalized ordinary differential equations. (English) Zbl 1090.34043

Summary: Effective necessary and sufficient conditions are established for the stability in the Lyapunov sense of solutions of the linear system of generalized ordinary differential equations \[ dx(t)=dA(t)\cdot x(t)+df(t), \] where \(A:\mathbb{R}_+\to \mathbb{R}^{n\times n}\) and \(f:\mathbb{R}_+\to\mathbb{R}^n\) \((\mathbb{R}_+[0,+\infty[)\) are, respectively, matrix- and vector-functions with bounded total variation components on every closed interval from \(\mathbb{R}_+\), having properties analogous to the case of systems of ordinary differential equations with constant coefficients. The results obtained are realized for linear systems of both impulsive equations and difference equations.

MSC:

34D20 Stability of solutions to ordinary differential equations
34A30 Linear ordinary differential equations and systems
34A37 Ordinary differential equations with impulses
39A12 Discrete version of topics in analysis
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