M. Ashordia, Sh. Akhalaia, and N. Kekelia

On Necessary and Sufficient Conditions for Stability of Linear Systems of Generalized Ordinary Differential Equations

abstract:
Necessary and sufficient conditions are established for stability in the Lyapunov sense of solutions of the linear system of generalized ordinary differential equations
\begin{eqnarray*}
dx(t)=dA(t)\cdot x(t)+d f(t),
\end{eqnarray*}
where $A: \mathbb{R}_{+}\to \mathbb{R}^{n\times n}$ and $f: \mathbb{R}_{+}\to \mathbb{R}^{n}$ $(\mathbb{R}_{+}=[0,+\infty[\,)$ are, respectively, continuous from the left matrix-and vector-functions with bounded total variation components on every closed interval from $\mathbb{R}_{+}$.

Mathematics Subject Classification: 34K20, 34A37, 34D20

Key words and phrases: Stability in the Lyapunov sense, linear system of generalized ordinary differential equations, Lebesgue-Stiltjes integral