Novel criteria for exponential stability of functional differential equations (Q2845458)

A linear functional differential equation of the form NEWLINE\[NEWLINE \dot x(t) = A(t)x(t) +\int\limits_{-h}^0 d[\eta(t,\theta)]x(t+\theta),\qquad t\geq \sigma\tag{1} NEWLINE\]NEWLINE is considered, where \(A(\cdot)\in L_{\mathrm{loc}}^\infty (\mathbb R,\mathbb R^{n\times n})\) and \(\eta(\cdot,\cdot)\:\mathbb R\times [-h,0]\to\mathbb R^{n\times n}\) is measurable in \((t,\theta)\) and has bounded variation in \(\theta\) for each \(t\in\mathbb R\). Under certain conditions on the parameters of system (1), conditions for exponential stability are expressed in terms of the properties of some Metzler matrix.