On exponential stability of linear delay equations with oscillatory coefficients and kernels

arXiv2208.09018MaRDI QIDQ6408238

E. Braverman, Leonid Berezansky

Publication date: 18 August 2022

Abstract: New explicit exponential stability conditions are presented for the non-autonomous scalar linear functional differential equation ${\displaystyle }$ dot{x}(t)+ sum_{k=1}^m a_k(t)x(h_k(t))+int_{g(t)}^t K(t,s) x(s)ds=0, ${\displaystyle }$ where $h_k(t)leqt$, $g(t)leqt$, $a_k(cdot)$ and the kernel $K(cdot,cdot)$ are oscillatory and, generally, discontinuous functions. The proofs are based on establishing boundedness of solutions and later using the exponential dichotomy for linear equations stating that either the homogeneous equation is exponentially stable or a non-homogeneous equation has an unbounded solution for some bounded right-hand side. Explicit tests are applied to models of population dynamics, such as controlled Hutchinson and Mackey-Glass equations. The results are illustrated with numerical examples, and connection to known tests is discussed.

Mathematics Subject Classification ID

Asymptotic theory of functional-differential equations (34K25) Transformation and reduction of functional-differential equations and systems, normal forms (34K17) Stability theory of functional-differential equations (34K20) Linear functional-differential equations (34K06)

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