On exponential stability in mean square of nonlinear delay differential equations with Markovian switching.

This article investigates the exponential stability in mean square of nonlinear delay differential equations with Markovian switching, introducing explicit stability criteria based on the equation coefficients and the generator of the Markovian switching process. Unlike traditional methods relying on Lyapunov functions, the approach employs the comparison principle to derive verifiable conditions for stability, including applications to switched neural networks with time-dependent delays. The main results provide matrix inequalities involving system parameters that guarantee exponential stability, supported by illustrative examples demonstrating the criteria’s applicability even when existing methods fail. The paper also discusses potential extensions to non-homogeneous Markov chains and stochastic delay differential equations with diffusion terms, noting challenges and directions for future research.