Exponential stability of solutions to nonlinear time-varying delay systems of neutral type equations with periodic coefficients

Abstract

We consider a class of nonlinear time-varying delay systems of neutral type differential equations with periodic coefficients in the linear terms, $$\begin{aligned} \frac{d}{dt} y(t) &= A(t) y(t) + B(t) y(t-\tau(t)) + C(t) \frac{d}{dt} y(t-\tau(t)) \cr &\quad + F\Big(t, y(t), y(t-\tau(t)), \frac{d}{dt} y(t-\tau(t)) \Big), \end{aligned}$$ where A(t), B(t), C(t) are T-periodic matrices, and $$ \|F(t,u,v,w)\| \le q_1\|u\| + q_2\|v\| + q_3 \|w\|, \quad q_1, q_2, q_3 \ge 0, \quad t > 0. $$ We obtain conditions for the exponential stability of the zero solution and estimates for the exponential decay of the solutions at infinity.

Keywords

• time-varying delay equation
• neutral equation
• periodic coefficient
• exponential stability