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As a crucial means for stability analysis in control systems, the Lyapunov equation is applied in many fields of science and engineering. There are some methods proposed and studied for solving the non-stationary Lyapunov equation, such as the zeroing neural network (ZNN) model. However, a common drawback these methods have is that they rarely tolerate noises. Therefore, given that the existence of various types of noises during computation, a noise-tolerant ZNN (NTZNN) model with anti-noise ability is proposed for solving the non-stationary Lyapunov equation in this paper. For comparison, the conventional ZNN (CZNN) model is also applied to solve the same problem. Furthermore, theoretical analyses are provided to prove the global and exponential convergence performance of the proposed NTZNN model in the absence of noises. On this basis, the anti-noise performance of the proposed NTZNN model is proven. Finally, by adopting the proposed NTZNN model and the CZNN model to solve the non-stationary Lyapunov equation, computer simulations are conducted under the noise-free case and the noisy case, respectively. The simulation results indicate that the proposed NTZNN model is practicable for solving the non-stationary Lyapunov equation and superior to the CZNN model at the existence of noises. INDEX TERMS Non-stationary Lyapunov equation, noise-tolerant zeroing neural network (NTZNN), conventional zeroing neural network (CZNN), global and exponential convergence.

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Solving dynamic Sylvester matrix equations is a prevalent research topic and many methods have been arisen to solve the dynamic Sylvester equation, but few of them consider the noise effect. To investigate the new approach which can suppress the noise effect, integration feedback is added in the conventional Newton-Raphson iterated (CNRI) algorithm to form the proposed integration-implemented Newton-Raphson iterated (IINRI) algorithm based on the control theorem. Besides, this paper transforms the dynamic Sylvester equation into a linear equation which turns into the zeroing finding problem in further by constructing the error function. According to the theoretical analyses and the simulation results, the IINRI algorithm has higher accuracy and strong robustness under different noises (e.g. the constant noise, the linear noise, and the bounded random noise) while the performance of the CNRI algorithm is seriously degraded by the noises, which reveals that the IINRI algorithm is an efficient and powerful approach to solve dynamic Sylvester equation under noise perturbations. INDEX TERMS Dynamic Sylvester equation, integration-implemented, Newton-Raphson iterated algorithm, noise suppression.