Exponential Input-to-state Stability Of Reaction-diffusion Systems

Reaction-diffusion systems are widely used in physics,chemical engineering and some biological phenomena,and its stability has become a research hotspot.The general stability requires the system state reach the zero-equilibrium point,but in real life,many systems only need the system state in a bounded range,so a new stability is proposed:input-to-state stability.The convergence speed of exponential input-to-state stability as a special form is faster.Therefore,this paper studies the exponential input to state stability of reaction-diffusion system.In addition,usually,a considered system cannot avoid timedelay and random noise in practical applications.In this paper,reaction-diffusion system with delay and random noise is also considered,and also study a special form of reactiondiffusion system: stochastic delay neural network with diffusion effect.In chapter 2,we consider the problem of exponential input-to-state stability for deterministic reaction-diffusion systems.Firstly,the distributed input and boundary input are both included in our considered model.By employing Lyapunov functional method and inequality techniques,sufficient conditions to guarantee the exponential input-tostate stability are obtained.Next,because of the universal existence of time-delay,we study the delay reaction-diffusion system with both distributed input and boundary input.Using Lyapunov-Krasovskii functional method and Wirtinger-type inequality,we obtain a delay-dependent sufficient condition to ensure the exponential input-to-state stability of delay reaction-diffusion systems.In addition,this sufficient condition not only shows the effect of time delay on the system reaching exponential input-to-state stability,but also shows the effect of spatial diffusion term on the system reaching exponential input-tostate stability,which is an important feature of reaction-diffusion system different from general differential system.Because it is difficult to avoid random interference in the actual phenomenon,in chapter 3,we consider the mean square exponential input-to-state stability for stochastic delay reaction-diffusion neural networks with distributed input and boundary input.In addition,constant delay and time-varying delay are considered.With the help of Lyapunov-Krasovskii functional method,It(?) formula and inequality techniques,delaydependent sufficient conditions on mean square exponential input-to-state stability of stochastic delay reaction-diffusion neural networks are presented.These sufficient conditions show that the smaller the delay and the larger the diffusion coefficient have a positive effect on the stability of the system.At the end of each chapter,the validity of the theoretical results is verified by numerical simulation.At the end of this paper,the research content is summarized and some new directions and suggestions are provided for the future research.