Analysis And Design Of Time-delay High-order Inertial Neural Network Based On Non-reduced-order Method

As a special kind of models of neural networks,the inertial neural network is described by second-order differential equations.The second-order term is called the inertial term representing inductance in a circuit system.In some animal semicircular tubes,the membrane of hair cells can be realized by an equivalent circuit containing inductance.Therefore,it is of great significance to add inertia term into the neural network model.Moreover,it is found that ordinary low-order neural networks can solve limited optimization problems,while high-order neural networks have better performance.Due to the limited speed and irregular mode of information transmission between neurons,time delay is a common phenomenon in the research of inertial neural networks.At the same time,time delay also leads to divergence and instability of inertial neural networks.Therefore,the delayed high-order inertial neural networks have gradually attracted the attention of many scholars.This paper mainly studies the problems of stability,synchronization and state estimation of delayed high-order inertial neural networks.The main research contents are as follows:Firstly,the global h-stability of the high-order inertial neural networks with proportional delay is studied.A new Lyapunov-Krasovskii(L-K)functional is constructed without variable substitution,and a delay-dependent global h-stability criterion is derived.In addition,the proposed method is also applicable to the global h-stability problem of the high-order inertial neural networks with multiple proportional delays.Two numerical examples demonstrate the effectiveness of the proposed method.Secondly,the definition of global h-synchronization is proposed for the first time,and the global h-synchronization problem of the high-order inertial neural networks with time-varying delays is studied.It is important to note that the definition of h-synchronization is flexible,by selecting different regulating functions,one can derive different types of synchronization definitions.By constructing a new regulation function-dependent L-K functional,a new delay-dependent global h-synchronization criterion is obtained.In addition,an adaptive control algorithm is designed to ensure the global h-synchronization performance as well as reduce the control cost.A typical example is given to verify the superiority of this method.Then,the L2-L? state estimation problem for the high-order inertial neural networks with time-varying delays is proposed.An appropriate state estimator is constructed by using the non-reduced order method,which not only ensures the global h-stability of undisturbed error dynamics,but also ensures that the peak value of the estimated error remains within a certain range.Based on Lyapunov theory and some simple matrix calculation,a delay-dependent criterion is given.A simulation example is given to verify the applicability of the proposed estimation method.Finally,the stochastic global h-synchronization of high-order delay inertial neural networks with Markov jump parameters is studied.Instead of reducing the order of the original drive system by variable substitution,the corresponding second-order response system is given directly.The global mean square h-synchronization of the systems is realized by using second-order response system method.This method can greatly reduce the amount of calculation and control cost.A typical example shows the superiority and effectiveness of this method.The innovation of this paper is as follows:(1)A new definition of global hsynchronization is proposed.Different types of synchronization definitions can be obtained by changing the form of regulation functions h;(2)The non-reduced order method is adopted,which can reduce the amount of calculation compared with the traditional reduced order method;(3)A series of parameters are introduced.By adjusting the parameter values,not only can guarantee the linear matrix inequalities have a solution,but also can improve the decay rate;(4)The obtained delay-dependent criterion can be easily solved by MATLAB and is less conservative than the existing results.