Study On The Stability And Impulsive Stabilization Of Time-delay Systems

Impulsive effects widely exist in the engineering practices,such as attitude adjustment of spacecraft,the lateral pulse jet control of an atmospheric rocket,pest harvest in agriculture and the vaccine injection in the ecology groups.Compare with the other control methods,impulsive control turns out to be an effective,economic and convenient method in the implementation.In some cases,the systems cannot endure continuous control or it is impossible to implement continuous control,impulsive control is an effective alternative method.As a consequence,to investigate the stability and stabilization of impulsive system is an important and meaningful issue.In this dissertation,the stability problems of three different impulsive systems,including impulsive neural networks,impulsively coupled complex networks and impulsive interconnected system have been investigated by employing Lyapunov function and Lyapunov—Krasovskii functional.The main results are outlined in the following parts.In order to investigate the stability of neural networks with delayed piecewise constant arguments and periodic coefficients,a new definition of global exponential stability and a novel differential inequality for differential equations with piecewise constant arguments have been proposed.The existence,uniqueness and global exponential stability of the periodic solution are obtained for this kind of neural networks.Based on the new definition of exponential stability and the introduced differential inequality,the stability criteria for the periodic solution can be obtained just according to the original differential equation.It is not necessary to establish any relationship between the norms of the states with/without piecewise constant arguments to obtain the main results.Additionly,our stability criteria are independent on the piecewise constant argument sequence.According to the results of the numerical example investigation,the stability criteria in the present disseration are less conservative in comparison with the existed results in the literature.Based on the same definition of exponential stability,the Razumikhin technique for differential equations with piecewise constant arguments has been proposed to investigate the exponential stability issue of the impulsive neural networks with piecewise constant arguments.By using the introduced Razumikhin technique,both the sufficient stability criteria dependent and independent on the piecewise constant arguments have been obtained for the periodic solution.Finally,combine with the linear matrix inequality method,sufficient stability conditions in terms of linear matrix inequality have been derived for the equilibrium of neural networks.Typical numerical examples have been used to illustrate that the results here are less conservative than the existed results in the literature.For the impulsively coupled complex networks,a hybrid control scheme is proposed to investigate the synchronization stabilization problem in this dissertation.A general delayed impulsively coupled network with nonidentical nodes is presented.A Lyapunov function associated with the impulsive instant sequence and the convex combination technique are employed to obtain the global exponential synchronization criteria in terms of linear matrix inequalities.The derived sufficient conditions for synchronization are closely related to the coupling structure of the network,the lower and upper bound of the adjacent impulsive instant difference,and the impulsive control input.The results can also be used to obtain the synchronization of the impulsively coupled complex network,rather than the stabilization design.Typical numerical examples are presented to demonstrate the less conservation of the results.For time delay interconnected systems,the impulsive control is propsoed to stabilize these systems.Using a Lyapunov function related to impulsive time sequence,the convex combination technique and matrix inequality,the delay independent stabilization criteria in terms of linear matrix inequality are presented to guarantee the global exponential stability of these interconnected systems.The control gain of the decentralized impulsive control can be obtained by solving the corresponding linear matrix inequalities.On the other hand,by employing the Lyapunov—Krasovskii functional method,convex combination technique and free weighting matrix method,the sufficient criteria in terms of linear matrix inequality are presented to guarantee the exponential and asymptotical stabilization of the time delay interconnected systems.In the global exponential stability framework,the stabilization criteria obtained here are dependent on the size of time delay whether the decay ration converges to zero or not.The obtained results are closely related to not only the size of the time delay,but also the lower and upper bound of the adjacent impulsive instant difference and the impulsive control input.Typical numerical examples have been given to illustrate that the stabilization criteria proposed here are less conservative than early results based on the continuous control method.