Studies Of Stability And Synchronization On Some Neural Networks Systems With Distributed Parameter

Neural networks have obtained their successful applications in many areas such aspattern recognition，image processing，control problem and secure communication etc.The problems of dynamical behaviors of neural networks with distributed parametersand time delays has long received extensive attention from researchers working insystems and intelligent control communist. However, there are few results, or even noresults concerning the dynamical behaviors issues for neural networks withreaction–diffusion or/and mixed time-varying delays or/and stochastic perturbations.Based on Lyapunov functional theory，free-weighting matrx，Green formula，L-operaterinequality and stochastic analysis etc., dynamical behaviors of neural networks withdistributed parameter are systematically and deeply investigated. The maincontributions of this dissertation are listed as follows:1. By constructing a more general type of Lyapunov–Krasovskii functionalcombined with free-weighting matrix approach and analysis techniques,delay-dependent exponential stability criteria are derived in the form of linear matrixinequalities. The obtained results are dependent on the size of the time-varyingdelays and the measure of the space, which are usually less conservative thandelay-independent and space-independent ones.2. The problem of dynamics analysis is proposed for a class of novel stochasticMarkovian jump reaction–diffusion neural networks with partial information ontransition probability and mixed time delays. The new criterion for the asymptoticalstability of the equilibrium point in the mean square sense is obtained based on linearmatrix inequality forms. An improved Lyapunov–Krasovskii functional andfree-connection weighting matrices are introduced to derive the condition. The obtainedresults are dependent on delays and the measure of the space.3. The dynamical behaviors of impulsive stochastic reaction–diffusion neuralnetworks (ISRDNNs) with mixed time delays are discussed. By combining awell-known L-operator differential inequality with mixed time delays and theLyapunov-Krasovkii functional approach, as well as linear matrix inequality technique,some novel sufficient conditions are derived to ensure the existence, uniqueness andglobal exponential stability of the periodic solutions and p moment global exponentialstability for ISRDNNs with mixed time delays in the mean square sense and p momentglobal exponential stability of ISRDCGNNs, respectively. The results are new andimprove some of the previously known results. The proposed model is quite general since many factors such as noise perturbations, impulsive phenomena and mixed timedelays are considered. Numerical examples are provided to verify the usefulness of theobtained results.4. Almost sure input-to-state stability definition is firstly proposed for the delayedreaction–diffusion neural networks with Markovian jump parameters and Dirichletboundary conditions. By constructing new Lyapunov functional and utilizing someinequality techniques, sufficient conditions ensuring the almost sure input-to-statestability are given. The criteria can also ensure almost sure global exponential stabilitywhen the input is equal to zero. Numerical example is given to demonstrate theeffectiveness of the proposed approach and derived results.5. Robust exponential stability and stabilization conditions are presented foruncertain linear distributed parameter delayed systems. Based on extension theLyapunov-Krasovskii method to a Hilbert space and linear matrix inequality technique,robust exponential stability and stabilization criteria are given, respectively. The criteriaare dependent on time delay. Being applied to a parabolic equation, exponential stabilitycriterion for parabolic equation is reduced to standard linear matrix inequalities.6. A delay-differential equation modelling a BAM neural networks withreaction-diffusion terms is investigated. A feedback control law is derived to achievethe state global exponential synchronization of drive and response BAM NNs withreaction-diffusion terms by constructing a suitable Lyapunov functional, using thedrive-response approach and some inequality technique. A novel global exponentialsynchronization criterion is given in terms of two identical inequalities, which can bechecked easily.7. The adaptive synchronization problems for delayed neural networks withreaction-diffusion terms are studied. At first, an approach combining Lyapunovfunctional theory with stochastic analysis and the adaptive feedback control technique istaken to investigate this problem. Some novel sufficient conditions in terms of linearmatrix inequalities are obtained to ensure the asymptotic synchronization for theproposed drive and response neural networks. Then, based on the LaSalle invariantprinciple of functional differential equations, a sufficient condition for the adaptivesynchronization of such a system is obtained. Numerical example is given to show the feasibility and effectiveness of the proposed scheme and derived results.