Stability Analysis And Synchronization Control Of Neural Networks With Lévy Noise

Artifcial neural network is a subject which studies the information pro-cessing through the imitation of behavior characteristics of biological neural net-works. It performs the distributed and parallel information processing dependingon the construction of a structure like brain synapse connections and the adjust-ment of the relationship between the mass internal node interconnection. Recentyears, artifcial neural networks have been successfully applied into the feldssuch as pattern recognition, signal processing, expert system, combinatorial op-timization, robot control, which has meanwhile improved the development of thetheoretical research including the dynamics analysis of neural networks.In previous studies, Brownian motion is conventionally adopted to describethe disturbances arise in neural networks. Problems of stability and synchro-nization control are usually discussed according to the stochastic model withGaussian white noise. However, Brownian motion is not suitable to representphenomena of impulse or jump emerging in networks. L′evy process is a betterchoice for stochastic modeling when diverse disturbances appear in networks.In this dissertation, several models of neural networks with L′evy noise arepresented and approaches such as stochastic analysis, infnitesimal operator, M-matrix and LMI are used to tackle the problem of almost surely exponentialstability, moment exponential or asymptotic stability and synchronization prob-lem via data sampling control and adaptive control.The overall innovation of this dissertation lies in two aspect. Firstly, Gaus-sian white noise is replaced in the studied models with L′evy noise, which makesneural networks have a expansion of noise types and adapt to the diversity char-acteristics of noise. Thus the neural networks models become more realisticanyhow. Secondly, techniques of stability research are expanded and extendedfrom difusion systems to jump-difusion systems.The main contents of this dissertation are as follows. (1) A class of model is presented for Markovian switching neural networkswith L′evy noise. The analysis and interpretation is given for the generation prin-ciple of L′evy noise in neural networks and the superiority of L′evy noise models.By utilizing the generalized Ito’s formula, strong law of large numbers for martin-gales and ergodicity of Markov chain, the problem of almost surely exponentialstability concerning aforementioned model is investigated and stability criteriaare then established. These criteria depend only on the stationary distributionof the Markov chain and some constants. The research method of this problemrealizes the expansion and extension from the stochastic analysis of Ito type tothat of L′evy type. The results also generalize the original conclusion in stochasticneural networks of Ito type.(2) Via the approach of infnitesimal operator, the result of exponential sta-bility in pth moment is generalized from stochastic hybrid systems of Ito type tothat of L′evy type. On the basis of this result, the pth (p≥2) moment exponen-tial stability criteria are derived for stochastic hybrid systems of L′evy type byusing M-matrix approach. Moreover, delayed Markovian switching neural net-works with L′evy noise are treated as a special case of hybrid systems mentionedabove and the mean square exponential stability criterion is then established forthis kind of neural networks by using M-matrix approach again. This criterionis fully operational and easy to test. It is verifed by simulation that the state ofneural networks tends to zero in mean square at a convergence rate of exponentialorder.(3) By the use of infnitesimal operator, the result of pth moment asymp-totic stability is generalized from stochastic hybrid systems of Ito type to that ofL′evy type. Based on this result, a conclusion concerning pth moment exponen-tial stability is derived for hybrid systems driven by L′evy noise, which is moregeneral than the above one obtained before. Afterwards the pth (p≥2) mo-ment asymptotic stability criterion is proposed for this kind of system throughM-matrix approach. Furthermore, according to this criterion, using M-matrixapproach again yields the mean square asymptotic stability criterion for delayedneural network with L′evy noise and Markovian switching which is a special caseof hybrid system driven by L′evy noise. A numerical simulation is done fnally to verify the mean square asymptotic stability and to evaluate the convergence rateof this type of neural networks.(4) The control strategy composed of data sampling and state feedback isadopted to discuss the problem of mean square synchronization for a kind ofdelayed neural networks with L′evy noise and Markovian switching. By the com-bination of infnitesimal operator and LMI, a sufcient condition is established toguarantee the synchronization of master system and slave system. This conditionis partly less conservatism because it depends on the time delay and Markovianswitching mode. The upper bound of sampling interval length can be solved fromthose LMIs as well. A simulation shows the synchronization behavior of mastersystem and slave system controlled by the design of sampling data.(5) An adaptive control is designed to investigate the mean square syn-chronization problem for a kind of delayed neural networks with L′evy noise andMarkovian switching. By the approach of infnitesimal operator and LMI, a suf-cient condition is derived to ensure the synchronization of neural networks. Thiscondition can weaken the conservatism because it is both mode and delay de-pendent. The synchronization control law can be solved from the LMIs and thecomputational complexity analysis is given for LMI. It is shown that our LMIshave a lower computational complexity than others ever before although a morecomplex model is considered here. A simulation result verifes that the slave sys-tem under adaptive control remains synchronized with the master system evenneural networks have chaotic behaviors.