Dynamical Behaviors Of Complex-valued Neural Networks And Its Applications

Complex-valued neural networks are the systems that deal with complex-valued informa-tion by resorting to the complex-valued parameters and variables.Therefore,when involving in complex-valued information such as the complex signals in practice,the complex-valued neural networks can work well.In the past decades,a large amount of works concerning on the complex-valued neural networks have been done,and it has been applied to various areas including pattern recognition,signal processing,engineering optimization,image processing,optoelectronics,computer vision and speech synthesis and so on.The facts show that such applications rely heavily on the dynamical behaviors of the complex-valued networks.As an extension of the real-valued neural networks,the states,connection weights and activation functions of the complex-valued neural networks are all complex-valued,hence there exist a lot of differences between them.Generally speaking,the complex-valued neural networks have much more complicated and better properties than the real-valued ones,which makes it pos-sible to solve some problems such as XOR problem and the detection of symmetry problem which cannot be solved by the real-valued networks.Based on such consideration,it is very important and necessary to investigate the dynamical behaviors of complex-valued neural net-works and apply them to the practical problems.In this thesis,the dynamical behaviors of the complex-valued neural networks are focused,including the monostability(?-stability,asymp-totic stability and exponential stability),multistability and state estimation.Moreover,the associative memory is analyzed and designed based on the complex-valued neural networks.The main contents of this dissertation can be summarised as follows:1.Monostability is investigated for several classes of complex-valued recurrent neural networks,including global ?-stability,global exponential stability and global asymptotic sta-bility.Firstly7a class of complex-valued recurrent neural networks with leakage delay and unbounded time-varying delay is considered,several sufficient delay-dependent conditions are established to ensure the global ?-stability of the addressed networks by constructing appropri-ate Lyapunov-Krasovskii functionals and utilizing the free weighting matrix method.Secondly,global exponential stability is analyzed for a class of complex-valued recurrent neural networks with time-varying delays,based on the matrix measure method and Halanay inequality with-out constructing any Lyapunov functions,some sufficient conditions are obtained to guarantee the global exponential stability of the addressed networks under different activation functions.Finally,by utilizing the nonlinear measure method and matrix inequality techniques,glob-al asymptotic stability is taken into account for a class of complex-valued neural networks with constant delays,in addition,the problem concerning on the robust stability of uncer-tain complex-valued neural networks with parameter uncertainties is also solved,and several sufficient conditions are derived to ascertain the existence,uniqueness and global stability of the equilibrium points for the addressed networks by constructing appropriate Lyapunov functional candidate.It should be pointed out that when investigating the monostability of the complex-valued neural networks,it is no longer required that the activation functions are derivable.Moreover,the results concerning on the monostability of complex-valued networks are given by utilizing different methods which can be solved easily by utilizing the Matlab toolbox.2.The multistability problem is discussed for several classes of delayed complex-valued neural networks.On the one hand,based on the geometrical properties of the activation func-tions and the fixed point theory,the multistability problem is addressed for the complex-valued neural networks with appropriate real-imaginary-type activation functions and distributed de-lays,several sufficient conditions are obtained which not only guarantee the existence of 9n equilibrium points but also assure the local exponential stability for the 4n equilibrium points among them,in addition,the attraction basins of the 4n equilibrium points are also estimat-ed and enlarged under some mild restrictions.On the other hand,based on the geometrical properties of the discontinuous activation functions and the Brouwer's fixed point theory,the multistability problem is considered for a class of complex-valued neural networks with discon-tinuous activation functions and time-varying delays.Firstly,the Filippov solution of delayed systems with discontinuous activation functions is defined.Furthermore,through rigorous analysis,several sufficient criteria are obtained to guarantee the existence of 25n equilibrium points(among them,there are 9n locally stable points and 16n-9n unstable points).Finally,the attraction basins of equilibrium points which are locally stable are estimated and enlarged.It is worth pointing out that compared with the existed results,in this paper not only the local stability and unstability of equilibrium points are analyzed for the complex-valued neu-ral networks,but also the attraction basins of the equilibrium points which are locally stable are estimated and enlarged.Moreover,it is the first time to consider the multistability prob-lem for the complex-valued networks under the assumption that the activation functions are discontinuous.3.State estimation problem of the deterministic/uncertain delayed complex-valued neu-ral networks is considered.Firstly,a class of delayed complex-valued neural networks with parameter uncertainties is studied,where the parameter uncertainties are assumed to be norm-bounded.Through available output measurements containing nonlinear Lipschitz-like terms,some sufficient delay-dependent criteria are derived to guarantee the existence of the desired estimator gains based on Lyapunov functional method and matrix inequality technique,and the designed estimator can make sure that,for all admissible parameter uncertainties and time delay,the error-state system is globally asymptotically stable.Secondly,by resorting to the sampled-data information from the available output measurements,the state estima-tion problem is analyzed for the complex-valued neural networks with parameter uncertainties,mixed delays and stochastic disturbances,where the parameter uncertainties are assumed to be norm-bounded and the stochastic disturbances are assumed to be Brownian motions.A state estimator is designed for the complex-valued model such that,for all admissible parameter uncertainties and sampled output measurements,the dynamics of the state estimation error system is guaranteed to be globally asymptotically stable in the mean square.It should be pointed out that compared with the existed references,this thesis not only considers the state estimation problem of the deterministic complex-valued neural networks but also takes into account the state estimation problem of the complex-valued networks with parameter uncer-tainties and stochastic disturbances,which can reflect much more realistic dynamical behaviors of the complex-valued networks.In addition,the sampled-data information from the available output measurements rather than the information from the available output measurements without sampling is utilized to design the state estimator for the complex-valued networks,which will cost less time and improve the performance of system.4.A novel method is proposed to design the associative memory algorithm for the complex-valued neural networks with asynchronous and unbounded delays,and several suf-ficient criteria are derived to guarantee the existence,uniqueness and global exponential sta-bility of the equilibrium points for the delayed complex-valued neural networks.The criteria are in terms of several inequalities,hence it would bring more robustness for the parameters of the model.It is worth pointing out that the desired store patterns are retrieved by feeding proper probes through external inputs rather than the initial conditions,which can avoid spu-rious memory patterns.Moreover,the derived results are much more general than the existed works,and numerical examples demonstrate the effectiveness of the obtained results.