Combining Self-Organizing Map and Lipschitz Condition for Estimation in Direction of Arrival

There are many DOA estimation methods based on different signal features, and these methods are often evaluated by experimental results, but lack the necessary theoretical basis. Therefore, a direction of arrival (DOA) estimation system based on self-organizing map (SOM) and designed for arbitrarily distributed sensor array is proposed. The essential principle of this method is that the map from distance difference of arrival (DDOA) to DOA is Lipschitz continuity, it indicates the similar topology between them, and thus Kohonen SOM is a suitable network to classify DOA through DDOA. The simulation results show that the DOA estimation errors are less than 1° for most signals between 0° to 180°. Compared to MUSIC, Root-MUSIC, ESPRIT, and RBF, the errors of signals under signal-to-noise ratios (SNR) declines from 20 dB to 2 dB are robust, SOM is better than RBF and almost close to MUSIC. Further, the network can be trained in advance, which makes it possible to be implemented in real-time.

Existence and Uniqueness of the Nonlinear BSDEs with a Small Parameter under Locally Lipschitz Condition

In this paper we study the following nonlinear BSDE：y（t） ＋ ∫1 t f（s,y（s）,z（s））ds ＋ ∫1 t [z（s） ＋ g 1 （s,y（s）） ＋ εg 2 （s,y（s）,z（s））]dW s=ξ,t ∈ [0,1],where ε is a small parameter.The coefficient f is locally Lipschitz in y and z,the coefficient g 1 is locally Lipschitz in y,and the coefficient g 2 is uniformly Lipschitz in y and z.Let L N be the locally Lipschitz constant of the coefficients on the ball B（0,N） of R d × R d×r.We prove the existence and uniqueness of the solution when L N ～ √ log N and the parameter ε is small.

THE CONVERGENCE OF TRUNCATED EULER-MARUYAMA METHOD FOR STOCHASTIC DIFFERENTIAL EQUATIONS WITH PIECEWISE CONTINUOUS ARGUMENTS UNDER GENERALIZED ONE-SIDED LIPSCHITZ CONDITION

In this paper,we consider the stochastic differential equations with piecewise continuous arguments(SDEPCAs)in which the drift coefficient satisfies the generalized one-sided Lipschitz condition and the diffusion coefficient satisfies the linear growth condition.Since the delay term t-[t]of SDEPCAs is not continuous and differentiable,the variable substitution method is not suitable.To overcome this dificulty,we adopt new techniques to prove the boundedness of the exact solution and the numerical solution.It is proved that the truncated Euler-Maruyama method is strongly convergent to SDEPCAs in the sense of L'(q≥2).We obtain the convergence order with some additional conditions.An example is presented to illustrate the analytical theory.

BSDEs with stochastic Lipschitz condition:A general result

We prove a general existence and uniqueness result of solutions for a backward stochastic differential equation(BSDE)with a stochastic Lipschitz condition.We also establish a continuous dependence property and a comparison theorem for solutions to this type of BSDEs,thus strengthening existing results.

Variational Approach for the Adapted Solution of Backw ard Stochastic Differential Equations with Locally Lipschitz Diffusion Coefficients

One existence integral condition was obtained for the adapted solution of the general backward stochastic differential equations(BSDEs). Then by solving the integral constraint condition, and using a limit procedure, a new approach method is proposed and the existence of the solution was proved for the BSDEs if the diffusion coefficients satisfy the locally Lipschitz condition. In the special case the solution was a Brownian bridge. The uniqueness is also considered in the meaning of "F0-integrable equivalent class" . The new approach method would give us an efficient way to control the main object instead of the "noise".

LIPSCHITZ TYPE CHARACTERIZATIONS FOR BERGMAN-ORLICZ SPACES AND THEIR APPLICATIONS

We give characterizations for Bergman-Orlicz spaces with standard weights via a Lipschitz type condition in the Euclidean, hyperbolic, and pseudo-hyperbolic metrics. As an application, we obtain the boundeness of the symmetric lifting operator from Bergman-Orlicz spaces on the unit disk into Bergman-Orlicz spaces on the bidisk.

Kantorovich’s theorem for Newton’s method on Lie groups

The convergence criterion of Newton’s method to find the zeros of a map f from a Lie group to its corresponding Lie algebra is established under the assumption that f satisfies the classical Lipschitz condition, and that the radius of convergence ball is also obtained. Furthermore, the radii of the uniqueness balls of the zeros of f are estimated. Owren and Welfert (2000) stated that if the initial point is close sufficiently to a zero of f, then Newton’s method on Lie group converges to the zero; while this paper provides a Kantorovich’s criterion for the convergence of Newton’s method, not requiring the existence of a zero as a priori.

A Criterion for Stability of Synchronization and Application to Coupled Chua's Systems

We investigate synchronization in an array network of nearest-neighbor coupled chaotic oscillators. By using of the Lyapunov stability theory and matrix theory, a criterion for stability of complete synchronization is deduced. Meanwhile, an estimate of the critical coupling strength is obtained to ensure achieving chaos synchronization. As an example application, a model of coupled Chua＇s circuits with linearly bidirectional coupling is studied to verify the validity of the criterion.

ON APPROXIMATION PROPERTIES OF NON-CONVOLUTION TYPE NONLINEAR INTEGRAL OPERATORS——Dedicated to Professor Paul L. Butzer on his 80^(th) Birthday

In the present paper we state some approximation theorems concerning point- wise convergence and its rate for a class of non-convolution type nonlinear integral opera- tors of the form：Tλ（f;x）=B∫AKλ（t,x,f（t））dr,x∈〈a,b〉λλA.In particular, we obtain the pointwise convergence and its rate at some characteristic points x0 off as （x,λ） → （x0, λ0） in LI 〈A,B 〉, where 〈 a,b 〉 and 〈A,B 〉 are is an arbitrary intervals in R, A is a non-empty set of indices with a topology and X0 an accumulation point of A in this topology. The results of the present paper generalize several ones obtained previously in the papers [191-[23]

RELATIONS BETWEEN TWO SETS OF FUNCTIONS DEFINEDBY THE TWO INTERRELATED ONE-SIDE LIPSCHITZCONDITIONS

In the theoretical study of numerical solution of stiff ODEs, it usually assumes that the righthand function f(y) satisfy one-side Lipschitz condition < f(y) - f(z),y - z >less than or equal to v(1)parallel to y - z parallel to(2),f : Omega subset of or equal to C-m --> C-m, or another related one-side Lipschitz condition [F(Y) - F(Z), Y - Z](D) less than or equal to v'parallel to Y - Z parallel to(D)(2), F : Omega(s) subset of or equal to C-ms --> C-ms, this paper demonstrates that the difference of the two sets of all functions satisfying the above two conditions respectively is at most that v' - v' only is constant independent of stiffness of function f.

A Periodic Solution of Cellular Neural Networks with Delay

It is shown that there is an unique ω-period solution x（t, φ^＊） for a delayed cellular network and its each solution x（t, φ） converges exponentially to x（t, φ^＊） if its each output function is bounded and satisfies Lipschitz condition when all input signals are at-periodic functions.

SMALLEST g-SUPERSOLUTION WITH CONSTRAINT

In this paper the existence and uniqueness of the smallest g-supersolution for BSDE is discussed in the case without Lipschitz condition imposing on both constraint function and drift coefficient in the different method from the one with Lipschitz condition.Then by considering (ξ,g) as a parameter of BSDE,and ( ξ α,g α) as a class of parameters for BSDE,where α belongs to a set A,for every α ∈A there exists a pair of solution { Y α,Z α } for the BSDE,the properties of sup α ∈A{ Y α } which is also a solution for some BSDE is studied.This result may be used to discuss optimal problems with recursive utility.

Subgradient Extragradient Methods for Equilibrium Problems and Fixed Point Problems in Hilbert Space

Inspired by inertial methods and extragradient algorithms,two algorithms were proposed to investigate fixed point problem of quasinonexpansive mapping and pseudomonotone equilibrium problem in this study.In order to enhance the speed of the convergence and reduce computational cost,the algorithms used a new step size and a cutting hyperplane.The first algorithm was proved to be weak convergence,while the second algorithm used a modified version of Halpern iteration to obtain strong convergence.Finally,numerical experiments on several specific problems and comparisons with other algorithms verified the superiority of the proposed algorithms.

On the Lipschitz Regularity in Signal Singularity Detection

The concepts of Lipschitz regularity and Hoelder one are reviewed. They arenot equivalent except for a < 1. A modification for Definition 1 on Lipschitz regularity in Ref.,which is not rigorous, is offered. Two propositions on Hoelder regularity are given and proven.

Global Practical Exponential Stabilization for One-Sided Lipschitz Systems with Time Delay

This paper addresses the practical stabilization problem for a class of one-sided Lipschitz nonlinear time delay systems with external disturbances.In case there is no perturbation,the exponential convergence of the observer was confirmed.When external disturbances appear in the system,a separation principle is established,and the authors show that the closed loop system is exponentially practical stable.By choosing a suitable Lyapunov-Krasovskii functional,the authors derive new sufficient conditions to guarantee the exponential stability of the systems.Finally,a physical model is performed to prove the efficiency and applicability of the suggested approach.

Convergence of Newton's Method and Uniqueness of the Solution of Equations in Banach SpacesⅡ

Some results on convergence of Newton's method in Banach spaces are established under the assumption that the derivative of the operators satisfies the radius or center Lipschitz condition with a weak L average.

RANDOM DIRICHLET TYPEFUNCTIONS ON THE UNIT BALL OF C^n

The random Dirichlet type functions on the unit ball of Cn are studied. Sufficient conditions of the multipliers of Dμ for 0 < μ≤1, if n = 1 or 0 < μ < 2 if n > 1 are given. The smoothness of random Dirichlet type functions is discussed.

Unknown Inputs Observer for a Class of Nonlinear Uncertain Systems: An LMI Approach

This paper deals with the simultaneous estimation of states and unknown inputs for a class of Lipschitz nonlinear systems using only the measured outputs. The system is assumed to have bounded uncertainties that appear on both the state and output matrices. The observer design problem is formulated as a set of linear constraints which can be easily solved using linear matrix inequalities (LMI) technique. An application based on manipulator arm actuated by a direct current (DC) motor is presented to evaluate the performance of the proposed observer. The observer is applied to estimate both state and faults.

Uncoupled PID Control of Coupled Multi-Agent Nonlinear Uncertain Systems

In this paper, PID（proportional-integral-derivative） controllers will be designed to solve the tracking problem for a class of coupled multi-agent systems, where each agent is described by a second-order high-dimensional nonlinear uncertain dynamical system, which only has access to its own tracking error information and does not need to communicate with others. This paper will show that a 3-dimensional manifold can be constructed based on the information about the Lipschitz constants of the system nonlinear dynamics, such that whenever the three parameters of each PID controller are chosen from the manifold, the whole multi-agent system can be stabilized globally and the tracking error of each agent approaches to zero asymptotically. For a class of coupled first-order multi-agent nonlinear uncertain systems, a PI controller will be designed to stabilize the whole system.

Strong Convergence Analysis of Split-Step q-Scheme for Nonlinear Stochastic Differential Equations with Jumps

In this paper,we investigate the mean-square convergence of the split-step q-scheme for nonlinear stochastic differential equations with jumps.Under some standard assumptions,we rigorously prove that the strong rate of convergence of the splitstep q-scheme in strong sense is one half.Some numerical experiments are carried out to assert our theoretical result.