In this survey we recall the results obtained in [16] where we gave a representation theorem for the solutions of stochastic di¤erential equations in Hilbert spaces. Using this representation theorem and the deterministic characterizations of exponential stability and uniform observability obtained in [16], [17], we will prove a result of Datko type concerning the exponential dichotomy of stoch...

The method of reconstruction for an n-dimensional system from observations is to form vectors of m consecutive observations, which for m > 2n, is generically an embedding. This is Takens’ result. Our analytical examples show that it is possible to obtain spurious Lyapunov exponents that are even larger than the largest Lyapunov exponent of the original system. Therefore, we present examples whe...

84 Abstract—In this paper, we analyze the convergence and stability properties of Hopfield Neural Networks (HNN). The global convergence and asymptotic stability of HNN have successful various applications in computing and optimization problems. After determining the mathematical model of the network, we do some analysis on the model. This analysis base on Lyapunov Stability Theorem. Firstly, w...

We prove the convexity and compactness of the closure of the lower partition range of an Rn-valued, nonatomic, supermodular capacity, employing a useful relationship between convex games and their Choquet integrals. The main result is applied to fair division problems, and the existence of Pareto optimal α-fair partitions is demonstrated for the case of nonadditive measures.

This article presents a small-gain theorem for networks composed of countably infinite number finite-dimensional subsystems. Assuming that each subsystem is exponentially input-to-state stable, we show if the gain operator, collecting all information about internal Lyapunov gains, has spectral radius less than one, overall network stable. The effectiveness our result illustrated through several...

This paper proposes new criteria for stability and performance of nonlinear interconnected systems by smoothly generalizing a recently-developed method of the state-dependent scaling. The criteria are directly related to Lyapunov functions so that they show how to construct Lyapunov functions which establish desired stability and performance explicitly. A set of popular Lyapunov stability crite...