Type Of Asymptotic Stability Research Articles

Strong Structural Controllability of Boolean Networks: Polynomial-Time Criteria, Minimal Node Control, and Distributed Pinning Strategies

In this paper, we initiate the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">strong structural controllability</i> of Boolean networks (BNs), coping with the trouble in identifying full nodal dynamics in practice, and utilize it to build up some existing results about BNs. The derived criteria for the strong structural controllability of BNs are polynomial-time checkable. As an interesting feature, controllability is shown to be equivalent to fixed-time controllability in this regard. We further consider the minimal strong structural controllability problem of BNs which, reduced from the minimal vertex cover problem of graphs, is turned out to be NP-hard. Proceeding forward, strong structural controllability provides useful insights to exclude the deficiencies of existing approaches. First, the network aggregation subject to the controllability of BNs is built by showing that the predecessor aggregated blocks can be dropped in turn if they are strongly structurally controllable. Second, the distributed pinning controllers are carried out to render arbitrary BN controllable through node-to-node message exchange, while traditional results only check the controllability with the preassigned controllers. Notably, the time complexity to design such controllers is only exponential with the maximal in-degree of pinned nodes rather than the node number of BNs. Third, in scenarios of BNs encountering probability disturbances, a basic theorem is put forward to elaborate the equivalence between several types of asymptotic stability. Together with this theorem, the criteria of strong structural controllability also facilitate the design of distributed controllers to asymptotically stabilize probabilistic BNs in probability.

Semiglobal exponential input-to-state stability of sampled-data systems based on approximate discrete-time models

Exact discrete-time models of nonlinear systems are difficult or impossible to obtain, and hence approximate models may be employed for control design. Most existing results provide conditions under which the stability of the approximate model in closed-loop carries over to the stability of the (unknown) exact model but only in a practical sense, i.e. the trajectories of the closed-loop system are ensured to converge to a bounded region whose size can be made as small as desired by limiting the maximum sampling period. In addition, some very stringent conditions exist for the exact model to exhibit exactly the same type of asymptotic stability as the approximate model. In this context, our main contribution consists in providing less stringent conditions by considering semiglobal exponential input-to-state stability (SE-ISS), where the inputs can successfully represent state-measurement and actuation errors. These conditions are based on establishing SE-ISS for an adequate approximate model and are applicable both under uniform and nonuniform sampling. As a second contribution, we show that explicit Runge–Kutta models satisfy our conditions and can hence be employed. An example of control design for stabilization based on approximate discrete-time models is also given.

Asymptotic stability and continuity of nonlinear hybrid stochastic differential equation with randomly occurring delay

This paper concentrates on discussing several types of asymptotic stability and continuity for the nonlinear hybrid stochastic differential equation (NHSDE) with randomly occurring delay. Different from the time delay considered in previous studies, we discuss the randomly occurring time-varying delay characterized by the random variable subject to the Bernoulli distribution. The existence and uniqueness of the maximal local solution and the global solution to the NHSDE are investigated by using some classical inequalities and the stochastic analysis theory. Then, by means of the delay-dependent Lyapunov functional method and the M-matrix technique, some novel sufficient conditions are given to ensure that the trivial solution can achieve the pth moment asymptotic stability, the almost surely asymptotic stability, the pth moment continuity and the continuity in probability. The stability and continuity criteria established in this paper include the upper bound of the time-varying delay and are less conservative. In the end, two numerical examples and corresponding computer simulations are provided to verify the validity of the presented theoretical results.

Likelihood adaptively modified penalties

A new family of penalty functions, ie, adaptive to likelihood, is introduced for model selection in general regression models. It arises naturally through assuming certain types of prior distribution on the regression parameters. To study the stability properties of the penalized maximum‐likelihood estimator, 2 types of asymptotic stability are defined. Theoretical properties, including the parameter estimation consistency, model selection consistency, and asymptotic stability, are established under suitable regularity conditions. An efficient coordinate‐descent algorithm is proposed. Simulation results and real data analysis show that the proposed approach has competitive performance in comparison with the existing methods.

Asymptotic Stability of a Class of Neutral Delay Neuron System in a Critical Case.

In this brief, the asymptotic stability properties of a neutral delay neuron system are studied mainly in a critical case when the exponential stability is not possible. If a critical value of the coefficient in the neutral delay neuron system is considered, then the difficulty for our investigation is caused by the fact that the spectrum of the linear operator is asymptotically approximated to the imaginary axis. It is obvious that, in such a case, the equation is not exponentially stable, and one needs more subtle methods in order to characterize this type of asymptotic stability. Hence, first, the local asymptotic stability for the neutral delay neuron system is studied, and the main tools involved are the asymptotic expansions of characteristic roots, Laplace transforms, and function series, and a complete analysis of the stability diagram is also presented. Then, based on the energy method, the globally asymptotic stability results for the neutral delay neuron system are derived.

Criteria for exponential asymptotic stability in the large of perturbations of linear systems with unbounded delays.

The purpose of this study is to provide necessary and sufficient conditions for exponential asymptotic stability in the large and uniform asymptotic stability of perturbations of linear systems with unbounded delays. A strong relationship is established between the two types of asymptotic stability. It is found that if the exponential estimate of the solution of a system tends to zero as t &#8594 &#8734 the system is said to be uniformly asymptotically stable. But if the solution of a system approaches the origin faster than any exponential function, then the system is said to be exponentially asymptotically stable. Utilizing the exponential estimate of the solution, stability criteria for the linear part of our system of interest is derived. With enough smoothness conditions on the perturbation function, and appeal made to Lyapunov's stability results and some Gronwall-type inequalities the required stability results are established for the linear perturbation. Journal of the Nigerian Association of Mathematical Physics Vol. 10 2006: pp. 529-536

Absolute Componentwise Stability of Interval Hopfield Neural Networks

The componentwise stability is a special type of asymptotic stability which ensures the individual monitoring of each state-space variable of a dynamical system. For an interval Hopfield neural network (IHNN), sufficient conditions are provided to analyze the absolute componentwise stability with respect to a class of activation functions (CAF). Both continuous- and discrete-time dynamics are considered. The conditions are formulated in terms of Hurwitz/Schur stability of a test matrix built from the information about the CAF and the interval matrices defining the IHNN. Some interesting results are derived as particular cases, which allow comparisons with several other works.

Remarks on three types of asymptotic stability

In this note, three types of asymptotic stability (Liapunov stability, Poincaré stability, and Zhukovkij stability) of motion in continuous autonomous dynamical systems are discussed. It is shown that each type of asymptotic stability can be geometrically characterized by omega limit set of an orbit of this type of stability.

Liapunov asymptotic stability and Zhukovskij asymptotic stability

In this paper the Liapunov stability and Zhukovkij stability of motion in continuous dynamical systems are discussed. It is shown that each type of asymptotic stability is typically characterized by omega limit set of an orbit of this type of stability.

Prolongations and Lyapunov functions in control systems

In contrast to the case of a single dynamical system, the asymptotic stability of orbits of control systems cannot be characterized in terms of suitably defined Lyapunov functions. It is shown that the existence of Lyapunov functions corresponds to a stronger type of asymptotic stability, which is defined by introducing higher prolongations.