Problem Of Asymptotic Stability Research Articles

On degenerate circular and shear flows: the point vortex and power law circular flows

We consider the problem of asymptotic stability and linear inviscid damping for perturbations of a point vortex and similar degenerate circular flows. Here, key challenges include the lack of strict monotonicity and the necessity of working in weighted Sobolev spaces whose weights degenerate as the radius tends to zero or infinity. By using a Fourier multiplier approach, we construct energy functionals to deduce stability in a perturbative setting. For sufficiently high spherical harmonics, we can handle any circular flows with power law singularities or zeros as or , while for low frequencies we can treat circular flows close to the Taylor–Couette flow. Similar results apply in the planar shear flow case close to Couette.

Optimal Control of Nonlinear Systems With Time Delays: An Online ADP Perspective

Drawing upon Lyapunov stability theories and online adaptive dynamic programming (ADP) technique, we propose a novel optimal control scheme for the nonlinear time delay system. Our contribution is twofold. First, we investigate the asymptotical stability problem and obtain a generalized stability condition in terms of linear matrix inequalities (LMIs). An explicit, easy-computing delay bound is presented by virtue of Gronwall's inequality. Second, we propose the neural network (NN)-based optimal control strategy by utilizing two approximate NNs. The NN-based optimal control law converges to the real optimal control law since that the estimation errors of NNs weights converge to zero. Numerical examples are presented to illustrate our results.

ABSOLUTE STABILITY CONDITIONS FOR DIFFERENTIAL EQUATIONS WITH DISTRIBUTED DELAY

The problem of asymptotic stability for autonomous functional differential equations is studied on the basis of the investigation of the roots of the characteristic function. We apply D-decomposition method for obtaining the sharp boundaries of stability domains. We obtain necessary and sufficient conditions of asymptotic stability for two families of linear autonomous differential equations with distributed delay and power kernels. These criteria of stability are formulated in terms of the parameters of the original problem. Based on the criteria, we find absolute stability conditions for each of the families.

Robust Impulsive Stabilization of Uncertain Nonlinear Singular Systems with Application to Transportation Systems

We consider the robust asymptotical stabilization problem for uncertain singular systems. We design a new impulsive control technique to ensure that the controlled singular system is robustly asymptotically stable and hence derive the corresponding stability criteria. These sufficient conditions are expressed in the form of algebra matrix inequalities and can be implemented numerically. We finally provide a numerical example of a transportation system to illustrate the effectiveness and usefulness of the proposed criteria.

Razumikhin and Krasovskii methods for asymptotic stability of nonlinear delay impulsive systems on time scales

In this paper, asymptotic stability problems of nonlinear delay impulsive systems on time scales are considered based on the Razumikhin and Krasovskii methods. First, a Krasovskii-type theorem is provided for uniform asymptotic stability and uniform exponential stability of delay impulsive systems on time scales. It is shown that this Krasovskii stability criterion does not require the time derivative of the Lyapunov functional to be necessarily nonpositive on each impulsive interval. Second, asymptotic stability problems of delay impulsive systems on time scales is investigated based on the Razumikhin approach. The advantage of this method is that the length of each impulse interval does not depend on the time delay, which results in the fact that the state trajectory may not decrease instantly and sharply at each impulsive point. Moreover, the idea of this paper provides a unified approach to study continuous system and its discrete counterpart simultaneously. An example is given for illustrating the effectiveness of the proposed results.

The Asymptotic Stability of a Stationary Solution with an Internal Transition Layer to a Reaction–Diffusion Problem with a Discontinuous Reactive Term

The problem of the asymptotic stability of a stationary solution with an internal transition layer of a one-dimensional reaction–diffusion equation is considered. What makes this problem peculiar is that it has a discontinuity (of the first kind) of the reactive term (source) at an internal point of the segment on which the problem is stated, making the solutions have large gradients in the narrow transition layer near the interface. The existence, local uniqueness, and asymptotic stability conditions are obtained for the solution with such an internal transition layer. The proof uses the asymptotic method of differential inequalities. The obtained existence and stability conditions of the solution should be taken into account when constructing adequate models that describe phenomena in media with discontinuous characteristics. One can use the results of this work to develop efficient methods for solving differential equations with discontinuous coefficients numerically.

Stabilization of a class of slow–fast control systems at non-hyperbolic points

In this document, we deal with the local asymptotic stabilization problem of a class of slow–fastsystems (or singularly perturbed Ordinary Differential Equations). The systems studied here have the following properties: (1) they have one fast and an arbitrary number of slow variables, and (2) they have a non-hyperbolic singularity at the origin of arbitrary degeneracy. Our goal is to stabilize such a point. The presence of the aforementioned singularity complicates the analysis and the controller design. In particular, the classical theory of singular perturbations cannot be used. We propose a novel design based on geometric desingularization, which allows the stabilization of a non-hyperbolic point of singularly perturbed control systems. Our results are exemplified on a didactic example and on an electric circuit.

Adaptive asymptotic stabilization of a class of unknown nonlinear systems with specified convergence rate

SummaryThis paper focuses on the asymptotic stabilization problem for a class of multivariable nonlinear systems with relative degree one, practical examples of which incorporate the liquid level control of water tanks, and the speed control of interconnected carts. The presence of the unknown (other than uncertain) additive and multiplicative nonlinearities renders asymptotic stability difficult to be achieved by the existing robust control methods. To conquer this obstacle, a novel adaptive control strategy is proposed. In the control design, the state constraint technique is newly introduced to the adaptive design to generate a proper control gain to compensate for unknown nonlinearities, instead of the use of extra approximating structures. By this means, both the prescribed transient performance regarding the convergence rate and the expected asymptotic stability are preserved. Finally, simulation results are given to illustrate the established theoretical findings.

Stability analysis of neural networks with time-varying delay using a new augmented Lyapunov–Krasovskii functional

This paper examines the problem of asymptotic stability of continuous neural networks with time-varying delay via a new Lyapunov–Krasovskii functional (LKF). First, a suitable quadratic functional is constructed, which coordinates with the use of the orthogonal-polynomials-based integral inequality. Second, the novel proposed LKF contains more state vectors of neural networks, so that more state information can be exploited adequately. By combining the new proposed LKF and orthogonal-polynomials-based integral inequality, novel delay-dependent stability criteria with less conservatism are established in the form of linear matrix inequalities (LMIs). Finally, two commonly-used numerical examples are provided to show the effectiveness and improvement of the proposed criteria.

Improved stability criteria for linear time-varying systems on time scales

ABSTRACT In this paper, asymptotic stability problems of linear time-varying (LTV) systems on time scales are considered based on a less conservative Lyapunov inequality, whose right side is not required to be necessarily negative. It is shown that the Lyapunov inequality covers not only the corresponding trivial (continuous and discrete) ones but also nontrivial ones. Based on this inequality, some necessary and sufficient conditions for asymptotic stability, exponential stability, uniformly exponential stability of LTV systems on time scales are obtained. An example about nontrivial systems is given for illustrating the effectiveness of the proposed results.

Stability of continuous-time positive switched linear systems: A weak common copositive Lyapunov functions approach

In this paper, we study the problem of asymptotic stability of continuous-time positive switched linear systems under both arbitrary and restricted switchings. It is well-known that asymptotic stability under arbitrary switching can be implied by several classes of strong common copositive Lyapunov functions (CLFs), i.e., functions whose derivative along the nontrivial system trajectories is negative. However, asymptotically stable positive switched systems may not admit strong common CLFs. The main contribution of this paper is to study the stability problem by requiring only weak common CLFs. Firstly, necessary and sufficient conditions are established for asymptotic stability under arbitrary switching. Among them, an easily verifiable graphical stability criterion, based on the connectivity of the digraphs associated with the subsystem matrices, is proposed. Secondly, we further relax the obtained graphical condition to derive a relaxed weak excitation condition for asymptotic stability under dwell-time switching. Finally, two examples are provided to illustrate the effectiveness of our theoretical results.

Algebraic Approach to the Stabilization of a Differential System of Retarded Type

For a spectrally controllable linear autonomous systems with commensurable delays, we construct state feedbacks ensuring the complete damping of the original system (finite stabilization) as well as the complete damping of the original system and the asymptotic stability of the closed-loop system (complete stabilization). The spectral reduction and asymptotic stabilization problems are considered as auxiliary problems. The argument is constructive, and the results are illustrated by an example.

Global asymptotic stability of stochastic reaction-diffusion recurrent neural networks with Markovian jumping parameters and mixed delays

In this paper, the problem of global asymptotic stability of stochastic Markovian jumping reaction-diffusion neural networks with discrete and distributed delays is investigated. By utilizing Lyapunov–Krasovskii functional method combined with linear matrix inequality approach, novel sufficient stability conditions for delayed stochastic reaction-diffusion recurrent neural networks with Markovian jumping parameters and mixed delays are derived. Finally, numerical examples with simulation results are given to illustrate the derived theoretical results.

TAKAGİ-SUGENO BULANIK COHEN-GROSSBERG TİPİ ZAMAN GECİKMELİ YAPAY SİNİR AĞLARINDA KARARLILIK ANALİZİ

This paper deals with the problem of the global asymptotic stability of the class of Takagi-Sugeno Fuzzy Cohen-Grossberg neural networks with multiple time delays. By constructing a suitable fuzzy Lyapunov functional, we present a new delay-independent sufficient condition for the global asymptotic stability of the equilibrium point for delayed Takagi-Sugeno Fuzzy Cohen-Grossberg neural networks with respect to the Lipschitz activation functions. The obtained condition simply relies on the network parameters of the neural system. Therefore, the equilibrium and stability properties of the neural network model considered in this paper can be easily verified by exploiting some basic properties of some certain classes of matrices.

Stabilization of Underactuated Surface Vessels: A Continuous Fractional Power Control Method

This paper investigates the problem of global asymptotic stabilization of underactuated surface vessels (USVs) whose dynamics features off-diagonal inertia and damping matrices. By using input and state transformations, the dynamic model of USV is converted into an equivalent system consisting of two cascade connected subsystems. For the transformed system, a continuous fractional power control framework is given to achieve global asymptotic stabilization of USVs. Then, the convergence under this framework is analyzed showing that the rate can be improved by adjusting the fractional power term. Finally, a continuous control algorithm is proposed to guarantee the global convergence rate of the USV system. Simulations are given to demonstrate the effectiveness of the presented method.

Lateral Control of an Autonomous Vehicle

The asymptotic stabilization problem for a class of nonlinear under-actuated systems is studied and solved. Its solution, together with the back-stepping and the forwarding control design methods, is exploited in the control of the nonlinear lateral dynamics of a vehicle. Even though the theoretical studies of the lateral control of autonomous vehicles are traditionally applied to lane keeping cases, the results can be applied to broader range of areas, such as lane changing cases. The comparison between the performances of the closed-loop systems with the given controller and a typical human driver is given and demonstrates the speediness and the effectiveness of the feedback controller.

On the Asymptotic Stability Problem for Solutions of Difference Switched Systems

We consider difference systems obtained by discretizing certain classes of differential systems. It is assumed that the system under consideration can operate in several modes. The problem is to establish conditions that guarantee the asymptotic stability of a given equilibrium position when switching regimes. We use the method of Lyapunov functions. We study the case when solutions of the system under various operating modes can have features of both linear and nonlinear behavior.

Global solutions to physical vacuum problem of non-isentropic viscous gaseous stars and nonlinear asymptotic stability of stationary solutions

This paper is concerned with spherically symmetric motions of non-isentropic viscous gaseous stars with self-gravitation. When the stationary entropy S‾(x) is spherically symmetric and satisfies a suitable smallness condition, the existence and properties of the stationary solutions are obtained for 65<γ<2 with weaker constraints upon S‾(x) compared with the one in [26], where γ is the adiabatic exponent. The global existence of strong solutions capturing the physical vacuum singularity that the sound speed is C12-Hölder continuous across the vacuum boundary to a simplified system for non-isentropic viscous flow with self-gravitation and the nonlinear asymptotic stability of the stationary solution are proved when 43<γ<2 with the detailed convergence rates, motivated by the results and analysis of the nonlinear asymptotic stability of Lane–Emden solutions for isentropic flows in [29,30].

Delay-independent stability conditions for a class of nonlinear difference systems

The paper addresses the asymptotic stability problem for a class of difference systems with nonlinearities of a sector type and time-delay. A new approach to Lyapunov–Krasovskii functionals constructing for considered systems is proposed. On the basis of the approach, delay-independent asymptotic stability conditions and estimates of the convergence rate of solutions are derived. In addition, stability of perturbed systems is investigated in the case where nonstationary perturbations admit zero mean values. Some examples are given to illustrate the obtained results.

An Analysis of Global Stability of Takagi–Sugeno Fuzzy Cohen–Grossberg Neural Networks with Time Delays

This paper deals with the problem of the global asymptotic stability of Takagi–Sugeno (T–S) fuzzy Cohen–Grossberg neural networks with multiple time delays. By using Lyapunov method and some basic properties of matrices, and employing the nondecreasing and slope-bounded activation functions, an easily verifiable sufficient criterion is derived to establish the asymptotic stability of a general class of (T–S) fuzzy Cohen–Grossberg neural networks with multiple time delays. The obtained stability condition establishes some relationships between the network parameters of the neural network model independently of the delay parameters. A numerical example is also given to illustrate the effectiveness of the theoretical results.