Exponential Stability In Terms Research Articles

Lyapunov Type Theorems for Exponential Stability of Linear Skew-Product Three-Parameter Semiflows with Discrete Time

For linear skew-product three-parameter semiflows with discrete time acting on an arbitrary Hilbert space, we obtain a complete characterization of exponential stability in terms of the existence of appropriate Lyapunov functions. As a nontrivial application of our work, we prove that the notion of an exponential stability persists under sufficiently small linear perturbations.

An LMI Approach to Stability Analysis of Coupled Parabolic Systems

We analyze the exponential stability of a class of distributed parameter systems. The system we consider is described by a coupled parabolic partial differential equation with spatially varying coefficients. We approximate the coefficients by splitting space domains but take into account approximation errors during stability analysis. Using a quadratic Lyapunov function, we obtain sufficient conditions for exponential stability in terms of linear matrix inequalities.

Observer-based exponential stabilization for time-delay systems via augmented weighted integral inequality

In this paper, we design observer-based feedback control for a class of linear systems. The novelty of the paper comes from the consideration of an augmented weighted based integral inequality involving quadratic functions with an exponential term which is less conservative than the celebrated weighted integral inequality employed in the context of time-delay systems. By using appropriately chosen Lyapunov–Krasovskii functional (LKF), together with the derived integral inequality, a new sufficient condition for exponential stability in terms of linear matrix inequalities (LMIs) is proposed for the delayed linear systems with state feedback control. Finally, the applicability and superiority of the proposed theoretical results over the existing ones are analyzed in virtue of numerical examples.

Admissible Banach function spaces for linear dynamics with nonuniform behavior on the half-line

For nonuniform exponentially bounded evolution families on the half-line we introduce a class of Banach function spaces on which we define nonuniform evolution semigroups. We completely characterize nonuniform exponential stability in terms of invertibility of the corresponding generators. We emphasize that in particular our results apply to all linear differential equations with bounded operator and finite Lyapunov exponent.

A complete characterization of exponential stability for discrete dynamics

For a discrete dynamics defined by a sequence of bounded and not necessarily invertible linear operators, we give a complete characterization of exponential stability in terms of invertibility of a certain operator acting on suitable Banach sequence spaces. We connect the invertibility of this operator to the existence of a particular type of admissible exponents. For the bounded orbits, exponential stability results from a spectral property. Some adequate examples are presented to emphasize some significant qualitative differences between uniform and nonuniform behaviour.

Stability analysis of 2D Roesser systems via vector Lyapunov functions * *This work is supported in part by Russian Foundation for Basic Research under grants 16-08-00916_a, 16-38-00192_mol_a and in part by National Science Centre in Poland under grant 2015/17/B/ST7/03703.

Abstract The paper gives new results that contribute to the development of a stability theory for 2D nonlinear discrete and differential systems described by a state-space model of the Roesser form using an extension of Lyapunov’s method. One of the main difficulties in using such an approach is that the full derivative or its discrete counterpart along the trajectories cannot be obtained without explicitly finding the solution of the system under consideration. This has led to the use of a vector Lyapunov function and its divergence or its discrete counterpart along the system trajectories. Using this approach, new conditions for asymptotic stability are derived in terms of the properties of two vector Lyapunov functions. The properties of asymptotic stability in the horizontal and vertical dynamics, respectively, are introduced and analyzed. This new properties arise naturally for repetitive processes where one of the two independent variables is defined over a finite interval. Sufficient conditions for exponential stability in terms of the properties of one vector Lyapunov function are also given as a natural follow on from the asymptotic stability analysis.

Nonuniform Stability of Arbitrary Difference Equations

We describe two approaches to study the robustness of the exponential stability of an arbitrary difference equation with finite delay on a Banach space. The first approach is based on a characterization of Perron-type of the exponential stability in terms of the invertibility of a certain linear operator between spaces of bounded sequences. The second approach is based on looking at the dynamics on a higher-dimensional space without delay and so for which one has the cocycle property. We emphasize that none of the results obtained with the two approaches implies the other. More precisely, the second approach allows to obtain the robustness of a weaker notion although also with a stronger hypothesis. We consider the general case of nonuniform exponential stability.

Asymptotic stability of thermoelastic systems of Bresse type

We provide a comprehensive stability analysis of the thermoelastic Bresse system (also known as the circular arch problem). In particular, assuming a temperature evolution of Gurtin–Pipkin type, we establish a necessary and sufficient condition for exponential stability in terms of the structural parameters of the problem. As a byproduct, a complete characterization of the longtime behavior of Bresse-type systems with Fourier, Maxwell–Cattaneo and Coleman–Gurtin thermal laws is obtained. Our main theorem also subsumes some recent achievements in the stability properties of thermoelastic Timoshenko systems with classical and nonclassical heat conduction.

Lyapunov’s direct method for linear systems of functional-differential equations in Sobolev space

We establish a criterion for exponential stability in the H1-topology in terms of operator inequalities for a linear FDE system of retarded type by Lyapunov’s direct method. As a corollary, some sufficient condition of exponential stability in terms of the matrix specifying the Stieltjes integral is obtained in the autonomous case. A few examples illustrating the results are exhibited.

Stability of nonlinear 2D systems described by the continuous-time Roesser model

This paper considers systems with two-dimensional dynamics (2D systems) described by the continuous-time nonlinear state-space Roesser model. The sufficient conditions of exponential stability in terms of vector Lyapunov functions are established. These conditions are then applied to analysis of the absolute stability of a certain class of systems comprising a linear continuous-time plant in the form of the Roesser model with a nonlinear characteristic in the feedback loop, which satisfies quadratic constraints. The absolute stability conditions are reduced to computable expressions in the form of linear matrix inequalities. The obtained results are extended to the class of continuous-time systems governed by the Roesser model with Markovian switching. The problems of absolute stability and stabilization via state- and output-feedback are solved for linear systems of the above class. The solution procedures for these problems are in the form of algorithms based on linear matrix inequalities.

Advances in neural network based learning

In recent years, the applications of diverse machine leaning algorithms and their fusion to the cybernetics and decisionmaking have been attracting more and more scholars from different disciplines. As a most frequently used technique of extracting knowledge from data, a feed-forwarded neural network (NN) has shown more advantages for the regression and classification problems. The main advantages come from the capability of good approximation of NNs and their corresponding highly nonlinear boundary. Due to the many merits and wide applications of NNs, the study on several fundamental issues of NNs, such as structure selection, overtraining, robustness and capability of resistance to noisy data, training on different types of data, and more importantly, the generalization ability, are still in progress although NNs have been an old topic in the areas of learning and reasoning. As a result, these extensive studies and significant improvements bring a number of new features anddevelopments to NNs. This issue makes an attempt to provide some latest advances of NN learning, the recent improvements of performance for NN learning systems, and new applications of NNs to different real fields. In this issue, 14 papers are accepted for publication. The 14 papers cover a variety of topics which include the stochastic stability NN with time delay, imbalanced and uncertain data, fuzzy NNs, bidirectional associative memory (BAM), deep learning of NNs, and other comparative and review studies. Categorization and brief description of these papers are given below. Four papers discuss the stability of NNs with delay time. Actually, the time delay is ubiquitous in most natural systems. Since the time delay is frequently encountered in NNs, the issue about stability of NNs with delay has been emerging as a challenging task for NN researchers. The paper authored by Cheng-De Zheng studied a class of stochastic reaction diffusion NNs with Markovian jumping parameters and time delays. Lyapunov functional method and stochastic analysis technique are used there to construct a new stochastic stability in the principle of the mean square. Continuing this topic, the paper authored by M. Syed Ali studied the stochastic stability analysis of uncertain recurrent NNs, also with Markovian jumping parameters and time varying delays. By using Lyapunov functional theory, Ito differential rule and matrix analysis techniques, a sufficient criterion in term of linear matrix inequality (LMI) is established such that, for all admissible parameter uncertainties and stochastic disturbances, the stochastic NN is perfectly stable. And then, a non-stationary problem on BAM NNs is discussed in Qingqing He’s paper. An exponential stability in terms of LMIs is achieved for a class of interval C-G type (BAM) NNs with the mixed time delays and non-smooth behaved functions by using Homomorphic mapping theory, non-smooth analysis, LMIs, free-weighting matrix and Lyapunov–krasovskii functional approach. Furthermore, the paper authored by R. Raja derived the stability analysis for a class of uncertain discrete-time stochastic BAM NNs with timevarying delays and impulses. Lyapunov–Krasovskii functional and stochastic analysis techniques were applied to obtain a set of sufficient conditions in term of LMI. Along this line, the non-stationary challenge in other class of NN can be tackled. The paper authored by Adel X.-Z. Wang A. B. Musa (&) Key Laboratory in Machine Learning and Computational Intelligence of Hebei Province, College of Mathematics and Computer Science, Hebei University, Baoding 071002, China e-mail: abdubashir20@yahoo.com

Exponential stability for linear evolutionary equations

We give an approach to exponential stability within the framework of evolutionary equations due to (R. Picard. A structural observation for linear material laws in classical mathematical physics. Math. Methods Appl. Sci., 32(14):1768-1803, 2009). We derive sufficient conditions for exponential stability in terms of the material law operator which is defined via an analytic and bounded operator-valued function and give an estimate for the expected decay rate. The results are illustrated by three examples: differential-algebraic equations, partial differential equations with finite delay and parabolic integro-differential equations.

Exponential stability in terms of two measures of impulsive stochastic functional differential systems via comparison principle

In this paper, based on like-Lyapunov functions and comparison principles, several criteria on the exponential stability in terms of two measures of impulsive stochastic functional differential systems with infinite or finite delays are obtained. The results improve and complement those in earlier publications. Two illustrative examples are also discussed to show the effectiveness and generality of our theorems.

Global exponential stability of high-order Hopfield-type neural networks with S-type distributed time delays

This paper studies the problems of global exponential stability of high-order Hopfield-type neural networks with s-type distributed time delays. By using the topological degree theory and differential inequality technique, we prove existence of the equilibrium point and global existence of the solutions. We obtain some sufficient conditions on global exponential stability in terms of intercept equation, which are easily verifiable and have a wider adaptive. An example is also discussed to illustrate our results.

Integral conditions for exponential dichotomy: A nonlinear approach

We give new necessary and sufficient integral conditions for the existence of exponential dichotomy of skew-product flows. Our methods are based on the structure of the associated stable subspace and unstable subspace. We propose a nonlinear approach, extending the study of exponential stability in terms of integral conditions to the case of uniform exponential dichotomy. We apply the main results to the study of uniform exponential dichotomy of non-autonomous systems.

Global exponential stability of delayed periodic dynamical systems

In this Letter, we discuss delayed periodic dynamical systems, compare capability of criteria of global exponential stability in terms of various L p (1⩽ p<∞) norms. A general approach to investigate global exponential stability in terms of various L p (1⩽ p<∞) norms is given. Sufficient conditions ensuring global exponential stability are given, too. Comparisons of various stability criteria are given. More importantly, it is pointed out that sufficient conditions in terms of L 1 norm are enough and easy to implement in practice.