Stability Analysis of Dynamic Systems in the Polynomial Vector-Matrix Representation

In two recent papers, Brayton and Tong developed some significant results which are the basis of a constructive approach in the stability analysis of dynamical systems. Although their algorithm is powerful, it taxes the capabilities of most modern computers when the dimension of a system is high or even when the dimension is moderate in size. In this paper we remove these difficulties to a certain extent by generalizing the work of Brayton and Tong to interconnected dynamical systems. In our approach, the analysis of complex dynamical systems (which may be of high dimension) is accomplished in terms of the qualitative properties of the free subsystems and the interconnections of such systems.

In this paper, we study the output consensus problems for heterogeneous multi‐agent systems (MASs) with a high‐dimensional leader and directed topology, where the followers are subject to unknown uncertainties. First, an observer is designed to estimate leader's unmeasurable states. Then, by using neural network (NN) approximation theory, a neuro‐adaptive controller consists of a compensation part and a cooperative term is proposed. Unlike most existing neuro‐adaptive controllers that utilize some discontinuous functions to eliminate the effects of NN approximation errors, a new class of continuous function is used and thereby the chattering effect is avoided. By using tools from graph theory and stability analysis of dynamical systems, it is proven that asymptotical output consensus can be achieved if the underlying network topology contains a directed spanning tree rooted at the leader and the control parameters are suitably chosen. Finally, a numerical simulation is presented to verify the theoretical result.

Optimal control theory deals with finding protocols to steer a system between assigned initial and final states, such that a trajectory-dependent cost function is minimized. The application of optimal control to stochastic systems is an open and challenging research frontier, with a spectrum of applications ranging from stochastic thermodynamics to biophysics and data science. Among these, the design of nanoscale electronic components motivates the study of underdamped dynamics, leading to practical and conceptual difficulties. In this work, we develop analytic techniques to determine protocols steering finite time transitions at a minimum thermodynamic cost for stochastic underdamped dynamics. As cost functions, we consider two paradigmatic thermodynamic indicators. The first is the Kullback–Leibler divergence between the probability measure of the controlled process and that of a reference process. The corresponding optimization problem is the underdamped version of the Schrödinger diffusion problem that has been widely studied in the overdamped regime. The second is the mean entropy production during the transition, corresponding to the second law of modern stochastic thermodynamics. For transitions between Gaussian states, we show that optimal protocols satisfy a Lyapunov equation, a central tool in stability analysis of dynamical systems. For transitions between states described by general Maxwell-Boltzmann distributions, we introduce an infinite-dimensional version of the Poincaré-Lindstedt multiscale perturbation theory around the overdamped limit. This technique fundamentally improves the standard multiscale expansion. Indeed, it enables the explicit computation of momentum cumulants, whose variation in time is a distinctive trait of underdamped dynamics and is directly accessible to experimental observation. Our results allow us to numerically study cost asymmetries in expansion and compression processes and make predictions for inertial corrections to optimal protocols in the Landauer erasure problem at the nanoscale.

On the convergence of two-step methods generated by point-to-point operators

We analyze the stability of a class of thin-shell wormholes with spherical symmetry evolving in flat FRW spacetimes. The wormholes considered here are supported at the throat by a perfect fluid with equation of state [Formula: see text] and have a physical radius equal to aR, where a is a time-dependent function describing the dynamics of the throat and R is the background scale factor. The study of wormhole stability is done by means of the stability analysis of dynamic systems.

Introduction. Minimization of ravine convex functions, both smooth and non-smooth, arises in many problems of planning, control, stability analysis of dynamic systems, artificial intelligence, and machine learning. Therefore, the development of new and improvement of existing methods is an important task, taking into account the fact that more and more frequently the functions to be minimized depend on a large number of variables. Special attention should be paid to the methods using the operation of linear transformation of space, which allow to improve the properties of the objective function and significantly accelerate its minimization. Known methods of this type, in particular, ellipsoid methods and r-algorithms, require recalculation of the transformation matrix at each iteration. Therefore, the development of methods with a one-time transformation of space, which have a lower cost of iteration, is definitely an urgent task. The purpose of the article is to review modifications of subgradient method with Polyak’s step that use a scalar parameter m(1 and one-time space transformation operation. Supplement the justification of the convergence and convergence rate of the modifications described. Give recommendations regarding determination of parameter m and space transformation matrix B. Results. Subgradient method with Polyak’s step in transformed space and parameter m>1 is an effective method for minimizing smooth and non-smooth convex functions that have a ravine structure. Determination of an appropriate value of the parameter m(1 and space transformation matrix B allows to significantly accelerate this method and use it for minimization functions that depend on a large number of variables. Conclusions. The development of fast methods for minimization of non-smooth convex functions of many variables with a ravine structure makes it possible to effectively solve modern problems of artificial intelligence, in particular, the problems of machine learning, image recognition, big data analysis, etc. Keywords: subgradient method, space transformation, ravine functions.

Algorithms that extract the principal or minor components of a signal are widely used in signal processing and control applications. This paper explores new frameworks for generating learning rules for iteratively computing the principal and minor components (or subspaces) of a given matrix. Stability analysis using Lyapunov theory and La Salle invariance principle is provided to determine regions of attraction of these learning rules. Among many derivations, it is specifically shown that Oja's rule and many variations of it are asymptotically globally stable. Lyapunov stability theory is also applied to weighted learning rules. Some of the essential features for the proposed MCA/PCA learning rules are that they are self normalized and can be applied to non-symmetric matrices. Exact solutions for some nonlinear dynamical systems are also provided

Eigenmode solutions are very important in stability analysis of dynamical systems. The set of eigenvalues of a non‐self‐adjoint differential operator originated from the linearization of some Cauchy problem is investigated. It is shown that the eigenvalues are purely imaginary, and that they are related to the eigenvalues of Heun's differential equation. These two results are used to derive the asymptotic behavior of the eigenvalues and to compute them numerically.

The stability analysis of dynamical systems is a critical aspect of understanding their behavior and ensuring reliable performance in various applications. Lyapunov functions play a pivotal role in this analysis, offering a powerful mathematical tool to assess the stability of equilibrium points. Despite the fact that at the moment there is no general method for finding the Lyapunov function, this review explores the diverse landscape of Lyapunov functions and their contributions to achieving global stability in dynamical systems. Examples of epidemic models and their Lyapunov functions with the graphical implementations in Python in different reproduction numbers are included for deeper understanding. In addition, the history of the Lyapunov global asymptotic stability theorem was considered shortly. Possible further research ideas in this area are included at the end.

Eigenvalue analysis is a well-established tool for stability analysis of dynamical systems. However, there are situations where eigenvalues miss some important features of physical models. For example, in models of incompressible fluid dynamics, there are examples where linear stability analysis predicts stability but transient simulations exhibit significant growth of infinitesimal perturbations. This behavior can be predicted by pseudo-spectral analysis. In this study, we show that an approach similar to pseudo-spectral analysis can be performed inexpensively using stochastic collocation methods and the results can be used to provide quantitative information about instability. In addition, we demonstrate that the results of the perturbation analysis provide insight into the behavior of unsteady flow simulations.

A framework for designing policies for networked systems with uncertainty

Stability analysis of discrete-time systems with arbitrary delay kernels based on kernel-related summation inequality and model transformation

Stability analysis of dynamical system is very useful and is able to classify the role of stable and unstable equilibrium points. In this work, Naiver–Stokes system has been studied by using KCC theory. The Jacobi stability and dynamics of the deviation vector near equilibrium points have been also studied. Further, the effect of bifurcation parameter on stability of Navier–Stokes system has been observed and found the limiting conditions for bifurcation. Numerical examples of particular interest have been taken to compare the results of Jacobi stability and linear stability. It is observed that Jacobi stability on the basis of KCC theory is more efficient than the linear stability.

Stability analysis of nonsmooth systems, using Lyapunov direct method, is a difficult task. An alternative way of studying the stability in such systems is to employ the concept of Lyapunov exponents. However, two difficulties in the calculation of Lyapunov exponents, i.e. numerical artifacts and low computational efficiency, often prohibit the application of this otherwise powerful method. In this paper, a case study of the stability analysis of a switching contact task control system using the concept of Lyapunov exponents is presented. The goal is to carry out a numerical investigation of the method of nonstandard finite difference discretization, for constructing discrete models of differential equations that describe the Lyapunov exponents for such a nonsmooth system. It is shown that, as compared with the standard fourth-order Runge-Kutta method, the nonstandard finite difference discretization scheme provides numerically stable results with a larger critical integration step-size and less computation time. Therefore, from numerical stability and computational efficiency viewpoints, the nonstandard finite difference discretization method is meritorious and should be given consideration when stability analysis of systems using Lyapunov exponents is of concern.

Geometrical method for stability analysis of dynamical systems and its application to hydrodynamics