Amœbas and structural stability of multidimensional systems: a test algorithm based on Monte-Carlo integration

Corrections to “Stability of Multidimensional Systems Using Genetic Algorithms”

Stability of multidimensional systems is a field of intensive research. In this context, different classes of Hurwitz polynomials (in the continuous case) and Schur polynomials (in the discrete case) are in the focus of interest. Although there exist various methods for testing whether a given polynomial belongs to a certain class of the afore mentioned. The type of converse problem, namely the design of stable polynomials is much more tedious. In this paper, a parametric model for the characterization of real or complex two-variable scattering Schur polynomials is given. In other words, the coefficients of the two-dimensional (2-D) polynomial model are functions of real parameters. The following features make it best suited for the design of 2-D systems: no dependencies between the real valued parameters, coverage of the whole class of 2-D scattering Schur polynomials, and the coefficients of the polynomial are rational functions of the parameters. The synthesis of 2-D lossless networks and unitary matrices play a key role in our considerations.

This paper deals with the stability of continuous-time multidimensional nonlinear systems in the Roesser form. The concepts from 1D Lyapunov stability theory are first extended to 2D nonlinear systems and then to general continuous-time multidimensional nonlinear systems. To check the stability, a direct Lyapunov method is developed. While the direct Lyapunov method has been recently proposed for discrete-time 2D nonlinear systems, to the best of our knowledge what is proposed in this paper are the first results of this kind on stability of continuous-time multidimensional nonlinear systems. Analogous to 1D systems, a sufficient condition for the stability is the existence of a certain type of the Lyapunov function. A new technique for constructing Lyapunov functions for 2D nonlinear systems and general multidimensional systems is proposed. The proposed method is based on the sum of squares (SOS) decomposition, therefore, it formulates the Lyapunov function search algorithmically. In this way, polynomial nonlinearities can be handled exactly and a large class of other nonlinearities can be treated introducing some auxiliary variables and constrains.

Stability of ecological systems: A theoretical review

The main purpose of this work is to provide a non-local approach to study aspects of structural stability of 3D Filippov systems. We introduce a notion of semi-local structural stability which detects when a piecewise smooth vector field is robust around the entire switching manifold, as well as, provides a complete characterization of such systems. In particular, we present some methods in the qualitative theory of piecewise smooth vector fields, which make use of geometrical analysis of the foliations generated by their orbits. Such approach displays surprisingly rich dynamical behavior which is studied in detail in this work. It is worth mentioning that this subject has not been treated in dimensions higher than two from a non-local point of view, and we hope that the approach adopted herein contributes to the understanding of structural stability for piecewise-smooth vector fields in its most global sense.

This paper presents a novel decentralized control strategy for a class of uncertain nonlinear large-scale systems with mismatched interconnections. First, it is shown that the decentralized controller for the overall system can be represented by an array of optimal control policies of auxiliary subsystems. Then, within the framework of adaptive dynamic programming, a simultaneous policy iteration (SPI) algorithm is developed to solve the Hamilton–Jacobi–Bellman equations associated with auxiliary subsystem optimal control policies. The convergence of the SPI algorithm is guaranteed by an equivalence relationship. To implement the present SPI algorithm, actor and critic neural networks are applied to approximate the optimal control policies and the optimal value functions, respectively. Meanwhile, both the least squares method and the Monte Carlo integration technique are employed to derive the unknown weight parameters. Furthermore, by using Lyapunov’s direct method, the overall system with the obtained decentralized controller is proved to be asymptotically stable. Finally, the effectiveness of the proposed decentralized control scheme is illustrated via simulations for nonlinear plants and unstable power systems.

Starting with the established concepts of multi-dimensional network resonance, practical-bounded-input-bounded-output (BIBO) stability of multi-dimensional discrete systems, and the integer order approximation of fractional order filters, a new method for the realization of integer-order approximations of 2-D fractional-order systems having frequency-planar beam shaped passbands is proposed. The method results in 2-D infinite impulse response (IIR) filters having nonseparable denominators in their transfer-functions while having guaranteed practical-BIBO stability for zero initial conditions in the corresponding discrete system. The resulting integer-order filters are shown to approximate 2-D fractional order frequency-planar beam shapes in both magnitude and phase leading to a new theoretical tool for use in the emerging area of multi-dimensional fractional-order circuits and systems. Although examples are provided for frequency-planar filters in two dimensions, the proposed methods are equally applicable to other types of transfer-functions having prototypes based on resistively-terminated passive networks defined over any integer number of dimensions.

Feedback boundary control of multi-dimensional hyperbolic systems with relaxation

In the paper we test robustness of particular dynamic systems in a compact regions of a plane and a weak structural stability of one dynamic system of high order in a compact region of its phase space. The test was carried out based on the fundamental theory of dynamical systems on a plane and based on the conditions for weak structural stability of high order dynamic systems. A numerical algorithm for testing the weak structural stability of high order dynamic systems has been proposed. Based on this algorithm we assess the weak structural stability of one computable general equilibrium model.

Understanding the factors that influence the persistence and stability of complex ecological networks is a central focus of ecological research. Recent research into these factors has predominantly attempted to unveil the ecological processes and structural constraints that influence network stability. Comparatively little attention has been given to the consequences of evolutionary events, despite the fact that the interplay between ecology and evolution has been recognized as fundamental to understand the formation of ecological communities and predict their reaction to change. We extend existing mutualistic population dynamical models by incorporating evolutionary adaptation events to address this critical gap. We relate ecological aspects of mutualistic community stability to the stability of persistent evolutionary pathways. Our findings highlight the significance of the structural stability of ecological systems in predicting biodiversity loss under both evolutionary and environmental changes, particularly in relation to species-level selection. Notably, our simulations reveal that the evolution of mutualistic networks tends to increase a network-dependent parameter termed critical competition, which places systems in a regime in which mutualistic interactions enhance structural stability and, consequently, biodiversity. This research emphasizes the pivotal role of natural selection in shaping ecological networks, steering them towards reduced effective competition below a critical threshold where mutualistic interactions foster stability in the face of environmental change.

Sway stability of steel scaffolding and formwork systems

In this paper, the stability problem of m-dimensional systems is tackled as an optimization problem. A neural network is proposed for the solution of this problem. Several numerical examples illustrate the validity and the efficiency of the method.

The open problem of the existence of double coprime factorizations (DCFs) for multi-input/multi-output (MIMO) multidimensional (nD) linear systems is considered in the paper. It is shown that DCFs exist for a class of MIMO nD linear systems. A simple and efficient method is proposed for the construction of DCFs. The main result of the paper is to show how to construct a coprime (over the ring of stable rational functions) matrix fraction description of a given MIMO nD system with a stable reduced minor. A parameterization of all stabilizing compensators is given for a MIMO nD system in this class. An example is illustrated.

Stabilization of nonstationary linear systems over noisy communication channels is considered. Stochastically stable sources, and unstable but noise-free or bounded-noise systems have been extensively studied in the information theory and control theory literature since the 1970s, with a renewed interest in the past decade. There have also been studies on noncausal and causal coding of unstable/nonstationary linear Gaussian sources. In this paper, tight necessary and sufficient conditions for stochastic stabilizability of unstable (nonstationary) possibly multidimensional linear systems driven by Gaussian noise over discrete channels (possibly with memory and feedback) are presented. Stochastic stability notions include recurrence, asymptotic mean stationarity and sample path ergodicity, and the existence of finite second moments. Our constructive proof uses random-time state-dependent stochastic drift criteria for stabilization of Markov chains. For asymptotic mean stationarity (and thus sample path ergodicity), it is sufficient that the capacity of a channel is (strictly) greater than the sum of the logarithms of the unstable pole magnitudes for memoryless channels and a class of channels with memory. This condition is also necessary under a mild technical condition. Sufficient conditions for the existence of finite average second moments for such systems driven by unbounded noise are provided.

N the paper1 by Shrivastava and Pradeep1 seven theorems on stability of multidimensional linear second-order systems with time-varying coefficients have been discussed. Unfortunately, all statements on asymptotic stability are useless because they are based on contradictory conditions. For example, theorem 1 is considered where, within the-sufficient conditions for asymptotic stability, a positive definite mass matrix and a negative definite time derivative of the mass matrix are simultaneously required: