Algebraic Sufficient Conditions Research Articles

Constructive proof of Lagrange stability and sufficient – Necessary conditions of Lyapunov stability for Yang–Chen chaotic system

This paper studies the stability problem of Yang–Chen system. By introducing different radial unbounded Lyapunov functions in different regions, global exponential attractive set of Yang–Chen chaotic system is constructed with geometrical and algebraic methods. Then, simple algebraic sufficient and necessary conditions of global exponential stability, global asymptotic stability, and exponential instability of equilibrium are proposed. And the relevant expression of corresponding parameters for local exponential stability, local asymptotic stability, exponential instability of equilibria are obtained. Furthermore, the branch problem of the system is discussed, some branch expressions are given for the parameters of the system.

Projections onto closed convex sets in Hilbert spaces

Let X be a real Hilbert space. We give necessary and sufficient algebraic conditions for a mapping $${F\colon X \to X}$$ with a closed image set to be the metric projection mapping onto a closed convex set. We provide examples that illustrate the necessity of each of the conditions. Our characterizations generalize several results related to projections onto closed convex sets.

Traveling waves for degenerate diffusive equations on networks

In this paper we consider a scalar parabolic equation on a star graph; the model is quite general but what we have in mind is the description of traffic flows at a crossroad. In particular, we do not necessarily require the continuity of the unknown function at the node of the graph and, moreover, the diffusivity can be degenerate. Our main result concerns a necessary and sufficient algebraic condition for the existence of traveling waves in the graph. We also study in great detail some examples corresponding to quadratic and logarithmic flux functions, for different diffusivities, to which our results apply.

Competition in periodic media:Ⅰ-Existence of pulsating fronts

This paper is concerned with the existence of pulsating front solutions in space-periodic media for a bistable two-species competition–diffusion Lotka–Volterra system. Considering highly competitive systems, a simple high frequency or small amplitudes algebraic sufficient condition for the existence of pulsating fronts is stated. This condition is in fact sufficient to guarantee that all periodic coexistence states vanish and become unstable as the competition becomes large enough.

Data processing for the sandwiched R\xe9nyi divergence: a condition for equality

The alpha -sandwiched Rényi divergence satisfies the data processing inequality, i.e. monotonicity under quantum operations, for alpha ge 1/2. In this article, we derive a necessary and sufficient algebraic condition for equality in the data processing inequality for the alpha -sandwiched Rényi divergence for all alpha ge 1/2. For the range alpha in [1/2,1), our result provides the only condition for equality obtained thus far. To prove our result, we first consider the special case of partial trace and derive a condition for equality based on the original proof of the data processing inequality by Frank and Lieb (J Math Phys 54(12):122201, 2013) using a strict convexity/concavity argument. We then generalize to arbitrary quantum operations via the Stinespring Representation Theorem. As applications of our condition for equality in the data processing inequality, we deduce conditions for equality in various entropic inequalities. We formulate a Rényi version of the Araki–Lieb inequality and analyze the case of equality, generalizing a result by Carlen and Lieb (Lett Math Phys 101(1):1–11, 2012) about equality in the original Araki–Lieb inequality. Furthermore, we prove a general lower bound on a Rényi version of the entanglement of formation and observe that it is attained by states saturating the Rényi version of the Araki–Lieb inequality. Finally, we prove that the known upper bound on the entanglement fidelity in terms of the usual fidelity is saturated only by pure states.

Robust stabilization of discrete-time periodic linear systems for tracking and disturbance rejection

In analogy to the Kucera–Youla parametrization, we construct and parametrize all stabilizing controllers of a stabilizable linear periodic discrete-time input/output system, the plant. We establish a necessary and sufficient algebraic condition for the existence of controllers among these for which the output of the plant tracks a given reference signal in spite of disturbance signals on the input and the output of the plant. With a minor additional assumption, the tracking stabilizing controllers are robust. As in the linear time-invariant (LTI) case, the reference and disturbance signals are assumed to be generated by an autonomous system. Our results are the analogs for periodic behaviors of the corresponding LTI results of Vidyasagar. A completely different approach to stabilization and control of discrete periodic systems was developed by Bittanti and Colaneri. We derive a categorical duality between periodic behaviors over the time-axis of natural numbers and finitely generated modules over a suitable noncommutative ring of difference operators and use this for the proof of the main stabilization and control results. Morita’s theory of equivalences between module categories is employed as an essential algebraic tool. All results of the paper are constructive.

Robust finite-time global synchronization of chaotic systems with different orders

This article proves that the robust finite-time synchronization behavior can be achieved for the chaotic systems with different orders. Based on the Lyapunov stability theory and using the nonlinear feedback control, sufficient algebraic conditions are derived to compute the linear controller gains. These gains are then used to achieve the robust finite time increasing order and reduced order synchronization of the chaotic systems. This study also discusses the design of a controller that accomplishes the finite time synchronization of two chaotic systems of different dimensions under the effect of both unknown model uncertainties and external disturbances. Numerical simulation results are furnished to validate the theoretical findings.

Sufficient Lie Algebraic Conditions for Sampled-Data Feedback Stabilizability of Affine in the Control Nonlinear Systems

For general nonlinear autonomous systems, a Lyapunov characterization for the possibility of semi-global asymptotic stabilizability by means of a time-varying sampled-data feedback is established. We exploit this result in order to derive a Lie algebraic sufficient condition for sampled-data feedback semi-global stabilizability of affine in the control nonlinear systems with non-zero drift terms. The corresponding proposition constitutes an extension of the “Artstein-Sontag” theorem on feedback stabilization.

Controllability improvement for multi-agent systems: leader selection and weight adjustment

ABSTRACTFor an uncontrollable system, adding leaders and adjusting edge weights are two methods to improve controllability. In this paper, the controllability of multi-agent systems under directed topologies is studied, especially on the leader selection problem and the weight adjustment problem. For a multi-agent system, necessary and sufficient algebraic conditions for controllability with fewest leaders are proposed. To improve controllability by adjusting edge weights, the system is supposed to be structurally controllable, which holds if and only if the communication topology contains a spanning tree. It is also proved that the number of the fewest edges needed to be assigned on new weights equals the rank deficiency of controllability matrix. In addition, a leader selection algorithm and a weight adjustment algorithm are presented. Simulation examples are provided to illustrate the theoretical results.

The synchronization of chaotic systems with different dimensions by a robust generalized active control

Active control strategy is a powerful control technique in synchronizing chaotic/hyperchaotic systems. Until now, active control techniques have been employed to synchronize chaotic systems with the same orders. The present study overcomes the limitations of synchronization of chaotic systems of similar dimensions using active control. In this article, the authors investigate the synchronization problem for a drive-response chaotic system with different orders under the effect of both unknown model uncertainties and external disturbance. Based on the Lyapunov stability theory and Routh–Hurwitz criterion, a robust generalized active control approach is proposed and sufficient algebraic conditions are derived to compute a suitable linear controller gain matrix that guarantees the globally exponentially stable synchronization. Two examples are presented to illustrate the main results, namely reduced-order synchronization between the hyperchaotic Lu and the unified chaotic systems and the increased-order synchronization between the unified chaotic and the hyperchaotic Lu systems. There are three main contributions of the present study: (a) generalization of the active control for synchronization of a class of chaotic systems with different orders; (b) a recursive approach is proposed to compute a suitable linear controller gain matrix and (c) reduced (increased) order synchronization under the effect of both unknown model uncertainties and external disturbances. A comparative study has been done with our results and previously published work in terms of synchronization speed and quality. Future applications of the proposed reduced (increased) order synchronization approach are discussed. Finally, numerical simulations are given to verify the effectiveness of the proposed reduced (increased) order active synchronization approach.

Discrete-Time Analysis of Multi-Agent Networks with Multiple Consensus Equilibria**This work was sponsored by BAP Project 9141 and Scientific and Technical Research Council of Turkey (TUBITAK) under Grant 114E613.

Abstract: In this paper, we investigate the multiple equilibrium consensus problem of first-order multi-agent networks. The problem is handled in a discrete time setting for networks with time invariant and time-varying topologies having undirected information flow. By using matrix theory and algebraic graph theory necessary and sufficient conditions are presented. The results are also illustrated with numerical examples.

A Research on Active Control to Synchronize a New 3D Chaotic System

This paper presents the robust synchronization problem of a 3D chaotic system by using the active control technique. Based on the Gershgorin theorem and Routh-Hurwitz criterion, sufficient algebraic conditions are derived to design a linear controller gain matrix. The conditions are then applied for the robust stability of the synchronization error dynamics in the presence of an unknown bounded smooth external disturbance. The proposed active control strategy with a suitable computation of the linear controller gain matrix is simple in design and establishes fast convergence rates of the synchronization error signals. Numerical simulation results further verified the analytical results.

On norm equivalence between the displacement and velocity vectors for free linear dynamical systems

As the main new result, under certain hypotheses, for free vibration problems, the norm equivalence of the displacement vector y(t) and the velocity vector y˙(t) is proven. The pertinent inequalities are applied to derive some two-sided bounds on y(t) and y˙(t) that are known so far only for the state vector x(t)=[yT(t),y˙T(t)]T. Sufficient algebraic conditions are given such that norm equivalence between y(t) and y˙(t) holds, respectively, does not hold, as the case may be. Numerical examples illustrate the results for vibration problems of n degrees of freedom with n∈{1,2,3,4,5} by computing the mentioned algebraic conditions and by plotting the graphs of y(t) and y˙(t). Some notations and definitions of References Kohaupt (2008b, 2011) are necessary and are therefore recapitulated. The paper is of interest to Mathematicians and Engineers.

Bipartite opinion forming: Towards consensus over coopetition networks

Within the framework of signed graph and multi-agent systems, this paper investigates the distributed bipartite opinion forming problem over coopetition networks. Several sufficient algebraic and geometric topology conditions that guarantee consensus, regardless of the magnitudes of individual coupling strengths among the agents, have been derived by exploring the interaction direction patterns. All the criteria presented do not require the global knowledge of the coupling weights of the entire network, and thus are easier to check. The effectiveness of the theoretical results are illustrated by numerical examples.

Adaptive control to synchronize and anti-synchronize two identical time delay Bhalekar-Gejji chaotic systems with unknown parameters

Based on the Lyapunov-Krasovskii functional theory and using the adaptive control, the work of this paper addresses the globally and asymptotically stable synchronisation and anti-synchronisation between two nearly identical time delay Bhalekar-Gejji chaotic systems in the presence of unknown parameters. Using the drive-response system synchronisation and anti-synchronisation schemes and based on the adaptive control, some algebraic sufficient conditions are derived that guarantee the globally and asymptotically stable synchronisation and anti-synchronisation between two identical time-delay Bhalekar-Gejji chaotic systems. Accordingly, appropriate update laws are designed to identify the fully unknown parameters. It has been shown that the proposed approach has an excellent transient performance in comparison with some noting existing results. Numerical simulations are given to verify and test the effectiveness of the current study.

Partial state stability and stabilization of Boolean control networks

The paper deals with problems on the stability and stabilization of Boolean networks with respect to a given part of variables characterizing the Boolean network. Using the semi-tensor product of matrices and the matrix expression of logic, the dynamics of a Boolean (control) network can be converted to discrete time linear (bilinear) dynamics called the algebraic form of the Boolean (control) network. The two main results are as follows: (i) From the algebraic form of a Boolean (control) network, two partial stability definitions are introduced. One is partial stability with respect to another variable set, and the other is partial stability with respect to arbitrary initial states. (ii) Based on the algebraic form, necessary and sufficient conditions for the stability of global partial states with respect to the variable set and arbitrary initial states are obtained. Examples are provided to illustrate the effectiveness of our results.

Metric problems for quadrics in multidimensional space

Given the equations of the first and the second order manifolds in Rn, we construct the distance equation, i.e. a univariate algebraic equation one of the zeros of which (generically minimal positive) coincides with the square of the distance between these manifolds. To achieve this goal we employ Elimination Theory methods. In the frame of this approach we also deduce the necessary and sufficient algebraic conditions under which the manifolds intersect and propose an algorithm for finding the coordinates of their nearest points. The case of parameter dependent manifolds is also considered.

Algebraic conditions for monotonicity of magnitude‐frequency responses in all‐pole fractional order systems

This study deals with the investigation of the magnitude-frequency responses of all-pole fractional order systems in the viewpoint of extrema existence in these responses. In this investigation, a sufficient algebraic condition, two necessary algebraic conditions, and a necessary and sufficient algebraic condition are obtained to guarantee the non-existence of extrema in the magnitude-frequency response of all-pole fractional order systems. Some examples are presented to show the effectiveness of the results of the paper.

On the nonlinear stability of ternary porous media via only one necessary and sufficient algebraic condition

Research on convective-diffusive fluid motions in porous media has a notable relevance (increasing with the number of salts dissolved in the fluid) either for geophysical applications (engineering geology, subsurface and structural geology, subsurface contaminant transport, underground water flow, ...) or because porous materials occur very frequently in real life (fiber materials for insulating purposes, metallic foams in heat transfer devices ([1,13]). In the present paper porous horizontal layers, heated from below and salted from above and below, in the Darcy-Boussinesq scheme, are investigated. By virtue of the absence of subcritical instabilities ([20]), the bifurcating competition of Rayleigh and Prandtl numbers for promoting or inhibiting the onset of convection, investigated through the linearized equations, allows to show ([20]) that this competition can be restricted to the inequalities (2) which are necessary and sufficient for inhibiting the onset of convection. But while the onset of convection requires that only one of (3) holds, the stability requires that, at least, all the reverse of (3) hold. Therefore, either for theoretical reasons or for practical use of stability conditions, the problem of overcoming this gap arises. We call <em>one stability condition (OSC) problem</em> the looking for inequalities of type (4) able to inhibit the onset of convection for large sets of values of the bifurcating parameters with special attention to the case {$\alpha=3,g=0$} since in this case (4) is necessary and sufficient for inhibiting the onset of convection. To this goal new fields for the salts densities are introduced. These transformations allow to: i) discover skew-symmetries hidden in the Darcy-Boussinesq equations; ii) obtain meaningful contributions to the OSC problem.

Matrix Quasi-Exactly Solvable Jacobi Elliptic Hamiltonian

We construct a new example of 2 × 2-matrix quasi-exactly solvable (QES) Hamiltonian which is associated to a potential depending on the Jacobi elliptic functions. We establish three necessary and sufficient algebraic conditions for the previous operator to have an invariant vector space whose generic elements are polynomials. This operator is called quasi-exactly solvable.