Polynomial Stability Research Articles

Some results related to Hurwitz stability of combinatorial polynomials

Many important problems are closely related to the zeros of certain polynomials derived from combinatorial objects. The aim of this paper is to observe some results and applications for the Hurwitz stability of polynomials in combinatorics and study other related problems.We first present a criterion for the Hurwitz stability of the Turán expressions of recursive polynomials. In particular, it implies the q-log-convexity or q-log-concavity of the original polynomials. We also give a criterion for the Hurwitz stability of recursive polynomials and prove that the Hurwitz stability of any palindromic polynomial implies its semi-γ-positivity, which illustrates that the original polynomial with odd degree is unimodal. In particular, we get that the semi-γ-positivity of polynomials implies their parity-unimodality and the Hurwitz stability of polynomials implies their parity-log-concavity. Those results generalize the connections between real-rootedness, γ-positivity, log-concavity and unimodality to Hurwitz stability, semi-γ-positivity, parity-log-concavity and parity-unimodality (unimodality). As applications of these criteria, we derive some Hurwitz stability results occurred in the literature in a unified manner. In addition, we obtain the Hurwitz stability of Turán expressions for alternating run polynomials of types A and B and the Hurwitz stability for alternating run polynomials defined on a dual set of Stirling permutations.Finally, we study a class of recursive palindromic polynomials and derive many nice properties including Hurwitz stability, semi-γ-positivity, non-γ-positivity, unimodality, strong q-log-convexity, the Jacobi continued fraction expansion and the relation with derivative polynomials. In particular, these properties of the alternating descents polynomials of types A and B can be implied in a unified approach.

Analyticity and stability results for a plate‐membrane type transmission problem

AbstractIn this paper, we consider a transmission problem for a system of a thermoelastic plate with (or without) a rotational inertia coupled with a membrane. On the plate a structural damping may or may not act, and on the membrane a damping of Kelvin–Voigt type may or may not be present; free boundary operators for the plate are considered in the coupling with the membrane. We prove well‐posedness of the problem and higher regularity of the solution. Depending on the damping and on the presence of the rotational term, we establish strong stability, exponential stability, lack of exponential stability, polynomial stability, and analyticity of the semigroup associated to the transmission problem.

Asymptotic behavior of the Rao–Nakra sandwich beam model with Kelvin–Voigt damping

The Rao–Nakra sandwich beam is a coupled system consisting of two wave equations for the longitudinal displacements of the top and bottom layers and an Euler–Bernoulli beam equation for the transversal displacement. This paper concerns the system’s stability when the Kelvin–Voigt damping terms act on the first and third equations. Using the semigroup theory of linear operators, we prove the global well-posedness of the associated initial boundary value problem. And then, we prove the lack of exponential stability of the system. Because of this lack of exponential stability, we study the polynomial stability and prove that the system decays with rate . We further prove that this decay rate is optimal.

( X , Y , φ ) -Stable semigroups, periodic solutions, and applications

ABSTRACT Motivated by the smoothing properties of heat semigroups on unbounded domains and the conditional stability of hyperbolic semigroups, we develop a unified approach toward the problems on the existence of periodic solutions to the evolution equation . Our method is based on the analysis of -stability of the semigroup generated by A, i.e. , t>0, for certain couple of Banach spaces and real-valued function φ satisfying . Our theory covers both cases corresponding to smoothing properties and the conditional stability of hyperbolic semigroups as well as some other important situations relating to the polynomial or exponential stability of semigroups. As illustrations for our theory, we give applications to the existence and uniqueness of periodic solutions to Navier–Stokes and damped wave equations.

Combinatorics and preservation of conically stable polynomials

Given a closed, convex cone Ksubseteq mathbb {R}^n, a multivariate polynomial fin mathbb {C}[textbf{z}] is called K-stable if the imaginary parts of its roots are not contained in the relative interior of K. If K is the nonnegative orthant, K-stability specializes to the usual notion of stability of polynomials. We develop generalizations of preservation operations and of combinatorial criteria from usual stability toward conic stability. A particular focus is on the cone of positive semidefinite matrices (psd-stability). In particular, we prove the preservation of psd-stability under a natural generalization of the inversion operator. Moreover, we give conditions on the support of psd-stable polynomials and characterize the support of special families of psd-stable polynomials.

Stability of a linear thermoelastic Bresse system with second sound under new conditions on the coefficients

AbstractIn this paper, we discuss the stability of a linear one‐dimensional thermoelastic Bresse system, where the coupling is given through the first component of the Bresse model with the heat conduction of second sound type. We state the well‐posedness and show the polynomial stability of the system, where the decay rate depends on the smoothness of initial data. Moreover, we prove the non exponential and the exponential decay depending on new conditions on the parameters of the system. The proof is based on a combination of the energy method and the frequency domain approach.

Asymptotic stability of neutral stochastic pantograph systems with Markovian switching

ABSTRACT This paper investigates the asymptotic stability of neutral stochastic pantograph systems with Markovian switching (NSPSMS). The stochastic LaSalle theorem has been set up to locate the attractive sets for NSPSMS without linear growth condition. Subsequently, combining the stochastic LaSalle theorem and uniformly continuous theory, almost surely asymptotic stability and pth moment asymptotic stability are analysed. Furthermore, we also present one new criterion on the pth moment polynomial stability and almost surely polynomial stability for NSPSMS, where the coefficients of the delay term are time invariable. Finally, three examples are provided to exhibit the efficiency of the proposed results.

INTRINSIC STABILIZER REDUCTION AND GENERALIZED DONALDSON–THOMAS INVARIANTS

AbstractLet $\sigma $ be a stability condition on the bounded derived category $D^b({\mathop{\mathrm {Coh}}\nolimits } W)$ of a Calabi–Yau threefold W and $\mathcal {M}$ a moduli stack parametrizing $\sigma $ -semistable objects of fixed topological type. We define generalized Donaldson–Thomas invariants which act as virtual counts of objects in $\mathcal {M}$ , fully generalizing the approach introduced by Kiem, Li and the author in the case of semistable sheaves. We construct an associated proper Deligne–Mumford stack $\widetilde {\mathcal {M}}^{\mathbb {C}^{\ast }}$ , called the $\mathbb {C}^{\ast }$ -rigidified intrinsic stabilizer reduction of $\mathcal {M}$ , with an induced semiperfect obstruction theory of virtual dimension zero, and define the generalized Donaldson–Thomas invariant via Kirwan blowups to be the degree of the associated virtual cycle $[\widetilde {\mathcal {M}}]^{\mathrm {vir}} \in A_0(\widetilde {\mathcal {M}})$ . This stays invariant under deformations of the complex structure of W. Applications include Bridgeland stability, polynomial stability, Gieseker and slope stability.

Energy decay of a mixture with local viscoelasticity

We consider a binary mixture system with local Kelvin–Voigt damping in one of the components of the mixture. The well‐posedness and the polynomial stability are proved by the semigroup theory and the frequency domain approach.

A unified approach to multivariate polynomial sequences with real stability

We give new sufficient conditions for a sequence of multivariate polynomials to be real stable. As applications, we obtain several known results, such as the real stability of multivariate Eulerian polynomials, multivariate Bell polynomials and multivariate polynomials over Stirling permutations, in a unified manner. And we also show some new results, such as the real stability of multivariate Lah polynomials and multivariate polynomials over Jacobi-Stirling permutations, and the proper position property of multivariate orthogonal polynomials and multivariate matching polynomials. Finally, we obtain a multivariate generalization of Fisk's result by presenting a 2×2 matrix, which preserves the proper position property.

Stability of matrix polynomials in one and several variables

The paper presents methods for the eigenvalue localisation of regular matrix polynomials, in particular, the stability of matrix polynomials is investigated. For this aim a stronger notion of hyperstability is introduced and widely discussed. Matrix versions of the Gauss-Lucas theorem and Szász inequality are shown. Further, tools for investigating (hyper)stability by multivariate complex analysis methods are provided. Several seconds- and third-order matrix polynomials with particular semi-definiteness assumptions on coefficients are shown to be stable.

Optimal explicit Runge–Kutta time stepping for density-based finite-volume solvers

In this paper, we generate optimal Runge–Kutta stability polynomials for several finite-volume spatial discretizations. From these stability polynomials we generate Butcher tableaus, and compare the accuracy and efficiency of these new schemes to commonly used existing classical temporal schemes. Theoretical analysis shows that these optimal schemes are expected to be significantly more efficient, allowing larger time-step sizes to be used relative to the number of Runge–Kutta stages. These predictions are then verified in practice for several test cases including an isentropic vortex, Taylor–Green vortex, and a T106A turbine cascade. Significant performance improvements were observed for all of these cases, with negligible impact on solution accuracy. Importantly, these improvements are achieved with only minor modifications to the solver.

Polynomial stability and weak stabilization for some partial functional differential equations with delay

Abstract The feedback stabilization of a class of delayed evolution equations in real Hilbert space is considered. By virtue of an observability-type inequality and a delayed control, sufficient conditions ensuring the strong and weak stabilization are provided. For the strong stabilization, the speed of convergence is successfully established. Various applications with numerical simulations are considered.

Stability of Complement Degree Polynomial of Graphs

A graph is a structure amounting to a set of objects in which some pairs of the objects are in some sense related. The objects correspond to mathematical abstractions called vertices (also called nodes or points) and each of the related pairs of vertices is called an edge (also called link or line). A directed graph is a graph in which edges have orientation. A simple graph is a graph that does not have more than one edge between any two vertices and no edge starts and ends at the same vertex. For a simple undirected graph G with order n, and let denotes its complement. Let δ(G), ∆(G) denotes the minimum degree and maximum degree of G respectively. The complement degree polynomial of G is the polynomial CD[G,x]= , where Cdi(G) is the cardinality of the set of vertices of degree i in . A multivariable polynomial f(x1,...,xn) with real coefficients is called stable if all of its roots lie in the open left half plane. In this paper, investigate the stability of complement degree polynomial of some graphs. 2020 Mathematics Subject Classification: 05C31, 05C50

Polynomial stability of a transmission problem involving Timoshenko systems with fractional Kelvin–Voigt damping

In this work, we study the stability of a one‐dimensional Timoshenko system with localized internal fractional Kelvin–Voigt damping in a bounded domain. First, we reformulate the system into an augmented model and using a general criteria of Arendt–Batty we prove the strong stability. Next, we investigate three cases: The first one when the damping is localized in the bending moment, the second case when the damping is localized in the shear stress, we prove that the energy of the system decays polynomially with rate in both cases. In the third case, the fractional Kelvin–Voigt is acting on the shear stress and the bending moment simultaneously. We show that the system is polynomially stable with energy decay rate of type , provided that the two dampings are acting in the same subinterval. The method is based on the frequency domain approach combined with multiplier technique.

Polynomial stability of a piezoelectric beam with magnetic effect and a boundary dissipation of the fractional derivative type

AbstractThis work studies the asymptotic behavior of a waves coupled system with a boundary dissipation of the fractional derivative type. We prove well-posedness and polynomial stability based on the semigroup approach, the energy method, and the result of stability.

Note on stability of an abstract coupled hyperbolic-parabolic system: Singular case

In this paper we try to complete the stability analysis for an abstract system of coupled hyperbolic and parabolic equations utt+Au−Aαw=0,wt+Aαut+Aβw=0,u(0)=u0,ut(0)=u1,w(0)=w0,where A is a self-adjoint, positive definite operator on a complex Hilbert space H, and (α,β)∈[0,1]×[0,1], which is considered in Ammar-Khodja et al. (1999), and after, in Hao et al. (2013). Our contribution is to identify a fine scale of polynomial stability of the solution in the region S3:=(α,β)∈[0,1]×[0,1];β<2α−1 taking into account the presence of a singularity at zero.

Stability and Stabilization of Polynomial Fuzzy-Model-Based Switched Nonlinear Systems Under MDADT Switching Signal

This article proposes a novel Lyapunov stability analysis for a class of switched nonlinear systems under the mode-dependent average dwell time (MDADT) switching signals. For the first time, a polynomial fuzzy model is used to describe the nonlinearity of the switched systems instead of a Takagi–Sugeno fuzzy model, which can represent a wider range of nonlinear plants. As existing advanced results are relatively conservative and lead to that tighter bounds on the dwell time are not easy to be obtained, therefore, the stability analysis for switched systems is still quite challenging. In this article, to tackle this problem, by developing a polynomial multiple Lyapunov function (MLF) approach and exploring the feature of the membership functions, relaxed stability conditions are derived for the switched polynomial fuzzy-model-based control systems. In particular, the polynomial MLF is guaranteed radially unbounded by a two-step procedure, making traditional quadratic MLF a specific case, thereby improved stability conditions can be established in the form of sum-of-squares. Moreover, the new algorithm for multivariate Chebyshev membership functions based on linear programming is proposed to introduce the information of membership functions and further relax the stability conditions to obtain tighter bounds on MDADT. Finally, a simulation example demonstrates the effectiveness of the developed techniques.

Stability for a 3D Ladyzhenskaya fluid model with unbounded variable delay

&lt;abstract&gt;&lt;p&gt;This paper is concerned with the stability of solutions to a Ladyzhenskaya fluid model with unbounded variable delay. We first prove the existence, uniqueness and regularity of global weak solutions to the Ladyzhenskaya model by using Galerkin approximations and the energy method based on some suitable assumptions about external forces. Then we obtain that the stationary solution is locally stable. Finally, we establish that the stationary solution has polynomial stability in a particular case of unbounded variable delay.&lt;/p&gt;&lt;/abstract&gt;

Robustness of polynomial stability with respect to sampling

We provide a partially affirmative answer to the following question on robustness of polynomial stability with respect to sampling: “Suppose that a continuous-time state-feedback controller achieves the polynomial stability of the infinite-dimensional linear system. We apply an idealized sampler and a zero-order hold to a feedback loop around the controller. Then, is the sampled-data system strongly stable for all sufficiently small sampling periods? Furthermore, is the polynomial decay of the continuous-time system transferred to the sampled-data system under sufficiently fast sampling?” The generator of the open-loop system is assumed to be a Riesz-spectral operator whose eigenvalues are not on the imaginary axis but may approach it asymptotically. We provide conditions for strong stability to be preserved under fast sampling. Moreover, we estimate the decay rate of the state of the sampled-data system with a smooth initial state and a sufficiently small sampling period.