Polynomial Stability Research Articles

Polynomial stability of the homology of Hurwitz spaces

Polynomial stability of the homology of Hurwitz spaces

Long-time behaviors of wave equations stabilized by boundary memory damping and friction damping

In this paper, we study the long-time behaviors of wave equations subject to boundary memory damping and friction damping. Different from assumptions that memory kernel is a nonnegative, monotone function in the previous literatures, we assume that the primitive of the memory kernel is a generalized positive definite kernel (abbreviated to GPDK), which may vary sign or oscillate. The key to the problem lies in establishing the connection between memory damping and energy terms. By combining the properties of the positive definite kernel with classical multiplier methods, and constructing auxiliary systems, we ultimately establish the asymptotic stability, exponential stability and polynomial stability of systems featuring boundary memory damping and friction damping. To illustrate our theoretical results, we provide some numerical simulations.

Some Combined Results from Eneström–Kakeya and Rouché Theorems on the Generalized Schur Stability of Polynomials and the Stability of Quasi-Polynomials-Application to Time-Delay Systems

This paper derives some generalized Schur-type stability results of polynomials based on several forms and generalizations of the Eneström–Kakeya theorem combined with the Rouché theorem. It is first investigated, under sufficiency-type conditions, the derivation of the eventually generalized Schur stability sufficient conditions which are not necessarily related to the zeros of the polynomial lying in the unit open circle. In a second step, further sufficient conditions were introduced to guarantee that the above generalized Schur stability property persists within either the same above complex nominal stability region or in some larger one. The classical weak and, respectively, strong Schur stability in the closed and, respectively, open complex unit circle centred at zero are particular cases of their corresponding generalized versions. Some of the obtained and proved results are further generalized “ad hoc” for the case of quasi-polynomials whose zeros might be interpreted, in some typical cases, as characteristic zeros of linear continuous-time delayed time-invariant dynamic systems with commensurate constant point delays.

Third order two-step Runge–Kutta–Chebyshev methods

The well-known high order stabilized codes (such as DUMKA and ROCK) have several drawbacks: numerically obtained stability polynomials (which do not have a closed analytic form), poor internal stability and convergence. RKC-type methods have much better computational properties. However, these types of methods currently have a second order maximum. In this paper, a family of third order stabilized methods with an explicit analytical solution of stability polynomials is presented. This was made possible by usage of two-step Runge–Kutta methods. A new code TSRKC3 is proposed, illustrated by several examples, and compared to existing programs.

Positivity and semi-global polynomial stability of high-order Cohen–Grossberg BAM neural networks with multiple proportional delays

In this paper, we study positivity and semi-global polynomial stability (PS) of higher-order Cohen-Grossberg BAM neural networks with multiple proportional time delays. The proportional delays considered here are unbounded and time-varying, differing from constant, bounded, and distributional time delays. The system model cannot be represented using vector and matrices, making certain approaches within the vector-matrix framework unsuitable for applying. To address this limitation, a direct method based on the solution of the system is proposed to provide sufficient conditions guaranteeing the positivity and semi-global polynomial stability (PS) of the model under consideration. Furthermore, the direct method is applied to establish global PS conditions for BAM neural networks with multiple proportional delays. The obtained conditions contain only a few simple linear scalar inequalities that are easily solved. The applicability of the obtained PS conditions is verified by two numerical examples, and the solution of a linear programming problem is also obtained based on these theoretical results. Notably, this method can be applied to many system models with proportional delays after minor modifications.

A memory-type thermoelastic laminated beam with structural damping and microtemperature effects: Well-posedness and general decay

Abstract In previous work, Fayssal considered a thermoelastic laminated beam with structural damping, where the heat conduction is given by the classical Fourier’s law and acting on both the rotation angle and the transverse displacements established an exponential stability result for the considered problem in case of equal wave speeds and a polynomial stability for the opposite case. This article deals with a laminated beam system along with structural damping, past history, and the presence of both temperatures and microtemperature effects. Employing the semigroup approach, we establish the existence and uniqueness of the solution. With the help of convenient assumptions on the kernel, we demonstrate a general decay result for the solution of the considered system, with no constraints regarding the speeds of wave propagations. The result obtained is new and substantially improves earlier results in the literature.

Stability analysis of polynomials with an approach of differential topology

SummaryIn this paper, we study the Hurwitz stability of polynomials. By considering that a degree Hurwitz polynomial has its corresponding Markov parameters, we define the set in Section 3, we also define . Based on properties of the Hankel matrices and the stability test, as well as by using ideas of differential topology, we show that is a fiber bundle with a base. This result allows us to obtain an interesting application: Given a Hurwitz polynomial, we can generate two families of positive definite Hankel matrices.

Optimal decay rate for a Rayleigh beam–string coupled system with frictional damping

This paper addresses the stability of a coupled system that consists of a conservative Rayleigh beam and a damped elastic string, the latter being equipped with frictional damping. By estimating the growth of the resolvent along the imaginary axis, we employ the frequency domain method to demonstrate that the coupled Rayleigh beam–string system exhibits polynomial stability, characterized by a decay rate t−1/4. Furthermore, we establish the optimality of the decay rate, t−1/4, through a rigorous spectral analysis. Finally, a numerical example is present to validate the theoretical findings.

Stability analysis of homogeneous cooperative positive differential systems with time-varying delays and its generalization

This article is devoted to studying the asymptotic behavior of differential systems with bounded delays. We first focus on the analysis of homogeneous cooperative positive systems. Under the assumption that the vector field is homogeneous of a degree greater than 1, we show that the non-trivial solutions of the system converge to the origin at a polynomial rate. In the case when the degree of homogeneity equals 1, we prove that the solutions will decay at an exponential rate. As a generalization of these results, we consider nonlinear non-autonomous differential systems with time-varying delays that are bounded above by stable homogeneous positive systems. By some additional imposed conditions, in light of the comparison principle, we obtain the locally exponential stability and polynomial stability of the equilibrium point to these systems. Finally, specific examples and discussions are provided to illustrate the validity of the proposed theoretical results.

An output feedback method to polynomial stabilization of hybrid pantograph stochastic systems with aperiodic sampling

This paper focuses on the polynomial stabilization problem of hybrid pantograph stochastic systems (HPSSs) with aperiodic sampling in the presence of randomly occurring transmission delay. Firstly, a sufficient condition is provided to ensure the existence, uniqueness and boundedness of solution to such systems. Next, taking advantage of the Lyapunov stability theory and the comparison method, the criteria of pth moment polynomial stability are established for the HPSSs under the feedback control and aperiodic intermittent control strategy, respectively. Finally, the proposed theoretical results of two control methods are simultaneously validated through a numerical example.

Polynomial stability of transmission system for coupled Kirchhoff plates

Polynomial stability of transmission system for coupled Kirchhoff plates

Indirect stabilization of locally weakly coupled second‐order evolution systems of hyperbolic type

The purpose of this work is to investigate the stabilization of locally weakly coupled second‐order evolution equations of hyperbolic type, where only one of the two equations is directly damped. As such system cannot be exponentially stable, we are interested in polynomial energy decay rates. Our main contributions concern strong abstract and polynomial stability properties based on stability properties of two auxiliary problems: the sole damped equation and the second equation with a damping related to the coupling operator. The main novelty is that the polynomial energy decay rates are obtained in different important situations not covered before; let us mention the case of a coupling operator not partially coercive and not necessarily bounded. The main tools are the use of the frequency domain approach combined with new multipliers technique based on the solutions of the resolvent equations of the two auxiliary problems mentioned before. Finally, our abstract framework is illustrated by several concrete examples not treated before.

Convergence and Almost Sure Polynomial Stability of Partially Truncated Split-Step Theta Method for Stochastic Pantograph Models with Lévy Jumps

This paper addresses a stochastic pantograph model with Lévy leaps where non-jump coefficients exceed linearity. The partially truncated split-step theta method is introduced and applied to the proposed model. The finite time Lϱ^(ϱ^≥2) convergence rate of the numerical scheme is obtained. Furthermore, the almost sure polynomial stability of the numerical scheme is investigated and numerical examples are presented to endorse the addressed theorems.

Multirate time-integration based on dynamic ODE partitioning through adaptively refined meshes for compressible fluid dynamics

In this paper, we apply the Paired-Explicit Runge-Kutta (P-ERK) schemes by Vermeire et al. [1,2] to dynamically partitioned systems arising from adaptive mesh refinement. The P-ERK schemes enable multirate time-integration with no changes in the spatial discretization methodology, making them readily implementable in existing codes that employ a method-of-lines approach.We show that speedup compared to a range of state of the art Runge-Kutta methods can be realized, despite additional overhead due to the dynamic re-assignment of flagging variables and restricting nonlinear stability properties. The effectiveness of the approach is demonstrated for a range of simulation setups for viscous and inviscid convection-dominated compressible flows for which we provide a reproducibility repository.In addition, we perform a thorough investigation of the nonlinear stability properties of the Paired-Explicit Runge-Kutta schemes regarding limitations due to the violation of monotonicity properties of the underlying spatial discretization. Furthermore, we present a novel approach for estimating the relevant eigenvalues of large Jacobians required for the optimization of stability polynomials.

Operator-valued multiplier theorems for causal translation-invariant operators with applications to control theoretic input-output stability

AbstractWe prove an operator-valued Laplace multiplier theorem for causal translation-invariant linear operators which provides a characterization of continuity from $$H^\alpha ({\mathbb {R}},U)$$ H α ( R , U ) to $$H^\beta ({\mathbb {R}},U)$$ H β ( R , U ) (fractional U-valued Sobolev spaces, U a complex Hilbert space) in terms of a certain boundedness property of the transfer function (or symbol), an operator-valued holomorphic function on the right-half of the complex plane. We identify sufficient conditions under which this boundedness property is equivalent to a similar property of the boundary function of the transfer function. Under the assumption that U is separable, the Laplace multiplier theorem is used to derive a Fourier multiplier theorem. We provide an application to mathematical control theory, by developing a novel input-output stability framework for a large class of causal translation-invariant linear operators which refines existing input-output stability theories. Furthermore, we show how our work is linked to the theory of well-posed linear systems and to results on polynomial stability of operator semigroups. Several examples are discussed in some detail.

Polynomial decay of a linear system of PDEs via Caputo fractional‐time derivative

An in‐depth study and analysis of the stability of one‐dimensional Linear System of PDEs via Caputo time fractional derivative (LSCFD) was presented. We proved some stability results for the LSCFD in different Hilbert spaces. Indeed, by using Fourier analysis method and the properties of the Mittag–Leffler Function (MLF), some polynomial stability results for LSCFD have been established. Finally, as an application, we used finite difference methods well suited to integer and fractional order derivatives, and performed some numerical experiments to confirm the theoretical results.

Polynomial stability for Lord–Shulman porous elasticity with microtemperature and strong time delay

AbstractIn this paper, we study a porous thermoelastic system with microtemperature and strong time delay acting on the volume fraction equation. The thermal effect of microtemperature is based on the Lord–Shulman theory (J Mech Phys Solids. 15(5) (1967), 299–309.), while the strong delay is motivated by Makheloufi's et al. recent work (Math Meth Appl Sci. 44 (2021), 6301–6317.). To prove the well‐posedness of the system, lack of exponential stability and the polynomial decay with optimal rate, we use the semigroup theory of linear operators.

Polynomial stabilization of the wave equation with a time varying delay term in the dynamical control

We consider the one-dimensional wave equation with a time-varying delay term in the dynamical control. Under suitable assumptions, we show the well posedness of the problem. These results are obtained by using semi-group theory. Combining the multiplier method with a non linear integral inequality, a rational energy decay result of the system is established. A fundamental aspect of this paper is that our involved operator is time dependent, therefore the standard frequential method cannot be invoked.

Robust Stability and Control of Fractional Polynomials Including Integer and Fractional Natural Exponential Functions

Robust Stability and Control of Fractional Polynomials Including Integer and Fractional Natural Exponential Functions

Polynomial stability of nonlinear Timoshenko system with distributed delay-time

Polynomial stability of nonlinear Timoshenko system with distributed delay-time