Strong Stability Result Research Articles

Strong Stability for a Viscoelastic Transmission Problem under a Nonlocal Boundary Control

The purpose of this paper is to consider a transmission problem of a viscoelastic wave with nonlocal boundary control. It should be noted that the present paper is based on the previous C. G. Gal and M. Warma works, together with H. Atoui and A. Benaissa. Namely, they focused on a transmission problem consisting of a semilinear parabolic equation in a general non-smooth setting with an emphasis on rough interfaces and nonlinear dynamic (possibly, nonlocal) boundary conditions along the interface, where a transmission problem in the presence of a boundary control condition of a nonlocal type was investigated in these papers. Owing to the semigroup theory, we prove the question of well-posedness. For the very rare cases, we combined between the frequency domain approach and the Borichev–Tomilov theorem to establish strong stability results.

Stability and Data Dependence Results for Jungck-type Iteration Scheme

We propose and study a new Jungck-type iteration scheme to approximate coincidence points of contractive mappings. The strong convergence, stability, and data dependency results have been discussed. Numerical experiments demonstrate that the newly introduced Jungck-type iteration scheme yields a higher convergence rate in comparison with other Jungck-type iteration schemes available in the literature.

On multi-species diffusion with size exclusion

We revisit a classical continuum model for the diffusion of multiple species with size-exclusion constraint, which leads to a degenerate nonlinear cross-diffusion system. The purpose of this article is twofold: first, it aims at a systematic study of the question of existence of weak solutions and their long-time asymptotic behaviour. Second, it provides a weak–strong stability estimate for a wide range of coefficients, which had been missing so far.In order to achieve the results mentioned above, we exploit the formal gradient-flow structure of the model with respect to a logarithmic entropy, which leads to best estimates in the full-interaction case, where all cross-diffusion coefficients are non-zero. Those are crucial to obtain the minimal Sobolev regularity needed for a weak–strong stability result. For meaningful cases when some of the coefficients vanish, we provide a novel existence result based on approximation by the full-interaction case.

On the stability of mean-field stochastic differential equations with irregular expectation functional

In this paper, we consider multidimensional mean-field stochastic differential equations where the coefficients depend on the law in the form of a Lebesgue integral with respect to the measure of the solution. Under the pathwise uniqueness property, we establish various strong stability results. As a consequence, we give an application for optimal control of diffusions. Namely, we propose a result on the approximation of the solution associated to a relaxed control.

Exponential stability of a mixture of two elastic solids with a degenerate weak damping mechanism

In this note, we investigate stability issues for a mixture of two elastic solids. The governing equations consist of two coupled systems of elastodynamic equations with a weak damping. The damping involves the difference of velocities, so that the damping operator is degenerate. First, we show that the underlying semigroup is strongly stable under a weaker parameters constraint, thereby improving an earlier strong stability result of Dafermos. Next, we show that under extra appropriate constraints on the parameters of the coupled system, the semigroup is exponentially stable. Our proof relies on the frequency domain method, the multipliers technique, and a seemingly new vector calculus identity (Lemma B). Our exponential stability result is new in the framework of mixtures involving elastodynamic equations with weak damping. We also show that for some sets of parameters, the system is asymptotically unstable; in particular, we use a concrete example to show that strong stability fails for a certain parameters constraint.

On the stability of a laminated Timoshenko problem with interfacial slip in the whole space under frictional dampings or infinite memories

Abstract The author of the present paper considered in [16] a model describing a vibrating strucure of an interfacial slip and consists of three coupled hyperbolic equations in one-dimensional bounded interval, where the dissipation is generated by either a frictional damping or an infinite memory, and it is acting only on one component. Some strong, polynomial, exponential and non exponential stability results were proved in [16] depending on the values of the parameters and the regularity of the initial data. The objective of the present paper is to compelete the study of [16] by considering this model in the whole line ℝ and under only one control given by a frictional damping or an infinite memory. When the system is controled via its second or third component (rotation angle displacement or dynamic of the slip), we show that this control alone is sufficient to stabilize our system and get different polynomial stability estimates in the L 2-norm of the solution and its higher order derivatives with respect to the space variable. The decay rate depends on the regularity of the initial data, the nature of the control and the parameters in the system. However, when the system is controled via its first component (transversal displacement), we found a new stability condition depending on the parameters in the system. This condition defines a limit between the stability and instability of the system in the sense that, when this condition is staisfied, the system is polynomially stable. Otherwise, when this condition is not satisfied, we prove that the solution does not converge to zero at all. The proofs are based on the energy method and Fourier analysis combined with judicious choices of weight functions.

Uniqueness of strong solutions and weak–strong stability in a system of cross-diffusion equations

Proving the uniqueness of solutions to multi-species cross-diffusion systems is a difficult task in the general case, and there exist very few results in this direction. In this work, we study a particular system with zero-flux boundary conditions for which the existence of a weak solution has been proven in Ehrlacher and Bakhta (ESAIM Math Model Numer Anal, 2017). Under additional assumptions on the value of the cross-diffusion coefficients, we are able to show the existence and uniqueness of non-negative strong solutions. The proof of the existence relies on the use of an appropriate linearized problem and a fixed-point argument. In addition, a weak–strong stability result is obtained for this system in dimension one which also implies uniqueness of weak solutions

A fairly strong stability result for parabolic quasiminimizers

AbstractIn this paper we consider parabolic Q‐quasiminimizers related to the p‐Laplace equation in . In particular, we focus on the stability problem with respect to the parameters p and Q. It is known that, if , then parabolic quasiminimizers with fixed initial‐boundary data on converge to the parabolic minimizer strongly in under suitable further structural assumptions. Our concern is whether or not we can obtain even stronger convergence. We will show a fairly strong stability result.

A degree version of the Hilton–Milner theorem

An intersecting family of sets is trivial if all of its members share a common element. Hilton and Milner proved a strong stability result for the celebrated Erdős–Ko–Rado theorem: when n>2k, every non-trivial intersecting family of k-subsets of [n] has at most (n−1k−1)−(n−k−1k−1)+1 members. One extremal family HMn,k consists of a k-set S and all k-subsets of [n] containing a fixed element x∉S and at least one element of S. We prove a degree version of the Hilton–Milner theorem: if n=Ω(k2) and F is a non-trivial intersecting family of k-subsets of [n], then δ(F)≤δ(HMn.k), where δ(F) denotes the minimum (vertex) degree of F. Our proof uses several fundamental results in extremal set theory, the concept of kernels, and a new variant of the Erdős–Ko–Rado theorem.

Infinite loop spaces from operads with homological stability

Motivated by the operad built from moduli spaces of Riemann surfaces, we consider a general class of operads in the category of spaces that satisfy certain homological stability conditions. We prove that such operads are infinite loop space operads in the sense that the group completions of their algebras are infinite loop spaces.The recent, strong homological stability results of Galatius and Randal-Williams for moduli spaces of even dimensional manifolds can be used to construct examples of operads with homological stability. As a consequence diffeomorphism groups and mapping class groups are shown to give rise to infinite loop spaces. Furthermore, the map to K-theory defined by the action of the diffeomorphisms on the middle dimensional homology is shown to be a map of infinite loop spaces.

Repeated Columns and an Old Chestnut

Let $t\ge 1$ be a given integer. Let ${\cal F}$ be a family of subsets of $[m]=\{1,2,\ldots ,m\}$. Assume that for every pair of disjoint sets $S,T\subset [m]$ with $|S|=|T|=k$, there do not exist $2t$ sets in ${\cal F}$ where $t$ subsets of ${\cal F}$ contain $S$ and are disjoint from $T$ and $t$ subsets of ${\cal F}$ contain $T$ and are disjoint from $S$. We show that $|{\cal F}|$ is $O(m^{k})$. Our main new ingredient is allowing, during the inductive proof, multisets of subsets of $[m]$ where the multiplicity of a given set is bounded by $t-1$. We use a strong stability result of Anstee and Keevash. This is further evidence for a conjecture of Anstee and Sali. These problems can be stated in the language of matrices. Let $t\cdot M$ denote $t$ copies of the matrix $M$ concatenated together. We have established the conjecture for those configurations $t\cdot F$ for any $k\times 2$ (0,1)-matrix $F$.

On Convergence and Stability of the Generalized Noor Iterations for a General Class of Operators

In this paper, we establish some strong convergence and stability results of multistep iterative scheme for a general class of operators introduced by Bosede and Rhoades [5] in a Banach space. As corollaries, some convergence and stability results for the Noor, Ishikawa, Mann and Picard iterative schemes are also established. Our convergence results generalize and extend the results of Berinde [3], Bosede [4], Olaleru [16], Rafiq [21, 22] among others, while our stability results are extensions and generalizations of multitude of results in the literature, including the results of Berinde [1], Bosede and Rhoades [5], Imoru and Olatinwo [9] and Osilike [18].

Robust Stability and Constrained Stabilization of Discrete-Time Delay Systems

This paper revisits the problem of robust stability and stabilization of uncertain time-delay systems. We focus on the class of non-negative discrete-time delay systems and show that it is asymptotically stable if and only if an associated non-negative system without delay is asymptotically stable*. This fact allows one to establish strong result on robust stability and stability radius for this class of systems. An alternative representation of delay systems is also constructed whereby its system matrix is in block companion from. Under the assumption of non-negativity for delay systems, this alternative form represents a conventional non-negative system and similar strong robust stability results are derived. Finally, we consider the problem of constrained stabilization and provide a new LMI feasibility solution for it. This makes it possible to stabilize a general discrete-time delay system such that the closed-loop system admits non-negative structure with desirable properties.

Determination of the insolation function in the nonlinear Sellers climate model

We are interested in the climate model introduced by Sellers in 1969 which takes the form of some nonlinear parabolic equation with a degenerate diffusion coefficient. We investigate here some inverse problem issue that consists in recovering the so-called insolation function. We not only solve the uniqueness question but also provide some strong stability result, more precisely unconditional Lipschitz stability in the spirit of the well-known result by Imanuvilov and Yamamoto (1998) [22]. The main novelties rely in the fact that the considered model is degenerate and above all nonlinear. Indeed we provide here one of the first result of Lipschitz stability in a nonlinear case.

D-Stability and Delay-Independent Stability of Homogeneous Cooperative Systems

We introduce a nonlinear definition of D-stability, extending the usual concept for positive linear time-invariant systems. We show that globally asymptotically stable, cooperative systems, homogeneous of any order with respect to arbitrary dilation maps are D-stable. We also prove a strong stability result for delayed cooperative homogeneous systems. Finally, we show that both of these results also hold for planar cooperative systems without the restriction of homogeneity.

Semigroup well-posedness in the energy space of a parabolic-hyperbolic coupled Stokes-Lamé PDE system of fluid-structure interaction

This paper is focused on an established, genuinely physical fluid-structure interaction model, herein the structure is immersed in a fluid with coupling taking place at the boundary interface between the two media. Mathematically, the model is a coupled parabolic-hyperbolic system of two partial differential equations in 3-d with nonstandard coupling at the boundary interface. Fluid and structure are mathematically expressed by the (dynamic) Stokes system (parabolic) and the Lame system (hyperbolic), respectively. The main claim presented is a contraction semigroup well-posedness result on the natural space of finite energy. There are two main features in the analysis: (i) a nonstandard elimination of the pressure term, as the boundary coupling between fluid and structure rules out application of the classical Leray/Helmoltz projection; (ii) a nonstandard usage of the Babuska-Brezzi inf-sup theory to assert maximal dissipativity of the candidate generator. A unified treatment includes both undamped and (perhaps, partially) damped boundary conditions at the interface. With the generator explicitly at hand, an analysis of its point spectrum on the imaginary axis is also included. In the undamped case, it depends on the geometry of the structure. In the case of full boundary damping, it implies as a by-product a strong stability result for the solutions by soft methods.

Dynamic behavior of a heat equation with memory

AbstractThis paper addresses the spectrum‐determined growth condition for a heat equation with exponential polynomial kernel memory. By introducing some new variables, the time‐variant system is transformed into a time‐invariant one. The detailed spectral analysis is presented. It is shown that the system demonstrates the property of hyperbolic equation that all eigenvalues approach a line that is parallel to the imaginary axis. The residual spectral set is shown to be empty and the set of continuous spectrum is exactly characterized. The main result is the spectrum‐determined growth condition that is one of the most difficult problems for infinite‐dimensional systems. Consequently, a strong exponential stability result is concluded. Copyright © 2008 John Wiley & Sons, Ltd.

Keynesian AD-AS Vadis?

We formulate a dynamic AD-AS model based on gradually adjusting wages and prices, perfect foresight of current inflation rates and adaptive expectations concerning the inflation climate in which the economy operates. The model consists of a wage and a price Phillips curve, a dynamic IS curve as well as a dynamic employment adjustment equation (Okun's law) and a Taylor interest rate rule. The model can be reduced to a 3D dynamical system by a suitable choice of the Taylor rule and implies strong stability results, in particular for an appropriately chosen interest rate policy rule. Through instrumental variables GMM system estimation with aggregate time series data for the U.K. economy, we obtain parameter estimates which support the specification of our theoretical model and its stability implications. We contrast these results with the standard (formally similarly structured) New Keynesian model with staggered wage and price setting where determinacy of the dynamics represents a severe problem and where (if determinacy can be achieved) inertia-free stability is obtained by the very choice of the solution method.

Stability and genericity for spde's driven by spatially correlated noise

We consider stochastic partial differential equations, on $\mathbb{R}^{d}$ $(d\geq 1)$, driven by a Gaussian noise white in time and colored in space. Assuming pathwise uniqueness holds, we establish various strong stability results. As consequence, we give an application to the convergence of the Picard successive approximation. Finally, we show that in the sense of Baire category, almost all stochastic partial differential equations with continuous and bounded coefficients have the properties of existence and pathwise uniqueness of solutions as well as the continuous dependence on the coefficients.

Uniform stabilization of a coupled PDE system arising in fluid-structure interaction with boundary dissipation at the interface

We consider a coupled parabolic-hyperbolic PDE system arising in fluid-structure interaction, with boundary dissipation at the interface, $d = 2,3$. Such a system is semigroup well-posed in the natural energy space [4]. We then establish that it is also uniformly (exponentially) stable, thus complementing the strong stability results of the undamped case [2], [4].