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Vilmos Komornik

Publications and source records attributed to Vilmos Komornik.

At least 19 recordsLinked to original sources
arXiv2026

The semilinear suspension bridge model with pointwise dissipation

This paper is devoted to the analysis of a semilinear suspension bridge model with pointwise localized dissipation. The main contribution of the work is the development of a robust semigroup framework that substantially simplifies the stability analysis of coupled systems with localized damping. The core of the approach relies on a comparison principle involving the essential growth bound of the associated semigroups together with the invariance of the essential spectrum under compact perturbations. This methodology allows us to reduce the stability analysis of the full coupled system to that of suitably chosen auxiliary systems, thereby avoiding the delicate multiplier techniques and observability arguments commonly used in the literature. As a first consequence, we establish the exponential stability of the corresponding linear homogeneous system. The semilinear problem is then treated as a locally Lipschitz perturbation of the exponentially stable linear semigroup, yielding global well-posedness and exponential decay of solutions. Finally, we investigate the long-time dynamics of the semilinear system. By combining the dissipative structure of the model with suitable regularization properties of the nonlinear semigroup, we prove the existence of a global compact attractor and an exponential attractor with finite fractal dimension. These results provide a complete description of the asymptotic behavior of solutions and show that the proposed framework is sufficiently flexible to be adapted to other coupled systems with localized dissipation.

math.AP↗
arXiv2025

A qualitative study of the generalized dispersive systems with time-delay: The unbounded case

We study the asymptotic behavior of the solutions of the time-delayed higher-order dispersive nonlinear differential equation \begin{equation*} u_t(x,t)+Au(x,t) +λ_0(x) u(x,t)+λ(x) u(x,t-τ)=0 \end{equation*} where \begin{equation*} Au=(-1)^{j+1}\partial_x^{2j+1}u+(-1)^m\partial_x^{2m}u+ \frac{1}{p+1}\partial_xu^{p+1} \end{equation*} with $m\le j$ and $1\le p<2j$. Under suitable assumptions on the time delay coefficients, we prove that the system is exponentially stable if the coefficient of the delay term is bounded from below by a suitable positive constant, without any assumption on the sign of the coefficient of the undelayed feedback. Additionally, in the absence of delay, general results of stabilization are established in $H^s(\mathbb{R})$ for $s\in[0,2j+1]$. Our results generalize several previous theorems for the Korteweg-de Vries type delayed systems in the literature.

math.AP↗
arXiv2025

Numerical stabilization for a mixture system with kind damping

In this paper, we conduct a numerical analysis of the strong stabilization and polynomial decay of solutions for the initial boundary value problem associated with a system that models the dynamics of a mixture of two rigid solids with porosity. This mathematical model accounts for the complex interactions between the rigid components and their porous structure, providing valuable information on the mechanical behavior of such systems. Our primary objective is to establish conditions under which stabilization is ensured and to rigorously quantify the rate of decay of the solutions. Using numerical simulations, we assess the effectiveness of different stabilization mechanisms and analyze the influence of key system parameters on the overall dynamics.

math.NA↗
arXiv2025

Numerical study of a Transmission Problem in Elasticity with kind damping

We investigate a transmission problem featuring a specific type of damping. Our primary focus is on analyzing the asymptotic behavior of the associated semigroup, $({\mathcal S}_{\mathcal A}(t))_{t\geq 0}$. We demonstrate that this semigroup exhibits a polynomial rate of decay towards zero when the initial data is taken over the domain ${\mathcal D}({\mathcal A})$. Furthermore, we establish that this decay rate is optimal. To support our theoretical findings, we present a comprehensive numerical study that validates and illustrates the sharpness of the obtained decay rates.

math.AP↗
arXiv2025

Topology of univoque sets in double-base expansions

Given two real numbers $q_0,q_1>1$ satisfying $q_0+q_1\geq q_0q_1$ and two real numbers $d_0\ne d_1$, by a {double-base expansion} of a real number $x$ we mean a sequence $(i_k)\in \{0,1\}^{\infty}$ such that \begin{equation*} x=\sum_{k=1}^{\infty}\frac{d_{i_k}}{q_{i_1}q_{i_2}\cdots q_{i_k}}. \end{equation*} We denote by $\mathcal{U}_{q_0,q_1}$ the set of numbers $x$ having a unique expansion. The topological properties of $\mathcal{U}_{q_0,q_1}$ have been investigated in the equal-base case $q_0=q_1$ for a long time. We extend this research to the case $q_0\neq q_1$. While many results remain valid, a great number of new phenomena appear due to the increased complexity of double-base expansions.

math.DS↗
arXiv2025

Observability of the linear Zakharov--Kuznetsov equation

We study the linear Zakharov--Kuznetsov equation with periodic boundary conditions. Employing some tools from the nonharmonic Fourier series we obtain several internal observability theorems. Then we prove various exact controllability and rapid uniform stabilization results by applying a duality principle and a general feedback construction. The method presented here introduces a new insight into the control of dispersive equations in two-dimensional cases and may be adapted to more general equations.

math.AP↗
arXiv2024

Stability of the Rao-Nakra sandwich beam with a dissipation of fractional derivative type: theoretical and numerical study

This paper is devoted to the solution and stability of a one-dimensional model depicting Rao--Nakra sandwich beams, incorporating damping terms characterized by fractional derivative types within the domain, specifically a generalized Caputo derivative with exponential weight. To address existence, uniqueness, stability, and numerical results, fractional derivatives are substituted by diffusion equations relative to a new independent variable, $ξ$, resulting in an augmented model with a dissipative semigroup operator. Polynomial decay of energy is achieved, with a decay rate depending on the fractional derivative parameters. Both the polynomial decay and its dependency on the parameters of the generalized Caputo derivative are numerically validated. To this end, an energy-conserving finite difference numerical scheme is employed.

math.NA↗
arXiv2024

Unique double base expansions

For two real bases $q_0, q_1 > 1$, we consider expansions of real numbers of the form $\sum_{k=1}^{\infty} i_k/(q_{i_1}q_{i_2}\cdots q_{i_k})$ with $i_k \in \{0,1\}$, which we call $(q_0,q_1)$-expansions. A sequence $(i_k)$ is called a unique $(q_0,q_1)$-expansion if all other sequences have different values as $(q_0,q_1)$-expansions, and the set of unique $(q_0,q_1)$-expansions is denoted by $U_{q_0,q_1}$. In the special case $q_0 = q_1 = q$, the set $U_{q,q}$ is trivial if $q$ is below the golden ratio and uncountable if $q$ is above the Komornik--Loreti constant. The curve separating pairs of bases $(q_0, q_1)$ with trivial $U_{q_0,q_1}$ from those with non-trivial $U_{q_0,q_1}$ is the graph of a function $\mathcal{G}(q_0)$ that we call generalized golden ratio. Similarly, the curve separating pairs $(q_0, q_1)$ with countable $U_{q_0,q_1}$ from those with uncountable $U_{q_0,q_1}$ is the graph of a function $\mathcal{K}(q_0)$ that we call generalized Komornik--Loreti constant. We show that the two curves are symmetric in $q_0$ and $q_1$, that $\mathcal{G}$ and $\mathcal{K}$ are continuous, strictly decreasing, hence almost everywhere differentiable on $(1,\infty)$, and that the Hausdorff dimension of the set of $q_0$ satisfying $\mathcal{G}(q_0)=\mathcal{K}(q_0)$ is zero. We give formulas for $\mathcal{G}(q_0)$ and $\mathcal{K}(q_0)$ for all $q_0 > 1$, using characterizations of when a binary subshift avoiding a lexicographic interval is trivial, countable, uncountable with zero entropy and uncountable with positive entropy respectively. Our characterizations in terms of $S$-adic sequences including Sturmian and the Thue--Morse sequences are simpler than those of Labarca and Moreira (2006) and Glendinning and Sidorov (2015), and are relevant also for other open dynamical systems.

math.NT↗
arXiv2022

Topology of univoque sets in real base expansions

Given a positive integer $M$ and a real number $q \in (1,M+1]$, an expansion of a real number $x \in \left[0,M/(q-1)\right]$ over the alphabet $A=\{0,1,\ldots,M\}$ is a sequence $(c_i) \in A^{\mathbb N}$ such that $x=\sum_{i=1}^{\infty}c_iq^{-i}$. Generalizing many earlier results, we investigate in this paper the topological properties of the set $U_q$ consisting of numbers $x$ having a unique expansion of this form, and the combinatorial properties of the set $U_q'$ consisting of their corresponding expansions. We also provide shorter proofs of the main results of Baker in [B] by adapting the method given in [EJK] for the case $M=1$.

math.CO↗
arXiv2021

Fibonacci Expansions

Expansions in the Golden ratio base have been studied since a pioneering paper of Rényi more than sixty years ago. We introduce closely related expansions of a new type, based on the Fibonacci sequence, and we show that in some sense they behave better.

math.NT↗
arXiv2021

Expansions in multiple bases over general alphabets

Expansions in non-integer bases have been extensively investigated since a pioneering work of Rényi. We introduce a more general framework of alphabet-base systems that also includes Pedicini's general alphabets and the multiple-base expansions of Neunhäuserer and Li. We extend the Parry type lexicographic theory to this setup, and we improve and generalize various former results on unique expansions.

math.CO↗
arXiv2020

Difference of Cantor sets and frequencies in Thue--Morse type sequences

In a recent paper, Baker and Kong have studied the Hausdorff dimension of the intersection of Cantor sets with their translations. We extend their results to more general Cantor sets. The proofs rely on the frequencies of digits in unique expansions in non-integer bases. In relation with this, we introduce a practical method to determine the frequency of any given finite block in Thue--Morse type sequences.

math.NT↗
arXiv2020

Hausdorff distance of univoque sets

Expansions in non-integer bases have been investigated abundantly since their introduction by Rényi. It was discovered by Erdős et al. that the sets of numbers with a unique expansion have a much more complex structure than in the integer base case. The present paper is devoted to the continuity properties of these maps with respect to the Hausdorff metric.

math.NT↗
arXiv2019

Univoque graphs and multiple expansions

Unique expansions in non-integer bases $q$ have been investigated in many papers during the last thirty years. They are often conveniently generated by labeled directed graphs. In the first part of this paper we give a precise description of the set of sequences generated by these graphs. Using the description of univoque graphs, the second part of the paper is devoted to the study of multiple expansions. Contrary to the unique expansions, we prove for each $j\ge 2$ that the set $U_q^j$ of numbers having exactly $j$ expansions is closed only if it is empty. Furthermore, generalizing an important example of Sidorov, we prove for a large class of bases that the Hausdorff dimension of $U_q^j$ is independent of $j$. In the last two sections our results are illustrated by many examples.

math.NT↗
arXiv2019

Pointwise control of the linearized Gear-Grimshaw system

In this paper we consider the problem of controlling pointwise, by means of a time dependent Dirac measure supported by a given point, a coupled system of two Korteweg-de Vries equations on the unit circle. More precisely, by means of spectral analysis and Fourier expansion we prove, under general assumptions on the physical parameters of the system, a pointwise observability inequality which leads to the pointwise controllability when we observe two control functions. In addition, with a uniqueness property proved for the linearized system without control, we are also able to show pointwise controllability when only one control function acts internally. In both cases we can find, under some assumptions on the coefficients of the system, the sharp time of the controllability.

math.AP↗
arXiv2019

Moving and oblique observations of beams and plates

We study the observability of the one-dimensional Schr{ö}dinger equation and of the beam and plate equations by moving or oblique observations. Applying different versions and adaptations of Ingham's theorem on nonharmonic Fourier series, we obtain various observability and non-observability theorems. Several open problems are also formulated at the end of the paper.

math.CA↗
arXiv2019

Energy decay for evolution equations with delay feedbacks

We study abstract linear and nonlinear evolutionary systems with single or multiple delay feedbacks, illustrated by several concrete examples. In particular, we assume that the operator associated with the undelayed part of the system generates an exponentially stable semigroup and that the delay damping coefficients are locally integrable in time. A step by step procedure combined with Gronwall's inequality allows us to prove the existence and uniqueness of solutions. Furthermore, under appropriate conditions we obtain exponential decay estimates.

math.AP↗