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Hoai-Minh Nguyen

Publications and source records attributed to Hoai-Minh Nguyen.

At least 37 records · Page 2Linked to original sources
arXiv2021

The completeness of the generalized eigenfunctions and an upper bound for the counting function of the transmission eigenvalue problem for Maxwell equations

Cakoni and Nguyen recently proposed very general conditions on the coefficients of Maxwell equations for which they established the discreteness of the set of eigenvalues of the transmission problem and studied their locations. In this paper, we establish the completeness of the generalized eigenfunctions and derive an optimal upper bound for the counting function under these conditions, assuming additionally that the coefficients are twice continuously differentiable. The approach is based on the spectral theory of Hilbert-Schmidt operators.

math.AP↗
arXiv2021

On the optimal controllability time for linear hyperbolic systems with time-dependent coefficients

The optimal time for the controllability of linear hyperbolic systems in one dimensional space with one-side controls has been obtained recently for time-independent coefficients in our previous works. In this paper, we consider linear hyperbolic systems with time-varying zero-order terms. We show the possibility that the optimal time for the null-controllability becomes significantly larger than the one of the time-invariant setting even when the zero-order term is indefinitely differentiable. When the analyticity with respect to time is imposed for the zero-order term, we also establish that the optimal time is the same as in the time-independent setting.

math.OC↗
arXiv2021

A dependence of the cost of fast controls for the heat equation on the support of initial datum

The controllability cost for the heat equation as the control time $T$ goes to 0 is well-known of the order $e^{C/T}$ for some positive constant $C$, depending on the controlled domain and for all initial datum. In this paper, we prove that the constant $C$ can be chosen to be arbitrarily small if the support of the initial data is sufficiently close to the controlled domain, but not necessary inside the controlled domain. The proof is in the spirit on Lebeau and Robbiano's approach in which a new spectral inequality is established. The main ingredient of the proof of the new spectral inequality is three-sphere inequalities with partial data.

math.OC↗
arXiv2020

Decay for the nonlinear KdV equations at critical lengths

We consider the nonlinear Korteweg-de Vries (KdV) equation in a bounded interval equipped with the Dirichlet boundary condition and the Neumann boundary condition on the right. It is known that there is a set of critical lengths for which the solutions of the linearized system conserve the $L^2$-norm if their initial data belong to a finite dimensional subspace $\M$. In this paper, we show that all solutions of the nonlinear KdV system decay to 0 at least with the rate $1/ t^{1/2}$ when $\dim \M = 1$ or when $\dim \M$ is even and a specific condition is satisfied, provided that their initial data is sufficiently small. Our analysis is inspired by the power series expansion approach and involves the theory of quasi-periodic functions. As a consequence, we rediscover known results which were previously established for $\dim \M = 1$ or for the smallest critical length $L$ with $\dim \M = 2$ by a different approach using the center manifold theory, and obtain new results. We also show that the decay rate is not slower than $\ln (t + 2) / t$ for all critical lengths.

math.AP↗
arXiv2020

Null-controllability, exact controllability, and stabilization of hyperbolic systems for the optimal time

In this paper, we discuss our recent works on the null-controllability, the exact controllability, and the stabilization of linear hyperbolic systems in one dimensional space using boundary controls on one side for the optimal time. Under precise and generic assumptions on the boundary conditions on the other side, we first obtain the optimal time for the null and the exact controllability for these systems for a generic source term. We then prove the null-controllability and the exact controllability for any time greater than the optimal time and for any source term. Finally, for homogeneous systems, we design feedbacks which stabilize the systems and bring them to the zero state at the optimal time. Extensions for the non-linear homogeneous system are also discussed

math.OC↗
arXiv2020

Cloaking property of a plasmonic structure in doubly complementary media and three-sphere inequalities with partial data

We investigate cloaking property of negative-index metamaterials in the time-harmonic electromagnetic setting for the so-called doubly complementary media. These are media consisting of negative-index metamaterials in a shell (plasmonic structure) and positive-index materials in its complement for which the shell is complementary to a part of the core and a part of the exterior of the core-shell structure. We show that an arbitrary object is invisible when it is placed close to a plasmonic structure of a doubly complementary medium as long as its cross section is smaller than a threshold given by the property of the plasmonic structure. To handle the loss of the compactness and of the ellipticity of the modeling Maxwell equations with sign-changing coefficients, we first obtain Cauchy's problems associated with two Maxwell systems using reflections. We then derive information from them, and combine it with the removing localized singularity technique to deal with the localized resonance. A central part of the analysis on the Cauchy's problems is to establish three-sphere inequalities with partial data for general elliptic systems, which are interesting in themselves. The proof of these inequalities first relies on an appropriate change of variables, inspired by conformal maps, and is then based on Carleman's estimates for a class of degenerate elliptic systems.

math.AP↗
arXiv2020

On the small-time local controllability of a KdV system for critical lengths

This paper is devoted to the local null-controllability of the nonlinear KdV equation equipped the Dirichlet boundary conditions using the Neumann boundary control on the right. Rosier proved that this KdV system is small-time locally controllable for all non-critical lengths and that the uncontrollable space of the linearized system is of finite dimension when the length is critical. Concerning critical lengths, Coron and Crépeau showed that the same result holds when the uncontrollable space of the linearized system is of dimension 1, and later Cerpa, and then Cerpa and Crépeau established that the local controllability holds at a finite time for all other critical lengths. In this paper, we prove that, for a class of critical lengths, the nonlinear KdV system is {\it not} small-time locally controllable.

math.AP↗
arXiv2020

The Weyl law of transmission eigenvalues and the completeness of generalized transmission eigenfunctions

The transmission problem is a system of two second-order elliptic equations of two unknowns equipped with the Cauchy data on the boundary. After four decades of research motivated by scattering theory, the spectral properties of this problem are now known to depend on a type of contrast between coefficients near the boundary. Previously, we established the discreteness of eigenvalues for a large class of anisotropic coefficients which is related to the celebrated complementing conditions due to Agmon, Douglis, and Nirenberg. In this work, we establish the Weyl law for the eigenvalues and the completeness of the generalized eigenfunctions for this class of coefficients under an additional mild assumption on the continuity of the coefficients. The analysis is new and based on the $L^p$ regularity theory for the transmission problem established here. It also involves a subtle application of the spectral theory for the Hilbert Schmidt operators. Our work extends largely known results in the literature which are mainly devoted to the isotropic case with $C^\infty$-coefficients.

math.AP↗
arXiv2020

Lyapunov functions and finite time stabilization in optimal time for homogeneous linear and quasilinear hyperbolic systems

Hyperbolic systems in one dimensional space are frequently used in modeling of many physical systems. In our recent works, we introduced time independent feedbacks leading to the finite stabilization for the optimal time of homogeneous linear and quasilinear hyperbolic systems. In this work, we present Lyapunov's functions for these feedbacks and use estimates for Lyapunov's functions to rediscover the finite stabilization results.

math.OC↗
arXiv2020

Finite-time stabilization in optimal time of homogeneous quasilinear hyperbolic systems in one dimensional space

We consider the finite-time stabilization of homogeneous quasilinear hyperbolic systems with one side controls and with nonlinear boundary condition at the other side. We present time-independent feedbacks leading to the finite-time stabilization in any time larger than the optimal time for the null controllability of the linearized system if the initial condition is sufficiently small. One of the key technical points is to establish the local well-posedness of quasilinear hyperbolic systems with nonlinear, non-local boundary conditions.

math.OC↗
arXiv2020

On the Discreteness of Transmission Eigenvalues for the Maxwell Equations

In this paper, we establish the discreteness of transmission eigenvalues for Maxwell's equations. More precisely, we show that the spectrum of the transmission eigenvalue problem is discrete, if the electromagnetic parameters $\eps, \, μ, \, \heps, \, \hmu$ in the equations characterizing the inhomogeneity and background, are smooth in some neighborhood of the boundary, isotropic on the boundary, and satisfy the conditions $\eps \neq \heps$, $μ\neq \hmu$, and $\eps/ μ\neq \heps/ \hmu$ on the boundary. These are quite general assumptions on the coefficients which are easy to check. To our knowledge, our paper is the first to establish discreteness of transmission eigenvalues for Maxwell's equations without assuming any restrictions on the sign combination of the contrasts $\eps-\heps$ and $μ- \hmu$ near the boundary, and allowing for all the electromagnetic parameters to be inhomogeneous and anisotropic, except for on the boundary where they are isotropic but not necessarily constant as it is often assumed in the literature.

math.AP↗
arXiv2019

Null-controllability of linear hyperbolic systems in one dimensional space

This paper is devoted to the controllability of a general linear hyperbolic system in one space dimension using boundary controls on one side. Under precise and generic assumptions on the boundary conditions on the other side, we previously established the optimal time for the null and the exact controllability for this system for a generic source term. In this work, we prove the null-controllability for any time greater than the optimal time and for any source term. Similar results for the exact controllability are also discussed.

math.OC↗
arXiv2019

Non-local, non-convex functionals converging to Sobolev norms

We study the pointwise convergence and the $Γ$-convergence of a family of non-local, non-convex functionals $Λ_δ$ in $L^p(Ω)$ for $p>1$. We show that the limits are multiples of $\int_Ω |\nabla u|^p$. This is a continuation of our previous work where the case $p=1$ was considered.

math.CA↗
arXiv2019

$Γ$-convergence of non-local, non-convex functionals in one dimension

We study the $Γ$-convergence of a family of non-local, non-convex functionals in $L^p(I)$ for $p \ge 1$, where $I$ is an open interval. We show that the limit is a multiple of the $W^{1, p}(I)$ semi-norm to the power $p$ when $p>1$ (resp. the $BV(I)$ semi-norm when $p=1$). In dimension one, this extends earlier results which required a monotonicity condition.

math.CA↗
arXiv2019

Exponential integrability in the spirit of Moser-Trudinger's inequalities of functions with finite non-local, non-convex energy

Let $d \ge 1$, $p \ge d$, and let $Ω$ be a smooth bounded open subset of $\mathbb{R}^d$. We prove some exponential integrability in the spirit of Moser-Trudinger's inequalities for measurable functions $u$ defined in $Ω$ such that $$ \mathop{\int_Ω \int_Ω}_{|u(x) - u(y)| > δ} \frac{1}{|x-y|^{d+p}} \, dx \, dy < + \infty, $$ for some $δ> 0$. This double integral appeared in characterizations of Sobolev spaces and involved in improvements of the Sobolev inequaliies, Poincaré inequalities, and Hardy inequalities.

math.FA↗
arXiv2019

The invisibility via anomalous localized resonance of a source for electromagnetic waves

We study the invisibility via anomalous localized resonance of a general source for electromagnetic waves in the setting of doubly complementary media. As a result, we show that cloaking is achieved if the power is blown up. We also reveal a critical length for the invisibility of a source that occurs when the plasmonic structure is complementary to an annulus of constant, isotropic medium.

math.AP↗
arXiv2019

Characterization of the traces on the boundary of functions in magnetic Sobolev spaces

We characterize the trace of magnetic Sobolev spaces defined in a half-space or in a smooth bounded domain in which the magnetic field $A$ is differentiable and its exterior derivative corresponding to the magnetic field $dA$ is bounded. In particular, we prove that, for $d \ge 1$ and $p>1$, the trace of the magnetic Sobolev space $W^{1, p}_A(\mathbb{R}^{d+1}_+)$ is exactly $W^{1-1/p, p}_{A^{\shortparallel}}(\mathbb{R}^d)$ where $A^{\shortparallel}(x) =( A_1, \dotsc, A_d)(x, 0)$ for $x \in \mathbb{R}^d$ with the convention $A = (A_1, \dotsc, A_{d+1})$ when $A \in C^1(\overline{\mathbb{R}^{d+1}_+}, \mathbb{R}^{d+1})$. We also characterize fractional magnetic Sobolev spaces as interpolation spaces and give extension theorems from a half-space to the entire space.

math.FA↗