arXiv subjects

Julian Edward

Publications and source records attributed to Julian Edward.

14 recordsLinked to original sources

arXiv2026

Boundary null controllability of weakly coupled structurally damped beams

We prove boundary null controllability in any positive time for a finite system of Euler--Bernoulli beams with possibly non-diagonalizable coupling. For $0<\rho<2$, a Fourier--Jordan reduction yields a vector moment problem with polynomial--exponential modes. Under the Fattorini--Hautus condition, global spectral non-collision, and nonvanishing modal transfer factors, a suitably estimated block-biorthogonal family produces an $H^2$ boundary control. Geometric multiplicities of the coupling matrix determine the minimal control dimension, and Jordan-block lengths determine the degrees of the generalized moments. At $\rho=2$, an exact heat-system reduction gives the same result for $A=A^*\geq0$, despite the time-differential linkage of the reduced boundary inputs.

math.OC↗

arXiv2026

Null-controllability for the beam equation with structural damping. Part 2: Integration by parts for fractional Laplacians and boundary control

Let $\Delta$ be the Neumann Laplacian on the interval $(0,\pi)$, and let $T>0$. An integration by parts formula is proven for the spectral fractional Laplacian, $(-\Delta)^\alpha$, for $\alpha \in (0,1)$. As an application, we prove well-posedness results for the structurally damped beam equation $$u_{tt}+\Delta^2 u+\rho (-\Delta)^\alpha u_t=0, x\in (0,\pi),t>0$$ with various boundary conditions including $$ u_x(0,t)=u_{xxx}(0,t)=0;\ u_x(\pi,t)=f(t),\ u_{xxx}(\pi,t)=0, $$ and $f\in L^2(0,T)$ and appropriate initial conditions. Viewing $f$ as a control, we prove null-controllability. Analagous results are proven for higher order controls, and for the Dirichlet Laplacian.

math.AP↗

arXiv2026

A symmetry formula for the spectral fractional Laplacian, and applications to boundary controllability for plate equation with structural damping

Let $\Delta$ be the Dirichlet Laplacian on a bounded domain $\Omega \subset \mathbb{R}^{N}$, and let $(-\Delta)^\alpha$ be the associated spectral fractional Laplacian with $\alpha \leq 1, \ \rho <2$. For general bounded domains with $C^2$ boundary, we prove a symmetry formula for $\alpha <1/2$, extending a result previously proven on rectangles for $\alpha <1$. As a consequence of this formula, well-posedness results are proven for the structurally damped plate equation $$u_{tt}+\Delta^2u+(-\Delta)^\alpha u_t=0$$ subject to Dirichlet or moment boundary control. For rectangular domains with $\alpha <1$, we prove boundary null-controllability results. For $\alpha <1/2, \ \rho \leq 2$, Dirichlet null controllability is proved for the unit disk in $\mathbb{R}^2$. This analysis then extended to the classical case, $\alpha =1$, on rectangles, where higher regularity is required for Dirichlet control.

math.AP↗

arXiv2026

Boundary null-controllability for the beam equation with classical structural damping

Let $\Delta$ be the Dirichlet Laplacian on the interval $(0,\pi)$, and let $T>0$. We prove a well-posedness results for the structurally damped beam equation $$u_{tt}+\Delta^2 u-\rho \Delta u_t=0, x\in (0,\pi),t>0$$ with various boundary conditions including $$ u(0,t)=u_{xx}(0,t)=0; u(\pi,t)=f(t),u_{xx}(\pi,t)=0, $$ and $f\in H_0^2(0,T)$ and appropriate initial conditions. Viewing $f$ as a control, we prove null controllability for all $\rho \leq 2$. For $\rho >2$, we show null controllability for arbitrary $T>0$ holds for almost all $\rho$, but fails for a dense subset of $(2,\infty)$. An analagous result is proven for Neumann control.

math.OC↗

arXiv2025

An inverse problem on a metric graph with cycle

Consider a quantum graph consisting of a ring with two attached edges, and assume Kirchhoff-Neumann conditions hold at the internal vertices. Associated to this graph is a Schr\"{o}dinger type operator $L=-\Delta +q(x)$ with Dirichlet boundary conditions at the two boundary nodes. Let $\{ \omega_n^2, \ \varphi_n(x)\}$ be the eigenvalues and associated normalized eigenfunctions. Let $v_1$ be a boundary vertex, and $v_2$ the adjacent internal vertex. Assume we know the following data: $\{ \omega_n^2,\partial_x \varphi_n(v_1),\partial_x\varphi_n(v_2)\}.$ Here $\partial_x\varphi_n(v_2)$ refers to an outward normal derivative at $v_2$ along one of the edges incident to the other internal vertex. From this data we determine the following unknown quantities: the lengths of edges and the potential functions on each edge.

math.AP↗

arXiv2024

Null-controllability for the beam equation with structural damping. Part 1. Distributed control

Let $\Delta$ be the Dirichlet Laplacian on the interval $(0,\pi)$. The null controllability properties of the equation $$u_{tt}+\Delta^2 u+\rho (\Delta)^\alpha u_t=F(x,t)$$ are studied. Let $T>0$, and assume initial conditions $(u^0,u^1)\in Dom(\Delta)\times L^2(0,\pi)$. We first prove finite dimensional null control results: suppose $F(x,t)=f^1(t)h^1(x)+f^2(t)h^2(x)$ with $h^1,h^2$ given functions. For $\alpha \in [0,3/2)$, we prove that there exist $h^1,h^2\in L^2(0,\pi)$ such that for any $(u^0,u^1)$, there exist $L^2$ null controls $(f^1,f^2).$ For $\alpha< 1$ and $\rho <2$, we prove null controllability with $f^2=0$ and $h^1$ belonging to a large class of functions. For $\alpha\in [3/2,2)$, we prove spectral and null controllability both generally fail, but two dimensional weak controllability holds. Our second set of results pertains to $F(x,t)=\chi_\Omega(x)f(x,t)$, with $\Omega$ any open subset of $(0,\pi)$. For any $\alpha \in [0,3/2),$ we prove there exists a null control $f\in L^2(\Omega\times(0,T))$ To prove our main results, we use the Fourier method to rewrite the control problems as moment problems. These are then solved by constructing biorthogonal sets to the associated exponential families. These constructions seem to be non-standard and may be of independent interest.

math.OC↗

arXiv2022

Shape, Velocity, and Exact Controllability for the Wave Equation on a Graph with Cycle

Exact controllability is proven on a graph with cycle. The controls can be a mix of controls applied at the boundary and interior vertices. The method of proof first uses a dynamical argument to prove shape controllability and velocity controllability, thereby solving their associated moment problems. This enables one to solve the moment problem associated to exact controllability. In the case of a single control, either boundary or interior, it is shown that exact controllability fails.

math.OC↗

arXiv2022

Exact Controllability for the Wave Equation on a Graph with Cycle and Delta-Prime Vertex Conditions

Exact controllability for the wave equation on a metric graph consisting of a cycle and two attached edges is proven. One boundary and one internal control are used. At the internal vertices, delta-prime conditions are satisfied. As a second example, we examine a tripod controlled at the root and the junction, while the leaves are fixed. These examples are key to understanding controllability properties in general metric graphs.

math.OC↗

arXiv2019

Regularity of solutions of quasi-linear elliptic equations with $L\log^m L$ coefficients

Let $D$ be an bounded region in ${\bf R}^n$. The regularity of solutions of a family of quasilinear elliptic partial differential equations is studied, one example being $Δ_nu=Vu^{n-1}$. The coefficients are assumed to be in the space $L\log^{m}L(D)$ for $m>n-1$. Using a Moser iteration argument coupled with the Moser-Trudinger inequality, a local $L^{\infty}$ bound on the solution $u$ is proven. A Harnack-type inequality is then proven. These results are shown to be sharp with respect to $m$. Then essential continuity of $u$ is proven, and away from the boundary a bound on the modulus of continuity.

math.AP↗

arXiv2018

Riesz bases from orthonormal bases by replacement

Given an orthonormal basis $ {\mathcal V}= \{v_j\} _{j\in N}$ in a separable Hilbert space $H$ and a set of unit vectors $ {\mathcal B}=\{w_j\}_{j\in N}$, we consider the sets $ {\mathcal B}_N$ obtained by replacing the vectors $v_1, ...,\, v_N$ with vectors $w_1,\, ...,\, w_N$. We show necessary and sufficient conditions that ensure that the sets $ {\mathcal B}_N$ are Riesz bases of $H$ and we estimate the frame constants of the $ {\mathcal B}_N$. Then, we prove conditions that ensure that $ {\mathcal B}$ is a Riesz basis. Applications to the construction of exponential bases on domains of $ R^d$ are also presented.

math.FA↗

arXiv2013

Existence problems for the $p$-Laplacian

We consider a number of boundary value problems involving the $p$-Laplacian. The model case is $-Δ_p u=V|u|^{p-2}u$ for $u\in W_0^{1,2}(D)$ with $D$ a bounded domain in ${\bf R}^n$. We derive necessary conditions for the existence of nontrivial solutions. These conditions usually involve a lower bound for a product of powers of the norm of $V$, the measure of $D$, and a sharp Sobolev constant. In most cases, these inequalities are best possible. Applications to non-linear eigenvalue problems are also discussed.

math.AP↗

arXiv2013

Minimal support results for Schrödinger equations

We consider a number of linear and non-linear boundary value problems involving generalized Schrödinger equations. The model case is $-Δu=Vu$ for $u\in W_0^{1,2}(D)$ with $D$ a bounded domain in ${\bf R^n}$. We use the Sobolev embedding theorem, and in some cases the Moser-Trudinger inequality and the Hardy-Sobolev inequality, to derive necessary conditions for the existence of nontrivial solutions. These conditions usually involve a lower bound for a product of powers of the norm of $V$, the measure of $D$, and a sharp Sobolev constant. In most cases, these inequalities are best possible.

math.AP↗

arXiv2002

Trapped modes for periodic structures in waveguides

The Laplace operator is considered for waveguides perturbed by a periodic structure consisting of N congruent obstacles spanning the waveguide. Neumann boundary conditions are imposed on the periodic structure, and either Neumann or Dirichlet conditions on the guide walls. It is proven that there are at least N (resp. N-1) trapped modes in the Neumann case (resp. Dirichlet case) under fairly general hypotheses, including the special case where the obstacles consist of line segments placed parallel to the waveguide walls. This work should be viewed as an extension of "Periodic structures on waveguides" by Linton and McIvor.

math-ph↗

arXiv2001

On the resonances of the Laplacian on waveguides

The resonances for the Dirichlet and Neumann Laplacian are studied on compactly perturbed waveguides. An upper bound on the number of resonances near the physical plane is proven. In the absence of resonances, an upper bound is proven for the localised resolvent. This is then used to prove that the existence of a quasimode whose asymptotics is bounded away from the thresholds implies the existence of resonances converging to the real axis.

math-ph↗