arXiv subjects

Sergei Avdonin

Publications and source records attributed to Sergei Avdonin.

10 recordsLinked to original sources

arXiv2026

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

Let $Δ$ be the Neumann Laplacian on the interval $(0,π)$, and let $T>0$. An integration by parts formula is proven for the spectral fractional Laplacian, $(-Δ)^α$, for $α\in (0,1)$. As an application, we prove well-posedness results for the structurally damped beam equation $$u_{tt}+Δ^2 u+ρ(-Δ)^αu_t=0, x\in (0,π),t>0$$ with various boundary conditions including $$ u_x(0,t)=u_{xxx}(0,t)=0;\ u_x(π,t)=f(t),\ u_{xxx}(π,t)=0, $$ 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 $Δ$ be the Dirichlet Laplacian on a bounded domain $Ω\subset \mathbb{R}^{N}$, and let $(-Δ)^α$ be the associated spectral fractional Laplacian with $α\leq 1, \ ρ<2$. For general bounded domains with $C^2$ boundary, we prove a symmetry formula for $α<1/2$, extending a result previously proven on rectangles for $α<1$. As a consequence of this formula, well-posedness results are proven for the structurally damped plate equation $$u_{tt}+Δ^2u+(-Δ)^αu_t=0$$ subject to Dirichlet or moment boundary control. For rectangular domains with $α<1$, we prove boundary null-controllability results. For $α<1/2, \ ρ\leq 2$, Dirichlet null controllability is proved for the unit disk in $\mathbb{R}^2$. This analysis then extended to the classical case, $α=1$, 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 $Δ$ be the Dirichlet Laplacian on the interval $(0,π)$, and let $T>0$. We prove a well-posedness results for the structurally damped beam equation $$u_{tt}+Δ^2 u-ρΔu_t=0, x\in (0,π),t>0$$ with various boundary conditions including $$ u(0,t)=u_{xx}(0,t)=0; u(π,t)=f(t),u_{xx}(π,t)=0, $$ and $f\in H_0^2(0,T)$ and appropriate initial conditions. Viewing $f$ as a control, we prove null controllability for all $ρ\leq 2$. For $ρ>2$, we show null controllability for arbitrary $T>0$ holds for almost all $ρ$, but fails for a dense subset of $(2,\infty)$. An analagous result is proven for Neumann control.

math.OC↗

arXiv2026

An inverse problem for semilinear wave equations on metric tree graphs

We study the inverse problem for a semilinear wave equation on metric tree graphs. From the Dirichlet-to-Neumann map defined at all but one of the boundary vertices, we recover unknown connectivity of the graph, lengths of the edges, the time-independent potential and the time-dependent coefficient of the nonlinear term of the equation.

math.AP↗

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 Schrödinger type operator $L=-Δ+q(x)$ with Dirichlet boundary conditions at the two boundary nodes. Let $\{ ω_n^2, \ φ_n(x)\}$ 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: $\{ ω_n^2,\partial_x φ_n(v_1),\partial_xφ_n(v_2)\}.$ Here $\partial_xφ_n(v_2)$ 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↗

arXiv2025

Inverse Problem for the Schrödinger Equation with Non-self-adjoint Matrix Potential

We consider the dynamical system with boundary control for the vector Schrödinger equation on the interval with a non-self-adjoint matrix potential. For this system, we study the inverse problem of recovering the matrix potential from the dynamical Dirichlet--to--Neumann operator. We first provide a method to recover spectral data for an abstract system from dynamic data and apply it to the Schrödinger equation. We then develop a strategy for solving the inverse problem for the Schrödinger equation using this method with other techniques of the Boundary control method.

math.AP↗

arXiv2024

Inverse Dynamic Problem for the Dirac System on Finite Metric Tree Graphs and the Leaf Peeling Method

In this paper, we consider the inverse dynamic problem for the Dirac system on finite metric tree graphs. Our main goal is to recover the topology (connectivity) of a tree, lengths of edges, and a matrix potential function on each edge. We use the dynamic response operator as our inverse data and apply the Leaf peeling method. In addition, we present a new dynamic algorithm to solve the forward problem for the Dirac system on general finite metric graphs.

math.AP↗

arXiv2024

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

Let $Δ$ be the Dirichlet Laplacian on the interval $(0,π)$. The null controllability properties of the equation $$u_{tt}+Δ^2 u+ρ(Δ)^αu_t=F(x,t)$$ are studied. Let $T>0$, and assume initial conditions $(u^0,u^1)\in Dom(Δ)\times L^2(0,π)$. 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 $α\in [0,3/2)$, we prove that there exist $h^1,h^2\in L^2(0,π)$ such that for any $(u^0,u^1)$, there exist $L^2$ null controls $(f^1,f^2).$ For $α< 1$ and $ρ<2$, we prove null controllability with $f^2=0$ and $h^1$ belonging to a large class of functions. For $α\in [3/2,2)$, we prove spectral and null controllability both generally fail, but two dimensional weak controllability holds. Our second set of results pertains to $F(x,t)=χ_Ω(x)f(x,t)$, with $Ω$ any open subset of $(0,π)$. For any $α\in [0,3/2),$ we prove there exists a null control $f\in L^2(Ω\times(0,T))$ 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↗