arXiv subjects

A. F. Vakulenko

Publications and source records attributed to A. F. Vakulenko.

4 recordsLinked to original sources

arXiv2025

On stability of triangular factorization of positive operators

Let $\mathfrak f=\{\mathscr F_s\}_{s>0}$ be a nest and $C$ a bounded positive operator in a Hilbert space $\mathscr F$. The representation $C=V^*V$ provided $V\mathscr F_s\subset\mathscr F_s$ is a triangular factorization (TF) of $C$ w.r.t. $\mathfrak f$. The factorization is stable if $C^α\underset{α\to\infty}\to C$ and $C^α=V^{α\,*}V^α$ implies $V^α\to V$. If $C$ is positive definite (isomorphism), then TF is stable. The paper deals with the case of positive but not positive definite $C$. We impose some assumptions on $C^α$ and $C$ which provide the stability of TF.

math.FA↗

arXiv2024

Three-dimensional inverse acoustic scattering problem by the BC-method

Let $Σ:=[0,\infty)\times S^2$, $\mathscr F:=L_2(Σ)$. The {\it forward} acoustic scattering problem under consideration is to find $u=u^f(x,t)$ satisfying \begin{align} \label{Eq 01} &u_{tt}-Δu+qu=0, && (x,t) \in {\mathbb R}^3 \times (-\infty,\infty); \\ \label{Eq 02} &u \mid_{|x|<-t} =0 , && t<0;\\ \label{Eq 03} &\lim_{s \to -\infty} s\,u((-s+τ)\,ω,s)=f(τ,ω), && (τ,ω) \in Σ; \end{align} for a real valued compactly supported potential $q\in L_\infty(\Bbb R^3)$ and a control $f \in\mathscr F$. The response operator $R: \mathscr F\to\mathscr F$, \begin{align*} & (Rf)(τ,ω)\,:= \lim_{s \to +\infty} s\, u^f((s+τ)\,ω,s), \quad (τ,ω) \in Σ\end{align*} depends on $q$ {\it locally}: if $ξ>0$ and $f\in\mathscr F^ξ:=\{f\in\mathscr F\,|\,\,\,f\!\mid_{[0,ξ)}=0\}$ holds, then the values $(Rf)\!\mid_{τ\geqslantξ}$ are determined by $q\!\mid_{|x|\geqslantξ}$ (do not depend on $q\!\mid_{|x|<ξ}$). The {\it inverse problem} is: for an arbitrarily fixed $ξ>0$, to determine $q\mid_{|x|\geqslantξ}$ from $X^ξR\upharpoonright\mathscr F^ξ$, where $X^ξ$ is the projection in $\mathscr F$ onto $\mathscr F^ξ$. It is solved by a relevant version of the boundary control method. The key point of the approach are recent results on the controllability of the system (\ref{Eq 01})--(\ref{Eq 03}).

math.AP↗

arXiv2024

On the M.Kac problem with augmented data

Let $Ω$ be a bounded plane domain. As is known, the spectrum $0<λ_1<λ_2\leqslant\dots$ of its Dirichlet Laplacian $L=-Δ{\upharpoonright}[H^2(Ω)\cap H^1_0(Ω)]$ does not determine $Ω$ (up to isometry). By this, a reasonable version of the M.Kac problem is to augment the spectrum with relevant data that provide the determination. To give the spectrum is to represent $L$ in the form $\tilde L=ΦLΦ^*={\rm diag\,}\{λ_1,λ_2,\dots\}$ in the space ${\bf l}_2$, where $Φ:L_2(Ω)\to{\bf l}_2$ is the Fourier transform. Let ${\mathscr K}=\{h\in L_2(Ω)\,|\,\,Δh=0\,\,{\rm in}\,\,Ω\}$ be the harmonic function subspace, $\tilde{\mathscr K}=Φ{\mathscr K}\subset{\bf l}_2$. We show that, in a generic case, the pair $\tilde L,\tilde {\mathscr K}$ determines $Ω$ up to isometry, what holds not only for the plain domains (drums) but for the compact Riemannian manifolds of arbitrary dimension, metric, and topology. Thus, the subspace $\tilde{\mathscr K}\subset{\bf l}_2$ augments the spectrum, making the problem uniquely solvable.

math-ph↗

arXiv2013

Non-smooth unobservable states in control problem for the wave equation in ${\mathbb R}^3$ (corrected)

The paper deals with a dynamical system \begin{align*} &u_{tt}-Δu=0, \qquad (x,t) \in {\mathbb R}^3 \times (-\infty,0) \\ &u \mid_{|x|<-t} =0 , \qquad t<0\\ &\lim_{s \to \infty} su((s+τ)ω,-s)=f(τ,ω), \qquad (τ,ω) \in [0,\infty)\times S^2\,, \end{align*} where $u=u^f(x,t)$ is a solution ({\it wave}), $f \in {\cal F} :=L_2\left([0,\infty);L_2\left(S^2\right)\right)$ is a {\it control}. For the reachable sets ${\cal U}^ξ:=\{u^f(\cdot, -ξ)\,|\,\, f \in {\cal F}\}\,\,(ξ\geqslant 0)$, the embedding ${\cal U}^ξ\subset {\cal H}^ξ:=\{y \in L_2({\mathbb R}^3)\,|\,\,\,y|_{|x|<ξ}=0\}$ holds, whereas the subspaces ${\cal D}^ξ:={\cal H}^ξ\ominus {\cal U}^ξ$ of unreachable ({\it unobservable}) states are nonzero for $ξ> 0$. There was a conjecture motivated by some geometrical optics arguments that the elements of ${\cal D}^ξ$ are $C^\infty$-smooth with respect to $|x|$. We provide rather unexpected counterexamples of $h\in {\cal D}^ξ$ with ${\rm sing\,supp\,}h \subset \{x\in{\mathbb R}^3|\,\,|x|=ξ_0>ξ\}$.

math-ph↗