Dynamic inverse problem for the one-dimensional system with memory

TL;DR

This work tackles the inverse dynamic problem of determining the potential $q(x)$ in a 1D system with memory driven by a boundary input, assuming a known relaxation kernel $K$. It develops a purely dynamic Boundary Control Method framework, deriving a Duhamel-type representation and introducing operators such as the control operator $W^T$, the connecting operator $C^T$, and the Blagoveschenskii function to connect inverse data $R^{2T}$ to system operators. The Gelfand–Levitan equations are then obtained through operator identities, yielding a kernel equation for $z(x,s)$ and the reconstruction formula $q(x)=2 rac{d}{dx} z(x,x)$. The approach extends beyond spectral methods to general potentials in $L^1_{loc}$ and memory kernels in $L^2_{loc}$, providing a path to characterize inverse data from boundary measurements.

Abstract

We study the inverse dynamic problem of recoverying the potential in the one-dimensional dynamical system with memory. The Gelfand--Levitan equations are derived for the kernel of the integral operator which is inverse to the control operator of the system. The potential is reconstructed from the solution of these equations.

Theorems & Definitions (7)

• Lemma 1
• proof
• Proposition 1
• proof
• Proposition 2
• proof
• Theorem 1