Dynamic inverse problem for the one-dimensional system with memory

TL;DR

This paper addresses the inverse problem of recovering the potential in a one-dimensional dynamical system with memory, using Gelfand-Levitan equations to reconstruct the potential from system data.

Contribution

It introduces a method to reconstruct the potential in a 1D system with memory via Gelfand-Levitan equations, advancing inverse problem techniques.

Findings

Derived Gelfand-Levitan equations for the kernel of the inverse control operator.

Proposed a reconstruction method for the potential from the integral equations.

Established a theoretical framework for inverse problems in systems with memory.

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.