Inverse problem for a one-dimensional dynamical Dirac system (BC-method)

Abstract

A forward problem for the Dirac system is to find obeying for ; for , and for , with the real . An input–output map is of the convolution form , where is a response function. By hyperbolicity of the system, for any , function is determined by . An inverse problem is the following: for an (arbitrary) fixed , given to recover . The procedure that determines is proposed, and the characteristic solvability conditions on r are provided. Our approach is purely time domain and is based on studying the controllability properties of the Dirac system. In itself the system is not controllable: the local completeness of states does not hold, but its relevant extension gains controllability. This is the fact that enables one to apply the boundary control method for solving the inverse problem.

Evolution of Weyl Functions and Initial-Boundary Value Problems

“…According to [14,Theorem 1], in the case where p, q, f (in (6.2) and in (6.3)) are continuously differentiable and f (0) = f ′ (0) = 0, there is a unique classical solution Y of (6.1), (6.3) and this solution admits representation…”

“…Representation (6.9)-(6.11) is proved in [14] using Duhamel formula. Let us also assume that V, f and f ′ are bounded:…”

“…Classical Dirac systems (2.1) are also called spectral Dirac systems and various new results on inverse problems for these systems appeared quite recently (see some references in Subsection 2.1). At the same time, a great and growing interest in dynamical systems and control methods is reflected in the active study of the dynamical inverse problems and, in particular, in the study of the inverse problems for dynamical Schrödinger and Dirac systems [7,8,13,14,33,42] (see also the references therein). Dynamical Dirac system (Dirac system in the time-domain setup) was studied in the important recent paper [14].…”

“…Here f is a complex-valued function (called boundary control by [14]) and the input-output map (response operator) R :…”

“…( Y stands in this section for the Fourier transformation of Y , see (6.16).) The inverse problem consists in recovery of the potential V from the response function r. This inverse problem was considered in [14] using boundary control methods.…”