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

Abstract

A forward problem for the Dirac system is to find $u=\begin{pmatrix}u_1(x,t)\\u_2(x,t)\end{pmatrix}$ obeying $iu_t+\begin{pmatrix}0&1\\-1&0\end{pmatrix}u_x+\begin{pmatrix}p&q\\q&-p\end{pmatrix}u=0$ for $x>0,\,t>0$;\,\,$u(x,0)=\begin{pmatrix}0\\0\end{pmatrix}$ for $x {\geqslant} 0 $, and $u_1(0,t)=f(t)$ for $t>0$, with the real $p=p(x), q=q(x)$. An input--output map $R: u_1(0,\cdot)\mapsto u_2(0,\cdot)$ is of the convolution form $Rf=if+r\ast f$, where $r=r(t)$ is a {\it response function}. By hyperbolicity of the system, for any $T>0$, function $r\big|_{0 {\leqslant} t {\leqslant} 2T}$ is determined by $p,q\big|_{0 {\leqslant} x {\leqslant} T}$. An inverse problem is: for an (arbitrary) fixed $T>0$, given $r\big|_{0 {\leqslant} t {\leqslant} 2T}$ to recover $p,q\big|_{0 {\leqslant} x {\leqslant} T}$. The procedure that determines $p,q$ 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. It is the fact, which enables one to apply the boundary control method for solving the inverse problem.