Characterization of the inverse problem data for one-dimensional two-velocity dynamical system

Abstract:

The evolution of the dynamical system in question is described by the wave equation $\rho u_{tt}-(\gamma u_{x}) _{x}+Au_{x}+Bu=0$, $x>0$, $t>0$, with the zero Cauchy data at $t=0$ and the Dirichlet boundary control at $x=0$. Here $\rho$, $\gamma$, $A$, $B$ are smooth real $2\times 2$-matrix-valued functions of $x$; $\rho =\mathrm {diag}\{\rho _1, \rho _2\}$ and $\gamma =\mathrm {diag}\{\gamma _1, \gamma _2\}$ are matrices with positive entries; and $u=u(x,t)$ is a solution (an ${\mathbb R}^2$-valued function). For $x\geq 0$, it is assumed that $\sqrt {\frac {\gamma _{2}}{\rho _{2}}}< \sqrt {\frac {\gamma _{1}}{\rho _{1}}}$ and $A^{\mathrm {tr}} =-A$, $A_x =B -B^{\mathrm {tr}}$. The “input-output” correspondence is realized by the response operator $R\colon u(0,t) \mapsto \gamma (0)u_x(0,t)$, $t\geq 0$, which plays the role of inverse problem data in applications. In the paper, a constructive characterization is given for the response operators of the systems of this type.