On an inverse problem for a one-dimensional two-velocity dynamical system

A. L. Pestov

Zapiski Nauchnykh Seminarov POMI, Mathematical problems in the theory of wave propagation. Part 44, Tome 426 (2014), pp. 150-188

Résumé

Evolution of the dynamical system under consideration is governed by the wave equation $\rho u_{tt}-(\gamma u_x)_x+Au_x+Bu=0$, $x>0$, $t>0$ with the zero initial Cauchy data and Dirichlet boundary control at $x=0$. Here, $\rho,\gamma,A,B$ are the smooth $2\times2$-matrix-functions of $x$; $\rho=\mathrm{diag}\{\rho_1,\rho_2\}$ и $\gamma=\mathrm{diag}\{\gamma_1,\gamma_2\}$ – the matrices with positive entries; $u=u(x,t)$ – a solution (an $\mathbb R^2$-valued function). In applications, the system corresponds to one-dimensional models, in which there are two types of the wave modes, which propagate with different velocities and interact to one another. The `input $\to$ state' correspondence is realized by a response operator $R\colon u(0,t)\mapsto\gamma(0)u_x(0,t)$, $t\geqslant0$, which plays the role of inverse data. The representations for the coefficients $A$ and $B$, which are used for their determination via the response operator, are derived. We provide an example of two systems with the same response operator, such that in the first system the wave modes do not interact, whereas in the second one the interaction does occur.

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