A stability estimate for a solution to a two-dimensional inverse problem of electrodynamics

We consider the problem of finding the three coefficients $c(x)$, $\sigma(x)$, and $q(x)$ in a hyperbolic equation. Here $c(x)$ is the coefficient at the Laplace operator, $\sigma(x)$ is the coefficient of the first time derivative, and $q(x)$ is the coefficient of the lower-order term. The problem results from the inverse electrodynamic problem of finding the electrodynamic parameters of an isotropic medium under the assumption that the properties of the medium and the exterior current are independent of one coordinate. We suppose that the coefficients $c(x)-1$, $\sigma(x)$, and $q(x)$ are small in some norm and their supports are contained in some disk $B$. This is equivalent to the assumption that the electrodynamic parameters of the medium are close to constants. We suppose that the source initiating oscillations has the form of the impulse function $\delta(t)\delta(x\cdot\nu)$ localized on the set $t=0$, $x\cdot \nu=0$. Here $\nu$ is a unit vector playing the role of a parameter of the problem. The electromagnetic field excited by this source applied outside $B$ is measured at points of the boundary of the domain $B$ on some time interval of a fixed length $T$ counted from the moment of arrival of the signal from the source for three different values of the parameter $\nu$. It is proven that, for a sufficiently large $T$, these data determine the sought coefficients uniquely. We obtain a conditional stability estimate for a solution to the problem.