Inverse problems for hyperbolic equations: the case of unknown time-dependent coefficients

Solvability is studied of the inverse problems of finding a solution $u(x,t)$ and the coefficients $q(t)$ of the equations
\begin{align*} u_{tt}-u_{xx}+q(t)a(x,t)u_t=&f(x,t),\\ u_{tt}-u_{xx}+q(t)a(x,t)u=&f(x,t) \end{align*}
In this case the overdetermination condition has the integral form
$$ \int_0^1K(x,t)u(x,t)dx=\mu(t). $$
Unique existence of regular solutions is established.