  This study investigates the inverse problem of determining the right-hand
side of a telegraph equation given in a Hilbert space. The main equation under
consideration has the form $(D_{t}^{\rho})^{2}u(t)+2\alpha
D_{t}^{\rho}u(t)+Au(t)=p( t)q+f(t)$, where $0<t\leq T$, $0<\rho<1$ and
$D_{t}^{\rho}$ is the Caputo derivative. The equation contains a self-adjoint
positive operator $A$ and a time-varying multiplier $p(t)$ in the source
function, which, like the solution of the equation, is unknown. To solve the
inverse problem, an additional condition $B[u(t)] = \psi(t)$ is imposed, where
$B$ is an arbitrary bounded linear functional. The existence and uniqueness of
a solution to the problem are established and stability inequalities are
derived. It should be noted that, as far as we know, such an inverse problem
for the telegraph equation is considered for the first time. Examples of the
operator $A$ and the functional $B$ are discussed.
