  In this paper, we address the problem of recovering constant source terms in
a discrete dynamical system represented by $x_{n+1} = Ax_n + w$, where $x_n$ is
the $n$-th state in a Hilbert space $\mathcal{H}$, $A$ is a bounded linear
operator in $\mathcal{B}(\mathcal{H})$, and $w$ is a source term within a
closed subspace $W$ of $\HH$. Our focus is on the stable recovery of $w$ using
time-space sample measurements formed by inner products with vectors from a
Bessel system $\mathcal{G} \subset \mathcal{H}$. We establish the necessary and
sufficient conditions for the recovery of $w$ from these measurements,
independent of the unknown initial state $x_0$ and for any $w \in W$. This
research is particularly relevant to applications such as environmental
monitoring, where precise source identification is critical.
