Frames for Source Recovery in Dynamical Systems

This paper addresses the problem of recovering constant source terms in discrete dynamical systems described by xn+1 = Axn + w, where xn represents the state in a Hilbert space $\mathcal{H}$, A is a bounded linear operator, and w is a source term within a closed subspace $W \subseteq \mathcal{H}.$ Using time-space sampling measurements, we establish necessary and sufficient conditions for stable recovery of w, independent of the unknown initial state x0. This work has practical applications in areas such as environmental monitoring, where precise source identification is critical.