Periodic source detection in discrete dynamical systems via space–time sampling

In this paper, we examine a discrete dynamical system defined by x ( n + 1 ) = A x ( n ) + w ( n ) , where x takes values in a Hilbert space H and w is a periodic source with values in a fixed closed subspace W of H . Our goal is to identify conditions on some spatial sampling system G = { g j } j ∈ J of H that enable stable recovery of the unknown source term w from space–time samples { 〈 x ( n ) , g j 〉 } n ≥ 0 , j ∈ J . We provide necessary and sufficient conditions on G = { g j } j ∈ J to ensure stable recovery of any w ∈ W . Additionally, we explicitly construct an operator R , dependent on G , such that R { 〈 x ( n ) , g j 〉 } n , j = w .