Estimate the spectrum of affine dynamical systems from partial observations of a single trajectory data

In this paper, we study the nonlinear inverse problem of estimating spectrum a system matrix, that drives finite-dimensional affine dynamical system, from partial observations single trajectory data. noiseless case, prove an annihilating polynomial whose roots are subset spectrum, can be uniquely determined We then which eigenvalues matrix recovered and derive various sufficient necessary conditions to characterize relationship between recoverability each eigenvalue observation locations. propose reconstruction algorithms, with theoretical guarantees, generalizing classical Prony method, ESPIRIT, pencil method. test algorithms over variety examples applications graph signal processing, disease modeling real-human motion dataset. The numerical results validate our demonstrate effectiveness proposed even when data did not follow exact linear system.