Robustness of Solutions of the Inverse Problem for Linear Dynamical Systems with Uncertain Data

The problem of estimation of parameters of a dynamical system from discrete data can be formulated as the problem of inverting the map from the parameters of the system to points along a corresponding trajectory. In this work, we focus on linear systems and derive necessary and sufficient conditions for single trajectory data to yield a matrix of parameters with specific dynamical properties. To address the key issue of robustness, we establish conditions that ensure that the desired properties of the solution to the inverse problem are maintained on an open neighborhood of the given data. We then build from these results to find bounds on the uncertainty in the data that can be tolerated without a change in the nature of the inverse problem. In particular, both analytical and numerical estimates are derived for the largest allowable uncertainty in the data for which the qualitative features of the inverse problem solution, such as uniqueness or attractor properties, persist. We also derive the conditions...