  Discrete-time linear time-varying (LTV) systems form a powerful class of
models to approximate complex dynamical systems with nonlinear dynamics for the
purpose of analysis, design and control. Motivated by inference of
spatio-temporal dynamics in breast cancer research, we propose a method to
efficiently solve an identification problem for a specific class of
discrete-time LTV systems, in which the states are fully observed and there is
no access to system inputs. In addition, it is assumed that we do not know on
which states the inputs act, which can change between time steps, and that the
total number of inputs is sparse over all states and over time. The problem is
formulated as a compressive sensing problem, which incorporates the effect of
measurement noise and which has a solution with a partially sparse support. We
derive sufficient conditions for the unique recovery of the system model and
input values, which lead to practical conditions on the number of experiments
and rank conditions on system outputs. Synthetic experiments analyze the
method's sensitivity to noise for randomly generated models.
