Observability for initial value problems with sparse initial data

In this work we introduce the concept of s-sparse observability for large systems of ordinary differential equations. Let be such a system. At time T>0, suppose we make a set of observations b=Ax(T) of the solution of the system with initial data x(0)=x0, where A is a matrix satisfying the restricted isometry property. The aim of this paper is to give answers to the following questions: Given the observations b, is x0 uniquely determined knowing that x0 is sufficiently sparse? Is there any way to reconstruct such a sparse initial data x0?