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 $\dot{x}=f(t,x)$ 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 $x(0)=x^0$, 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 $x^0$ uniquely determined knowing that $x^0$ is sufficiently sparse? Is there any way to reconstruct such a sparse initial data $x^0$?