Observability for Initial Value Problems with Sparse Initial Data

Details Observability for Initial Value Problems with Sparse Initial Data Observability for Initial Value Problems with Sparse Initial Data

2010-04-20

arxiv.org/abs/1004.3583v1

Mathematics

Optimization and Control

Submitted to Applied Mathematics Letters in November 2009 (status: under review).

Scientific paper

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 $x^0$?

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