Sparse object supports are often encountered in many imaging problems. For such sparse objects, recent theory of compressed sensing tells us that accurate reconstruction of objects are possible even from highly limited number of measurements drastically smaller than the Nyquist sampling limit by solving L1 minimization problem. This paper employs the compressed sensing theory for cryo-electron microscopy (cryo-EM) single particle reconstruction of virus particles. Cryo-EM single particle reconstruction is a nice application of the compressed sensing theory because of the following reasons: 1) in some cases, due to the difficulty in sample collection, each experiment can obtain micrographs with limited number of virus samples, providing undersampled projection data, and 2) the nucleic acid of a viron is enclosed within capsid composed of a few proteins; hence the support of capsid in 3-D real space is quite sparse. In order to minimize the L1 cost function derived from compressed sensing, we develop a novel L1 minimization method based on the sliding mode control theory. Experimental results using synthetic and real virus data confirm that the our algorithm provides superior reconstructions of 3-D viral structures compared to the conventional reconstruction algorithms.