The paper presents an analysis of continuous Petri nets using linear algebra of matrices. It provides a link between the classical results of linear systems and Petri nets via the continuous model. A class of variable speed continuous Petri nets (VCPNs), namely Extended State Machines VCPNs (ESMVCPNs) is studied. A state-variable representation and its associated matrix algebra are used to describe systems modeled by ESMVCPNs. The state variable formulation introduced here is that of a linear continuous time-invariant system with nonnegative state (marking) and control vector. The stability and asymptotic stability concepts of linear dynamic systems are applied to continuous Petri nets. A necessary and sufficient condition of conservativeness of ESMVCPNs, based on eigenvalues, is proved. These results are used for the control of a production system.