Drazin ODE

So, the linear homogeneous problem can be solved by first transforming to its Drazin ODE, the solution of which can be given in terms of the transition matrix... 

In this paragraph we give expressions for E2E2 and E2A2, the system matrices for the Drazin ODE of a mechanical system with constraints formulated on velocity level. 

We claim that Eqs. (2.6.25) together with Eq. (2.6.26) constitute the Drazin ODE for the velocity constrained mechanical problem... 

In the last paragraph we distinguished two different types of velocity variables p and V, The velocity constraint was based on p, whereas the Drazin ODE was formulated in terms of v. This scheme will be repeated in this paragraph, when constructing the Drazin ODE of the position constrained problem. There, we will introduce two types of position variables p and q. The position constraint will be formulated in terms of p, whereas the Drazin ODE is based on g-position variables. Let us start again from the second order formulation of the problem ... 

In the last chapters we transformed linear DAEs to ODEs the linear state space form and the Drazin ODE. Both are related to the equations of motion in DAE form in the sense, that they lead to the same solution, when initialized with corresponding initial conditions. One might wonder, if these ODEs share also other properties like stability properties How are the eigenvalues of the various forms related ... 

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