Linearization of DAEs along trajectories (Q1805094)

Linearization of differential algebraic equations, DAEs, \(F(x',x,t) = 0\) and of control DAEs \(F(x',x,t,u) = 0\) are considered. A linearization is called linear time invariant if the linearization is performed in the time variable also, resulting in a time invariant DAE and control DAE respectively. It is shown on two examples that this type of linearization may change crucial properties of the problem, e.g. stability or the dimension of the solution manifold. A linearization not involving time leads to a linear DAE where the coefficient matrices are time dependent. If this linearization is done with respect to a constant solution it is shown using additional assumptions that the linearized DAE has a solution manifold with the same dimension. Using this result a similar result is shown for a linearization along a trajectory. With the same arguments linearization of control DAEs are investigated.