Stability Loss in Quasilinear DAEs by Divergence of a Pencil Eigenvalue

The divergence through infinity of certain eigenvalues of a linearized differential- algebraic equation (DAE) may result in a stability change along an equilibrium branch. This behavior cannot be exhibited by explicit ODEs, and its analysis in index one contexts has been so far unduly restricted to semiexplicit systems. By means of a geometric reduction framework we extend the characterization of this phenomenon to quasilinear DAEs, which comprise semiexplicit problems as a particular case. Our approach clarifies the nature of the singularities which are responsible for the stability change and also accommodates rank deficiencies in the leading system matrix. We show how to address this problem in a matrix pencil setting, an issue which leads to certain results of independent interest involving the geometric index of a quasilinear DAE and the Kronecker index of its linearization. The results are shown to be of interest in electrical circuit theory, since the differential-algebraic network models actually used in circuit simulation are not in semiexplicit but in quasilinear form.