Geometric analysis of differential-algebraic equations via linear\n control theory

Abstract

We consider linear differential-algebraic equations DAEs and the Kronecker\ncanonical form KCF of the corresponding matrix pencils. We also consider linear\ncontrol systems and their Morse canonical form MCF. For a linear DAE, a\nprocedure named explicitation is proposed, which attaches to any linear DAE a\nlinear control system defined up to a coordinates change, a feedback\ntransformation and an output injection. Then we compare subspaces associated to\na DAE in a geometric way with those associated (also in a geometric way) to a\ncontrol system, namely, we compare the Wong sequences of DAEs and invariant\nsubspaces of control systems. We prove that the KCF of linear DAEs and the MCF\nof control systems have a perfect correspondence and that their invariants are\nrelated. In this way, we connect the geometric analysis of linear DAEs with the\nclassical geometric linear control theory. Finally, we propose a concept named\ninternal equivalence for DAEs and discuss its relation with internal\nregularity, i.e., the existence and uniqueness of solutions.\n