Geometric Analysis of Differential-Algebraic Equations via Linear Control Theory

We consider linear differential-algebraic equations (DAEs) of the form and the Kronecker canonical form (KCF) [L. Kronecker, Sitzungsberichte der Königlich Preußischen Akademie der Wissenschaften zu Berlin, 1890, pp. 1225--1237] of the corresponding matrix pencils . We also consider linear control systems and their Morse canonical form (MCF) [A. Morse, SIAM J. Control, 11 (1973), pp. 446--465; B. P. Molinari, Internat. J. Control, 28 (1978), pp. 493--510]. For a linear DAE, a procedure called explicitation is proposed, which attaches to any linear DAE a linear control system defined up to a coordinates change, a feedback transformation, and an output injection. Then we compare subspaces associated to a DAE in a geometric way with those associated (also in a geometric way) to a control system, namely, we compare the Wong sequences of DAEs and invariant subspaces of control systems. We prove that the KCF of linear DAEs and the MCF of control systems have a perfect correspondence and that their invariants are related. In this way, we connect the geometric analysis of linear DAEs with the classical geometric linear control theory. Finally, we propose a concept called internal equivalence for DAEs and discuss its relation with internal regularity, i.e., the existence and uniqueness of solutions.