A geometric index reduction method for DAE systems

LISI D'ALFONSO; GABRIELA JERONIMO; FRANÇOIS OLLIVIER; ALEXANDRE SEDOGLAVIC; PABLO SOLERNÓ

Beijing

Workshop; Differential Algebra and Related Topics IV; 2010

We will address the index reduction problem for quasi-regular DAE systems.We will show that any of these systems can be transformed into agenerically equivalent first order DAE system consisting of a single purelyalgebraic (polynomial) equation plus an under-determined ODE (that is, asemi-explicit DAE system with differentiation index 1). Finally, we will describe a Kronecker-type algorithm with bounded complexity which computesthis associated system.Our approach makes use of the computation of successive derivatives ofthe equations, as many as the differentiation index of the input system, but,unlike previous methods, we deal with them in a purely algebraic way. Usinga construction originally introduced by Kronecker, we parametrize the pointsof an algebraic variety associated with the system of all the new equationsby means of the points of a hypersurface. In order to keep track of thedifferential structure, we use the parametrizations to construct a vector fieldover the hypersurface defining the semi-explicit DAE system.