DAE

Author
Affiliation

Enrico Bertolazzi

University of Trento, Department of Industrial Engineering

Differential-algebraic equations (DAEs) combine differential dynamics with algebraic constraints. They arise naturally in constrained mechanical systems, electrical circuits, chemical processes, and optimal control.

TipPrerequisites, objectives, and expected outcomes

Prerequisites: ODE initial-value problems, multivariable calculus, and basic matrix factorizations (Appendix A — Gaussian Elimination and Echelon Forms).

Objectives: recognize DAE structure, determine and reduce index, construct consistent initial data, and select a constraint-aware numerical method.

Expected outcomes: the reader can distinguish differential and algebraic variables, diagnose hidden constraints and drift, perform a convergence study, and justify a solver choice from index, stiffness, and geometry.

The part follows a single line of argument. The linear constant-coefficient case is treated first, because there the whole theory is available in closed form; the reduction is then rebuilt with Gaussian elimination, which is what a computer can actually do; the index is defined in general; any DAE is brought to semi-explicit form and classified; the reduction is extended to an arbitrary state-dependent system; and dummy variables are introduced to keep its cost bounded.

Contents

  1. Linear DAEs — matrix pencils, the Weierstrass canonical form, the Kronecker index.
  2. Index Reduction for Linear DAEs — separating the algebraic equations by elimination, differentiating them, and filing them as invariants of the resulting ODE.
  3. The Differential Index — the general definition and the other notions of index.
  4. Semi-Explicit Form and Its Classification — the transformation \dot{\bm x}=\bm y, \bm0=\bm F(\bm x,\bm y,t), the index-1, index-2 and index-3 conditions, Hessenberg form.
  5. General Index Reduction — the form \bm E(\bm x,t)\dot{\bm x}=\bm G(\bm x,t), the generalized LU that separates the algebraic part, and the reduction algorithm.
  6. Dummy Variables — keeping the size of the reduced system bounded.
  7. DAEs from Lagrangian Mechanics — where index-three systems come from.
  8. Numerical Methods for DAEs — consistent initialization, drift, stabilization, projection, order reduction.
  9. Case Studies and Exercises, in the companion volume.