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The $k$-Compound of a Difference-Algebraic System

Ron Ofir, Michael Margaliot

TL;DR

This work introduces the $k-compound system corresponding to a differential-algebraic system, and describes several applications to the analysis of discrete-time dynamical systems described by difference-al algebraic equations.

Abstract

The multiplicative and additive compounds of a matrix have important applications in geometry, linear algebra, and the analysis of dynamical systems. In particular, the $k$-compounds allow to build a $k$-compound dynamical system that tracks the evolution of $k$-dimensional parallelotopes along the original dynamics. This has recently found many applications in the analysis of non-linear systems described by ODEs and difference equations. Here, we introduce the $k$-compound system corresponding to a differential-algebraic system, and describe several applications to the analysis of discrete-time dynamical systems described by difference-algebraic equations.

The $k$-Compound of a Difference-Algebraic System

TL;DR

This work introduces the $k-compound system corresponding to a differential-algebraic system, and describes several applications to the analysis of discrete-time dynamical systems described by difference-al algebraic equations.

Abstract

The multiplicative and additive compounds of a matrix have important applications in geometry, linear algebra, and the analysis of dynamical systems. In particular, the -compounds allow to build a -compound dynamical system that tracks the evolution of -dimensional parallelotopes along the original dynamics. This has recently found many applications in the analysis of non-linear systems described by ODEs and difference equations. Here, we introduce the -compound system corresponding to a differential-algebraic system, and describe several applications to the analysis of discrete-time dynamical systems described by difference-algebraic equations.

Paper Structure

This paper contains 16 sections, 10 theorems, 57 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Multiplicative compound of a matrix
  4. Geometric interpretation of the multiplicative compound
  5. Matrix pencils
  6. Difference-algebraic equations
  7. The $k$-multiplicative compound DAE
  8. The $k$-multiplicative compound matrix pencil
  9. Regularity of $(A,B)^{(k)}$
  10. Spectral properties of $(A,B)^{(k)}$
  11. Analysis of the $k$-multiplicative compound DAE
  12. Tractability and asymptotic stability of the $k$-multiplicative compound DAE
  13. Consistent and non-consistent initial conditions of the $k$-multiplicative compound DAE
  14. An Application: the set of stable initial conditions of a DAE
  15. Application to a 3D Leslie model
  16. ...and 1 more sections

Key Result

Proposition 1

gene_inverse The DAE eq:DAE_TINV is tractable iff there exists $\lambda\in\mathbb C$ such that $\det (A- \lambda B ) \ne 0$, that is, iff $(A,B)$ is regular.

Figures (4)

  • Figure 1: 2D parallelotope with vertices $0$, $x^1$, and $x^2$.
  • Figure 2: Trajectories of the 3-dimensional DAE in Example \ref{['exa:periodic']} projected onto the 2-dimensional subspace $\text{range}(\hat{B}_0)$. One trajectory is shown with asterisks, and the other with circles. The areas defined by the two trajectories are outlined with dashed borders.
  • Figure 3: Left: two trajectories of the Leslie model \ref{['eq:lde_leslie']}, projected onto the 2-dimensional space $\mathcal{V}^1$ using a projection matrix $P$. Initial values are shown with asterisks ($*$). Right: corresponding solution of the 2-compound system, projected onto the one-dimensional space $\mathcal{V}^2$ (see the definition in Prop. \ref{['prop:consistent_dim_k']}).
  • Figure :

Theorems & Definitions (30)

  • Definition 1
  • Definition 2
  • Proposition 1
  • Proposition 2
  • Definition 3
  • Proposition 3
  • Definition 4
  • Remark 1
  • Proposition 4
  • proof
  • ...and 20 more