Contraction Analysis of Nonlinear DAE Systems

This article studies the contraction properties of nonlinear differential-algebraic equation (DAE) systems. Such systems typically appear as a singular perturbation reduction of a multiple-time-scale differential system. In addition, a given DAE may result from the reduction of many synthetic differential systems. We show that an important property of a contracting DAE system is that the reduced system always contracts faster than any synthetic counterpart. At the same time, there always exists a synthetic system, whose contraction rate is arbitrarily close to that of the DAE. Synthetic systems are useful for the analysis of attraction basins of nonlinear DAE systems. As any rational DAE system can be represented in quadratic form, the Jacobian of the synthetic system can be made affine in the system variables. This allows for scalable techniques to construct attraction basin approximations, based on uniformly negative matrix measure conditions for the synthetic system Jacobian.

Automated summary

This article explores the contraction properties of nonlinear differential-algebraic equation systems, demonstrating that the reduced system contracts faster than any synthetic counterpart and highlighting the role of synthetic systems in analyzing attraction basins of nonlinear DAE systems.