Implicit function theorem for nonlinear time-delay systems with algebraic constraints

TL;DR

This paper generalizes the implicit function theorem to nonlinear time-delay systems with algebraic constraints, introducing an iterative method for coordinate transformations and applying it to reduce DDAE indices.

Contribution

It extends the implicit function theorem to time-delay systems and provides an iterative algorithm for coordinate changes, enabling solutions for delayed differential-algebraic equations.

Findings

Iterative algorithm effectively checks conditions for bicausal transformations.

Application to DDAEs reduces their indices and facilitates solutions.

Numerical examples demonstrate the practical utility of the theoretical results.

Abstract

In this note, we discuss a generalization of the well-known implicit function theorem to the time-delay case. We show that the latter problem is closely related to the bicausal changes of coordinates of time-delay systems. An iterative algorithm is proposed to check the conditions and to construct the desired bicausal change of coordinates for the proposed implicit function theorem. Moreover, we show that our results can be applied to delayed differential-algebraic equations (DDAEs) to reduce their indices and to get their solutions. Some numerical examples are given to illustrate our results.