has just appeared in the journal Nonlinear Analysis: Hybrid Systems (NAHS). For the next few weeks, you can access the manuscript directly from Elsevier.
Chen, Yahao; Trenn, Stephan
In: Nonlinear Analysis: Hybrid Systems, vol. 46, no. 101238, pp. 1-17, 2022, ISSN: 1751-570X.
In this paper, we propose a novel notion called impulse-free jump solution for nonlinear differential-algebraic equations (DAEs) of the form E(x)x' = F(x) with inconsistent initial values. The term “impulse-free” means that there are no Dirac impulses caused by jumps from inconsistent initial values, i.e., the directions of jumps stay in ker E(x). We find that the existence and uniqueness of impulse-free jumps are closely related to the notion of geometric index-1 and the involutivity of the distribution defined by ker E(x). Moreover, a singular perturbed system approximation is proposed for nonlinear DAEs; we show that solutions of the perturbed system approximate both impulse-free jump solutions and C1-solutions of nonlinear DAEs. Finally, we show by some examples that our results of impulse-free jumps are useful for the problems like consistent initializations of nonlinear DAEs and transient behavior simulations of electric circuits.