Impulse-free jump solutions of nonlinear differential–algebraic equations

On impulse-free solutions and stability of switched nonlinear differential–algebraic equations

“…In our recent contributions (Chen & Trenn, 2021a, 2022a, we studied impulse-free jumps for DAEs with inconsistent initial values. The main idea behind the following definition of impulsefree jump (solutions) is that we view a jump not only as an instant change between an inconsistent point and a consistent one but also as a parametrized curve J(τ ) 2 whose derivatives with respect to τ satisfy a certain rule, i.e., staying in ker E, such a rule ensures that the jump does not cause any impulse.…”

“…Definition 2.4 (Impulse-free Jump (Chen & Trenn, 2022a)). Consider a DAE Ξ = (E, F ), let C be the consistency space of Ξ , fix an initial point x − 0 ∈ X .…”

“…Remark 2.6. It is important to note that the ex-equivalence preserves both C 1 -solutions and IFJ solutions (and thus IFJs) of DAEs (Chen & Respondek, 2022;Chen & Trenn, 2022a). Note that the jump rule (3) shown in Liberzon and Trenn (2012) is not invariant under the ex-equivalence, i.e., given a jump…”

“…However, we have shown in our recent works (Chen & Trenn, 2021a, 2022a) that nonlinear coordinate transformations do not preserve the jump rule (3), namely, we may get different consistent points x + 0 from (3) depending on which coordinates are chosen for the DAE Ξ (see also Remark 2.6). To have a coordinatesfree jump rule, the notion of impulse-free jump solution is proposed in Chen and Trenn (2022a) (see also Definition 2.4).…”

“…However, we have shown in our recent works (Chen & Trenn, 2021a, 2022a) that nonlinear coordinate transformations do not preserve the jump rule (3), namely, we may get different consistent points x + 0 from (3) depending on which coordinates are chosen for the DAE Ξ (see also Remark 2.6). To have a coordinatesfree jump rule, the notion of impulse-free jump solution is proposed in Chen and Trenn (2022a) (see also Definition 2.4). Because inconsistent initialization can be frequently triggered by switching behaviors in switched DAEs, the main purpose of the present paper is to extend the impulse-free jump rule to switched nonlinear DAEs and to discuss their solutions and stability.…”