16:00 - 17:00
Title: Nonlinear (switched) DAEs: normal forms, impulse-free jumps and stability
Abstract: In the first part of this talk, we deal with inconsistent initial value problems of nonlinear DAEs of the form $E(x)\dot x= F(x)$. We define impulse-free jumps of nonlinear DAEs as parameterized curves with derivatives in the distribution $\ker E(x)$. Then with the help of a proposed nonlinear Weierstrass form, we study the existence and uniqueness of the impulse-free jumps. After that, a singular perturbed system approximation is proposed for nonlinear DAEs; we show that solutions of the perturbed system approximate both impulse-free jumps and $\mathcal C^1$-solutions of nonlinear DAEs. In the second part of the talk, we extend the jump rule in the first part to the switched case, which generalizes the impulse-free condition of switched linear DAEs to the nonlinear case. Moreover, a novel notion called the jump-flow explicitation is used to simply the common Lyapunov function condition for the stability analysis of switched nonlinear DAEs. Finally, we generalize the well-known commutativity condition of switched nonlinear ODEs to the DAEs case. We show that to guarantee the stability of nonlinear switched DAEs with all stable models, not only the commutativity of the flow vector fields but also some extra invariant conditions are needed.