Impulse-free jump solutions of nonlinear differential-algebraic equations
Real nonlinear DAEs like \[ E(x)\dot{x} = F(x) \] \(E(x)\) being a matrix), are known to feature impasse points, when \(\det E(x_0)=0\). Here the authors focus on how define jump solutions with no Dirac impulsion, from a non-consistent initial value. It allows to model transient situations, e.g. for nonlinear electrical circuits. A consistent initial value \(x_0\) is such that the DAE has a \(C^1\) solution at its location. A jump solution is a trajectory \(\tau \mapsto J(\tau)\) defined on a bounded interval \([0,a]\) from an inconsistent value \(J(0)=x_1\) to a consistent one \(J(a)=x_0\) and it is said impulse-free when for any \(\tau\): \[ E(J(\tau))\frac{dJ(\tau)}{d\tau} = 0. \] This way of defining jump solutions is also coordinate-free. The main tools used by the authors are first the geometric approach of nonlinear DAEs (see 90's works of Rabier and Rheinboldt or Reich), then the use of a singularly perturbed associated system, to obtain the impulse-free jump solution as a limit. The main theorems are obtained for geometric index-1 DAEs: \begin{itemize} \item Theorem \(4.6\): existence and uniqueness of an impulse-free solution, in the neighborhood of a consistent point, upon specific regularity conditions. \item Theorem \(5.3\): existence and stability of an impulse-free jump solution as a limit of the solutions of the singularly perturbed system. \end{itemize} The paper is clear with non trivial examples from electricial circuits, and interesting considerations on the numerical validation of the results (in particular, Matlab is used to compute consistent initial values).
nonlinear differential-algebraic equations
scientific article; zbMATH DE number 7601765
scientific article