The new article Nonsmooth Hessenberg Differential-Algebraic Equations by Peter Stechlinski and Paul Barton (click here for 50 days of free access) studies Hessenberg differential-algebraic equations (DAEs) of size ν with nonsmooth right-hand side functions.
Well-posedness theory established in this article includes consistent initialization robust to parametric perturbations, and local existence and uniqueness of solutions in the classical senses. Regular solutions on an interval of interest are shown to have maximal continuations and to be lexicographically smooth with respect to problem parameters. Consequently, lexicographic directional differentiation can be applied to yield an auxiliary, nonsmooth “high-index” sensitivity system whose solution characterizes generalized derivative information of the reference solution of interest with respect to parameters. This allows for numerically implementable dynamic optimization methods; open-loop optimal control theory is given which provides subdifferential elements of objective functions via solving a nonsmooth equation system involving the parametric sensitivity functions. As a corollary of the results in this article, foundational theory for smooth Hessenberg DAEs having classical differentiation index ν is established in a rigorous manner.