Generalized Derivatives of Differential–Algebraic Equations

Nonsmooth equation-solving and optimization algorithms which require local sensitivity information are extended to systems with nonsmooth parametric differential–algebraic equations embedded. Nonsmooth differential–algebraic equations refers here to semi-explicit differential–algebraic equations with algebraic equations satisfying local Lipschitz continuity and differential right-hand side functions satisfying Carathéodory-like conditions. Using lexicographic differentiation, an auxiliary nonsmooth differential–algebraic equation system is obtained whose unique solution furnishes the desired parametric sensitivities. More specifically, lexicographic derivatives of solutions of nonsmooth parametric differential–algebraic equations are obtained. Lexicographic derivatives have been shown to be elements of the plenary hull of the Clarke (generalized) Jacobian and thus computationally relevant in the aforementioned algorithms. To accomplish this goal, the lexicographic smoothness of an extended implicit function is proved. Moreover, these generalized derivative elements can be calculated in tractable ways thanks to recent advancements in nonsmooth analysis. Forward sensitivity functions for nonsmooth parametric differential–algebraic equations are therefore characterized, extending the classical sensitivity results for smooth parametric differential–algebraic equations.