Differential-algebraic equations from a functional-analytic viewpoint: A survey

Abstract

The purpose of this paper is to provide an overview on the state of the art concerning functional-analytic properties associated with differential-algebraic equations (DAEs). We summarize the relevant literature and develop a basic theory of linear and nonlinear differential-algebraic operators. In particular, we consider Fredholm properties, normal solvability, generalized inverses, least-squares solutions, splittings of regular linear differential-algebraic operators, bounded outer inverses, local solvability of equations with regular nonlinear differential-algebraic operators, Newton–Kantorovich iterations, and regularizations of the ill-posed problems arising from higher-index operators.