Differential algebraic equations (DAEs), including so-called descriptor systems, began to attract significant research interest in applied and numerical mathematics in the early 1980s, no more than about three decades ago. In this relatively short time, DAEs have become a widely acknowledged tool to model processes subjected to constraints, in order to simulate and to control processes in various application fields such as network simulation, chemical kinematics, mechanical engineering, system biology.DAEs and their more abstract versions in infinite-dimensional spaces comprise a great potential for future mathematical modeling of complex coupled processes.The purpose of the book is to expose the impressive complexity of general DAEs from an analytical point of view, to describe the state of the art as well as open problems and so to motivate further research to this versatile, extra-ordinary topic from a broader mathematical perspective.The book elaborates a new general structural analysis capturing linear and nonlinear DAEs in a hierarchical way. The DAE structure is exposed by means of special projector functions. Numerical integration issues and computational aspects are treated also in this context.

Notations.- Introduction.- Part I. Projector based approach.- 1 Linear constant coefficient DAEs.-.2 Linear DAEs with variable coefficients.- 3 Nonlinear DAEs.- Part II. Index-1 DAEs: Analysis and numerical treatment.- 4 Analysis.- 5 Numerical integration.- 6 Stability issues.- Part III. Computational aspects.- 7 Computational linear algebra aspects.- 8 Aspects of the numerical treatment of higher index DAEs.- Part IV. Advanced topics.- 9 Quasi-regular DAEs.- 10 Nonregular DAEs.- 11 Minimization with constraints described by DAEs.- 12 Abstract differential algebraic equations.- A. Linear Algebra - Basics.-.B. Technical Computations.- C Analysis.- References.- Index.