The index of an infinite dimensional implicit system

The idea of the index of a differential algebraic equation (DAE) (or implic it differential equation) has played a fundamental role in both the analysi s of DAEs and the development of numerical algorithms for DAEs. DAEs freque ntly arise as partial discretizations of partial differential equations (PD Es). In order to relate properties of the PDE to those of the resulting DAE it is necessary to have a concept of the index of a possibly constrained P DE. Using the finite dimensional theory as motivation, this paper will exam ine what one appropriate analogue is for infinite dimensional systems. A ge neral definition approach will be given motivated by the desire to consider numerical methods. Specific examples illustrating several kinds of behavio r will be considered in some detail. It is seen that our definition differs from purely algebraic definitions. Numerical solutions, and simulation dif ficulties, can be misinterpreted if this index information is missing.