General Index Reduction by Embedding for Integro-differential-algebraic Equations

Abstract

Integro differential algebraic equations (idae) are widely used in applications. The existing definition of the signature matrix for dae is insufficient, which leads to the failure of structural methods. Moreover existing structural methods may fail for an idae if its Jacobian matrix after differentiation is still singular due to symbolic cancellation or numerical degeneration.In this paper, for polynomially nonlinear systems of idae, a global numerical method is given to solve both problems above using numerical real algebraic geometry. Firstly, we redefine the signature matrix. Secondly, we introduce a definition of degree of freedom for idae. This can help to ensure termination of the index reduction algorithm by the embedding. Thirdly, combined with numerical real algebraic geometry, we give a global numerical method to detect all possible solution components of idae. An example of two stage drive system is used to demonstrate our method and its advantages.