A projection-based formulation of the Implicit Function Theorem and its application to time-varying manifolds

In this paper, we derive a projection-based formulation of the Implicit Function Theorem. We give conditions, when an algebraic, time-parameterized equation G(t,x) = 0 is solvable for components P^c x that are selected by a projection P^c and we derive an implicit function g that specializes P^c x in terms of the complementary components P x, where P = I - P^c. We apply this result to construct a projection-based parametric description of time-varying submanifolds and to generalize the concept of projections to these sets. We illustrate our results by several examples. The results are motivated by the positivity analysis of differential-algebraic equations (DAEs). These are implicit systems F(t,x,\dot x)=0 whose solutions x are supposed to remain componentwise nonnegative whenever the initial value is nonnegative. To entangle the differential and algebraic components in F(t,x,\dot x)=0 without changing the coordinate system, we pursue the presented projection-based solution of implicit algebraic equations.