High-Order Open Mapping Theorems

Abstract

The well-known finite-dimensional first-order open mapping theorem says that a continuous map with a finite-dimensional target is open at a point if its differential at that point exists and is surjective. An identical result, due to Graves, is true when the target is infinite-dimensional, if "differentiability" is replaced by "strict differentiability." We prove general theorems in which the linear approximations involved in the usual concept of differentiability is replaced by an approximation by a map which is homogeneous relative to a one-parameter group of dilations, and the error bound in the approximation involves a "homogeneous pseudonorm" or a "homogeneous pseudodistance," rather than the ordinary norm. We outline how these results can be used to derive sufficient conditions for openness involving higher-order derivatives, and carry this out in detail for the second-order case.

Third order open mapping theorems and applications to the end-point map

“…Notice that the two statements are different in nature: indeed the first one does not use the notion of regular zero. Also, point (ii) can be seen as a more geometric version of the third order open mapping theorem proved by Sussmann in [35]. Its rephrasement in algebraic terms can be found in Theorem 2.8.…”

Third order open mapping theorems and applications to the end-point map

“…The two statements are different in nature: indeed, the first one does not use the notion of regular zero. Also, point (b) can be seen as a more geometric version of the third order open mapping theorem proved by Sussmann in [32]. Its rephrasement in algebraic terms can be found in theorem 2.8.…”

Necessary conditions involving Lie brackets for impulsive optimal control problems

“…As for higher-order necessary conditions -see e.g. [6], [11], [13], [14], [23], [21] for the bounded control case-we are aware only of results for the commutative case, i.e. when [g i , g j ] ≡ 0 for all i, j = 1, .…”