Around Problem 8.1: Augmenting an Injective Immersion into a Diffeomorphism
This chapter addresses the question of the extension of an injective immersion into a diffeomorphism. This is done by complementing continuously the full-rank rectangular Jacobian of the injective immersion into an invertible square matrix. Indeed, when this is possible, an explicit formula for the diffeomorphism is given. Several sufficient conditions for a Jacobian complementation are given with either explicit formulas or constructive algorithms and are illustrated in the example of an oscillator with unknown frequency.
- 1.Andrieu, V., Eytard, J.B., Praly, L.: Dynamic extension without inversion for observers. In: IEEE Conference on Decision and Control, pp. 878–883 (2014)Google Scholar
- 2.Bernard, P., Praly, L., Andrieu, V.: Expressing an observer in given coordinates by augmenting and extending an injective immersion to a surjective diffeomorphism (2018). https://hal.archives-ouvertes.fr/hal-01199791v6
- 5.Dugundgi, J.: Topology. Allyn and Bacon, Boston (1966)Google Scholar
- 6.Eckmann, B.: Mathematical Survey Lectures 1943–2004. Springer, Berlin (2006)Google Scholar