On Morse Theory for Piecewise Smooth Functions

Authors:
A. A. Agrachev;D. Pallaschke;S. Scholtes
Affiliations:
Steklov Mathematical Institute, Russian Academy of Sciences, ul. Vavilova 42, 117966 Moscow, Russia;Institute for Statistics and Mathematical Economics, University of Karlsruhe, Kaiserstr. 12, D-76128 Karlsruhe, Germany;Department of Engineering, University of Cambridge, Trumpington Street, Cambridge CB 2 1AG, Great Britain
Venue:
Journal of Dynamical and Control Systems
Year:
1997

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Abstract

Lower level sets of continuous selections of C2-functions defined on a smooth manifold in the vicinity of a nondegenerate critical point in the sense of [11] are studied. It is shown that the lower level set is homotopy equivalent to the join of the lower level sets of the smooth and the nonsmooth part, respectively, of the corresponding normal form. Some generalized Morse inequalities are deduced from this result.