Swallowtail, Whitney Umbrella and Convex Hulls
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Swallowtail, Whitney Umbrella and Convex Hulls Vyacheslav D. Sedykh1 Received: 17 March 2020 / Revised: 30 July 2020 / © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract One of the singularities of the convex hull of a generic hypersurface in R4 leads to a generic sewing of two famous surfaces, the swallowtail and the Whitney umbrella, along their selfintersection lines. We prove that germs of all such sewings at the common endpoint of the self-intersection lines are diffeomorphic to each other with respect to diffeomorphisms of the ambient space. Keywords Convex hull · Singularities · Swallowtail · Whitney umbrella
1 Introduction The optimization problem of a controlled system leads to the replacement of the indicatrix of admissible velocities of the system by its convex hull (the relaxation procedure). The convex hull of a subset in Rn is the intersection of all closed half-spaces containing this subset. The transition from the indicatrix to its convex hull is explained by a possibility to apply mixed strategies, that is, to move with different velocities on rapidly changing segments of the trajectory (see [3]). The boundary of the convex hull of an indicatrix is not a C ∞ -smooth manifold in general. If an indicatrix is bounded by a smooth compact generic nonconvex hypersurface in Rn , then the boundary of its convex hull is continuously differentiable and homeomorphic to the sphere S n−1 (see [1]). Let zero velocity lie inside the convex hull. Then, singularities of the boundary of the convex hull of an indicatrix define the maximal limit velocity of the movement in a given direction when mixed strategies are used. There is a general problem (1972-12 in [4]) related to the study of singularities of convex hulls of submanifolds of any dimension in Rn . Some results of this study can be found in papers [6, 10–15, 19]. Applications of the theory of convex hull singularities to the investigation of the boundary of local controllability (attainability) of a generic controlled system
Vyacheslav D. Sedykh
[email protected] 1
National University of Oil and Gas “Gubkin University”, Leninsky prospect 65, Moscow, 119991, Russia
Vyacheslav D. Sedykh
are described in the survey [7]. As an example of a practical use of a classification of singularities of convex hulls, see the paper [18] which is devoted to the study of phase transitions of two-component mixtures in thermodynamics. So, let M be a smooth compact m-dimensional submanifold (without boundary) in Rn . Denote by the boundary of its convex hull. Then, each point of lies in such a tangent hyperplane to the manifold M that M lies on one side of it. Such hyperplanes are called support hyperplanes to M. A support hyperplane to a generic manifold M is tangent to M at n different points at most. These points are vertices of a simplex. This simplex is called the support simplex of a support hyperplane. The hypersurface is the union of the support simplices of all support hyperplanes to the manifold M. Definition A hyperplane π i
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