Generic structure of the Lagrangian manifold in chattering problems

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• 作者：L. V. Lokutsievskii (1)
  
• 关键词：chattering problem ; Lagrangian manifold ; singular extremal ; Fuller chattering problem ; singular extremal ; Hamiltonian system ; singularity of conic type
• 刊名：Mathematical Notes
• 出版年：2014
• 出版时间：May 2014
• 年：2014
• 卷：95
• 期：5-6
• 页码：786-794
• 全文大小：607 KB
• 参考文献：1. M. I. Zelikin and V. F. Borisov, / Theory of Chattering Control, in / Systems Control Found. Appl. (Birkh盲user Boston, Boston, MA, 1994).
  2. A. A. Agrachev and Yu. L. Sachkov, / Geometric theory of control (Fizmatlit, Moscow, 2005) [in Russian].
  3. A. F. Filippov, / Differential Equations with Discontinuous Right-Hand Side (Nauka, Moscow, 1985) [in Russian].
  4. I. A. K. Kupka, 鈥淭he ubiquity of Fuller鈥檚 phenomenon,鈥?in / Nonlinear Controllability and Optimal Control, Monogr. Textbooks Pure Appl. Math. (Dekker, New York, 1990), Vol. 133, pp. 313鈥?50.
  
• 作者单位：L. V. Lokutsievskii (1)
  
  1. Moscow State University, Moscow, Russia
  
• ISSN：1573-8876

文摘

This paper studies the structure of the singularity of Lagrangian manifolds in a neighborhood of the surface of singular extremals of second order in optimal control problems. For the Fuller classical problem, the structure of the Lagrangian manifold is explicitly constructed: it is shown that it has a singularity of conic type at the origin of coordinates. In the general case, it is proved that the Lagrangian manifold is a locally trivial fiber bundle over the surface of singular extremals with each fiber having a singularity of a similar conic type at the point of exit of the singular extremals.