Extremal synthesis for generic planar systems

Ugo Boscain, Benedetto Piccoli

Research output: Contribution to journalArticlepeer-review

20 Scopus citations


For a generic single-input planar control system we analyze the structure of the set of extremals for the time-optimal problem. Generically all extremals are finite concatenations of regular arcs that are bang or correspond to a smooth feedback. Moreover, the support of extremals is a Whitney stratified set. We collect these information in the definition of extremal synthesis. In the cotangent bundle, we give a topological classification of the singularities of the extremal synthesis and study the projections of the support of extremals (regarded as a two-dimensional object, after normalization) from ℝ2 × S1 to the plane. With respect to the Whitney classical singularities here we deal with a stratified set with "edges" and "corners," and along with cusps and folds, we find other stable singularities.

Original languageEnglish (US)
Pages (from-to)209-258
Number of pages50
JournalJournal of Dynamical and Control Systems
Issue number2
StatePublished - Apr 2001
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Control and Systems Engineering
  • Algebra and Number Theory
  • Numerical Analysis
  • Control and Optimization


  • Optimal control
  • Projection singularities
  • Synthesis theory


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