Generalized solutions in systems with active unilateral constraints

Boris M. Miller, Joseph Bentsman

Research output: Contribution to journalArticlepeer-review


The concept of the generalized solution that admits well-posed representation of controlled complex behavior in systems with active unilateral phase constraints is proposed. Based on this concept, the definition of the generalized solution for this class of problems is introduced that encompasses Zeno type behavior and sliding modes along the constraint boundary. The general representation of such solutions in terms of nonlinear differential equations with a measure is derived. The latter is shown to solve a long-standing problem of providing unique extensibility of a trajectory beyond accumulation points in systems with Zeno-type behavior. An example is given, showing that the representation proposed completely captures Zeno-type behavior and provides unique extensibility of solutions without the need to truncate infinite sequences and/or switch system coefficients depending on system motion relative to the generalized coordinates of the accumulation point.

Original languageEnglish (US)
Pages (from-to)510-526
Number of pages17
JournalNonlinear Analysis: Hybrid Systems
Issue number4
StatePublished - Dec 2007


  • Differential equations with a measure
  • Generalized solutions
  • Phase constraints

ASJC Scopus subject areas

  • Control and Systems Engineering
  • Analysis
  • Computer Science Applications


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