## Abstract

An application in robotics motivates us to characterize the evolution of a subset in state space due to a compact neighborhood of an arbitrary dynamical system - an instance of a differential inclusion. Earlier results of Blagodat-skikh and Filippov (1986) and Butkovskii (1982) characterize the boundary of the attainable set and the forward projection operator of a state. Our first result is a local characterization of the boundary of the forward projection of a compact regular subset of the state space. Let the collection of states such that the differential inclusion contains an equilibrium point be called a singular invariant set. We show that the fields at the boundary of the forward projection of a singular invariant set are degenerate under some regularity assumptions when the state-wise boundary of the differential inclusion is smooth. Consider instead those differential inclusions such that the state-wise boundary of the problem is a regular convex polytope - a piecewise smooth boundary rather than smooth. Our second result gives conditions for the uniqueness and existence of the boundary of the forward projection of a singular invariant set. They characterize the bundle of unstable and stable manifolds of such a differential inclusion.

Original language | English |
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Pages (from-to) | 105-138 |

Number of pages | 34 |

Journal | Journal of Nonlinear Science |

Volume | 6 |

Issue number | 2 |

DOIs | |

State | Published - 1 Jan 1996 |

Externally published | Yes |

## Keywords

- Control uncertainty
- Differential inclusions
- Fine-motion planning
- Multivalued differential equations
- Robotics
- Stability boundaries
- Unstable and stable manifolds

## ASJC Scopus subject areas

- Modeling and Simulation
- Engineering (all)
- Applied Mathematics