Title | Switching behavior of solutions of ordinary differential equations with abs-factorable right-hand sides |

Publication Type | Journal Article |

Year of Publication | 2015 |

Authors | Khan KA, Barton PI |

Journal | Systems & Control Letters |

Volume | 84 |

Pagination | 27-34 |

Date Published | 10/2015 |

Keywords | non-Zeno behavior, nonsmooth analysis, ordinary differential equations, switching systems |

Abstract | We consider nonsmooth dynamic systems that are formulated as the unique solutions of ordinary differential equations (ODEs) with right-hand side functions that are finite compositions of analytic functions and absolute-value functions. Various non-Zenoness results are obtained for such solutions: in particular, any absolute-value function in the ODE right-hand side can only switch between its two linear pieces finitely many times on any finite duration, even when a discontinuous control input is included. These results are extended to obtain numerically verifiable necessary conditions for the emergence of “valley-tracing modes”, in which the argument of an absolute-value function is identically zero for a nonzero duration. Such valley-tracing modes can create theoretical and numerical complications during sensitivity analysis or optimization. We show that any valley-tracing mode must begin either at the initial time, or when another absolute-value function switches between its two linear pieces. |

URL | http://www.sciencedirect.com/science/article/pii/S016769111500153X |

DOI | 10.1016/j.sysconle.2015.07.007 |

# Switching behavior of solutions of ordinary differential equations with abs-factorable right-hand sides

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