## Abstract

We study the time optimal control of the system x ̇_{1} = x_{1}f{hook}_{1}(x_{1}, x_{2}) + u_{1}(t) g_{1}(x_{1}), x ̇_{2} = x_{2}f{hook}_{2}(x_{1}, x_{2}) + u_{2}(t)g_{2}(x_{2}), where x_{1} is the size of the population of one species, x_{2} is the population size of the second species, f{hook}_{1} and f{hook}_{2} are the fractional growth rates of the respective species, g_{1} and g_{2} are nowhere vanishing functions of class C^{1}(0, + ∞), and the control u(t) = (u_{1}(t), u_{2}(t)) takes on values in a closed rectangle. The functions f{hook}_{1} and f{hook}_{2} are chosen to represent prey-predator, competitive, and symbiotic interactions. We show, for the various interactions, that a time optimal control, if it exists, must be "bang-bang," and give sufficient conditions for the controllability, and for the existence, of time optimal controls of the above system.

Original language | English (US) |
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Pages (from-to) | 578-603 |

Number of pages | 26 |

Journal | Journal of Mathematical Analysis and Applications |

Volume | 53 |

Issue number | 3 |

DOIs | |

State | Published - Mar 1976 |

## ASJC Scopus subject areas

- Analysis
- Applied Mathematics