On the continuous dependence of trajectories of bilinear systems on controls and its applications.

*(English)*Zbl 0654.93028Bilinear, continuous-time, finite-dimensional, nonstationary control systems are studied. Under some assumptions concerning the smoothness of the matrices of coefficients, an analytical representation of dependence of trajectories of systems on control is established. On the basis of this representation, an estimate for continuous dependence of trajectories of a system with single input on control is developed.

Moreover, a numerical method for searching the optimal control of time- independent bilinear systems is described, which is a modification of the well-known gradient projection method. This modification is based on an estimate proved before. Some properties of the attainable set of bilinear systems are also analyzed. Finally, an illustrative numerical example is given. The results presented in the paper extend to the case of time- dependent bilinear systems the results given by the author [Kybernetika 23, 198-213 (1987; Zbl 0638.93035)].

Moreover, a numerical method for searching the optimal control of time- independent bilinear systems is described, which is a modification of the well-known gradient projection method. This modification is based on an estimate proved before. Some properties of the attainable set of bilinear systems are also analyzed. Finally, an illustrative numerical example is given. The results presented in the paper extend to the case of time- dependent bilinear systems the results given by the author [Kybernetika 23, 198-213 (1987; Zbl 0638.93035)].

Reviewer: J.Klamka

##### MSC:

93C10 | Nonlinear systems in control theory |

93B05 | Controllability |

90C52 | Methods of reduced gradient type |

93B03 | Attainable sets, reachability |

93C35 | Multivariable systems, multidimensional control systems |

93B40 | Computational methods in systems theory (MSC2010) |

65K10 | Numerical optimization and variational techniques |

##### References:

[1] | L. S. Pontrjagin: Ordinary Differential Equations. Nauka, Moscow 1970. In Russian. |

[2] | P. Lancaster: Theory of Matrices. Academic Press, New York-London 1969. · Zbl 0186.05301 |

[3] | S. Čelikovský: On the representation of trajectories of bilinear systems and its applications. Kybernetika 23 (1987), 3, 198-213. · Zbl 0638.93035 |

[4] | J. Doležal, P. Černý: Methods of optimal control for practical determination of multifunctional catalysts. Automatizace 21 (1978), 1, 3-8. In Czech. |

[5] | E. P. Hofer: Optimierung eines katalytischen Rohrreaktors. Regelungstechnik 23 (1975), 109-117. |

[6] | F. P. Vasiljev: Methods of solving of extremal problems. Moscow 1981. In Russian. |

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