Mathematical methods to find optimal control of oscillations of a hinged beam (deterministic case).

*(English. Russian original)*Zbl 07190727
Cybern. Syst. Anal. 55, No. 6, 1009-1026 (2019); translation from Kibern. Sist. Anal. 2019, No. 6, 145-164 (2019).

Summary: We consider several problem statements for the optimal controlled excitation of oscillations of a hinged beam. Oscillations occur under the influence of several external periodic forces. In the simplest statement, it is assumed that the structure of the beam is homogeneous. In a more complex formulation, inhomogeneities (defects) on the beam are allowed. The goal of controlling the oscillations of the beam is to provide a predetermined shape and a predetermined pointwise phase of oscillations in a given frequency range. The problem is to determine the number of forces and their characteristics (application, amplitude, and phase of oscillations), which provide the desired waveform with a given accuracy. With the help of analytical mathematical methods, the problems in question are reduced to simpler multiextremum problems of minimizing basic functionals, which can be numerically solved using the multifunctional package AORDA PSG.

##### MSC:

49J20 | Existence theories for optimal control problems involving partial differential equations |

49M99 | Numerical methods in optimal control |

74K10 | Rods (beams, columns, shafts, arches, rings, etc.) |

74K30 | Junctions |

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\textit{G. Zrazhevsky} et al., Cybern. Syst. Anal. 55, No. 6, 1009--1026 (2019; Zbl 07190727); translation from Kibern. Sist. Anal. 2019, No. 6, 145--164 (2019)

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##### References:

[1] | G. M. Zrazhevsky, “Determination of the optimal parameters of the beam waveform actuation,” Bulletin of Taras Shevchenko National University of Kyiv, Ser. Physics & Matematics, Issue 3 (2013), pp. 138-141. URL: http://nbuv.gov.ua/UJRN/VKNU_fiz_mat_2013_3_34. |

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[5] | N. I. Akhiezer, Lectures on the Approximation Theory [in Russian], Nauka, Moscow (1965). |

[6] | AORDA Portfolio Safeguard (PSG). URL: http://www.aorda.com/html/PSG_Help_HTML/index.html?bpoe.htm. |

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