Multi-objective optimal control for eigen-frequencies of a torsional shaft using Pontryagin’s maximum principle.

*(English)*Zbl 1325.74081Summary: The present work investigates through Pontryagin’s maximum principle the multi-objective optimal control for eigen frequencies of a torsional shaft. Control variables are diameters of the shaft’s segments. Maier objective functional is used to control the final state of the objective functional for solving multi-objective optimal problem, in which maximizing eigen frequencies and minimizing system’s weight are simultaneously involved. The analogy coefficient \(k\) in the necessary optimality condition is explicitly determined by considering eigen frequencies as state variables. Numerical simulations demonstrate the relationship between the optimal configuration of the shaft and their eigen modes. The Pareto fronts and the boundary of the feasible region are constructed for the objectives. The Pareto fronts and the feasible region facilitate the estimation of the level of the trade-off between objectives, and the selection of a suitable solution among a set of competitive objectives.

##### MSC:

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

90C29 | Multi-objective and goal programming |

49K15 | Optimality conditions for problems involving ordinary differential equations |

49N90 | Applications of optimal control and differential games |

##### Keywords:

eigenfrequencies; multi-objective optimal control; torsional shaft; Pontryagin’s maximum principle
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\textit{H.-L. Bui} et al., Meccanica 50, No. 9, 2409--2419 (2015; Zbl 1325.74081)

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