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**Control of nonlinear underactuated systems.**
*(English)*
Zbl 1022.70010

From the abstract: “We introduce a new method to design control laws for nonlinear, underactuated systems. Our method produces an infinite-dimensional family of control laws…”

It is not clear that this new method always produces any non-trivial control law. By means of Riemannian geometry there is given an equation of motion containing a potential function. The authors try to find admissible (“underactuated system”) control forces by constructing a new metric tensor \(g\) and a new potential \(V\). For \(g\) and \(V\) they derive systems of partial differential equations which merely are necessary conditions. Proposition 1.4 partially deals with existence, but the proof is not understandable.

It is not clear that this new method always produces any non-trivial control law. By means of Riemannian geometry there is given an equation of motion containing a potential function. The authors try to find admissible (“underactuated system”) control forces by constructing a new metric tensor \(g\) and a new potential \(V\). For \(g\) and \(V\) they derive systems of partial differential equations which merely are necessary conditions. Proposition 1.4 partially deals with existence, but the proof is not understandable.

Reviewer: Peter Kraut (Erlabrunn)

### MSC:

70H03 | Lagrange’s equations |

70Q05 | Control of mechanical systems |

37N35 | Dynamical systems in control |

93D15 | Stabilization of systems by feedback |

93C10 | Nonlinear systems in control theory |

93B29 | Differential-geometric methods in systems theory (MSC2000) |

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\textit{D. Auckly} et al., Commun. Pure Appl. Math. 53, No. 3, 354--369 (2000; Zbl 1022.70010)

### References:

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