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**Modelling analysis and control of thin plates.**
*(English)*
Zbl 0662.73039

Recherches en Mathématiques Appliquées, 6. Paris etc.: Masson. vi, 175 p. FF 180.00 (1988).

In Chapter I derivations of several mathematical models of a homogeneous isotropic thin plate are presented. The following models are derived: the classical Kirchhoff model, the Mindlin-Timoshenko-Hencky model (M-T-model), the geometrical nonlinear von Kármán model, a nonlinear model incorporating transverse shear (analog to the Mindlin-Timoshenko-model). - The unknown variables in the M-T-model are the angles of rotation of the filaments of the plate (\(\psi\),\(\phi)\) and the vertical displacement (W). The governing equations are coupled by terms which are multiplied by the shear modulus \(K\). Letting \(K\to \infty\) leads to a higher order equation for \(W\). This is the Kirchhoff model, and the equation for \(W\) is Ẅ-\(\gamma\) \(\Delta\) Ẅ\(+\Delta^ 2W=0\).

In Chapter II a rigorous study of the situation when \(K\to \infty\) is given. Chapter III begins the study of exact controllability of the various plate models (which is the main purpose of this book). In Chapter III the exact controllability of the M-T-model is examined. Exact controllability consists in proving the following: starting from any given initial state the system may be steered to rest (by a proper choice of boundary control) at a given time. This is done for the M-T-model, by systematic use of HUM (Hilbert Uniqueness Model). - In Chapter IV the one dimensional M-T-model (beams) is considered. Chapter V deals with similar questions which arise for the Kirchhoff model, the exact controllability of the model under various boundary conditions, and the behavior of the controlled system when \(\gamma\) \(\to 0.\)

In Chapter VI thermal effects are incorporated into the Kirchhoff model. In the case when the edge temperature is held at the “reference temperature” and assuming the thermal stresses to be “small”, partial exact controllability of the system is proved. Chapter VII is devoted to the reachability problem for viscoelastic plates with “long range memory”. For one type of boundary control action it is possible to establish that the system may be steered from an arbitrary initial state to a state of rest in a time T (of order \(\gamma^{1/2})\).

In Chapter II a rigorous study of the situation when \(K\to \infty\) is given. Chapter III begins the study of exact controllability of the various plate models (which is the main purpose of this book). In Chapter III the exact controllability of the M-T-model is examined. Exact controllability consists in proving the following: starting from any given initial state the system may be steered to rest (by a proper choice of boundary control) at a given time. This is done for the M-T-model, by systematic use of HUM (Hilbert Uniqueness Model). - In Chapter IV the one dimensional M-T-model (beams) is considered. Chapter V deals with similar questions which arise for the Kirchhoff model, the exact controllability of the model under various boundary conditions, and the behavior of the controlled system when \(\gamma\) \(\to 0.\)

In Chapter VI thermal effects are incorporated into the Kirchhoff model. In the case when the edge temperature is held at the “reference temperature” and assuming the thermal stresses to be “small”, partial exact controllability of the system is proved. Chapter VII is devoted to the reachability problem for viscoelastic plates with “long range memory”. For one type of boundary control action it is possible to establish that the system may be steered from an arbitrary initial state to a state of rest in a time T (of order \(\gamma^{1/2})\).

Reviewer: F. A. Emmerling (Neubiberg)

### MSC:

74K20 | Plates |

74-02 | Research exposition (monographs, survey articles) pertaining to mechanics of deformable solids |

93B05 | Controllability |

70Q05 | Control of mechanical systems |

35B30 | Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs |

35B37 | PDE in connection with control problems (MSC2000) |

93B99 | Controllability, observability, and system structure |