Boundary stabilization of thin plates.

*(English)*Zbl 0696.73034
SIAM Studies in Applied Mathematics, 10. Philadelphia: Society for Industrial and Applied Mathematics. viii, 176 p. $ 36.50 (1989).

This book is intended to provide a comprehensive and unified treatment of asymptotic stability of the motion of a thin plate when appropriate stabilizing feedback mechanisms acting through forces and moments are introduced along a part of the edge of the plate.

In Chapter 1 a specific example of the sort of problems to be studied is presented. In Chapter 2 five different models of thin plates are derived. Three are purely elastic: the fourth-order Kirchhoff model, the sixth- order Mindlin-Timoshenko (MT) model and that of Kármán (large deflection). The fourth model is a viscoelastic plate and the last a thermoelastic plate. In Chapter 3 the M-T-system is studied. The plate is assumed to be clamped along a portion of its edge, while forces and moments are applied on the remainder of the boundary. The influence of the shear modulus K of the M-T-system is discussed in the two limiting situations \(K\to 0\) and \(K\to \infty\) in Chapter 4. Uniform stabilization in nonlinear plate problems is the subject of Chapter 5. The author considers first the problem of stabilizing a linear Kirchhoff plate by using a nonlinear velocity feedback in the shear force at the boundary. The second topic is boundary stabilization of the Kármán model. In Chapter 6 a viscoelastic plate with long-range memory is studied. At last in Chapter 7 a uniform decay rate is established for the thermoelastic energy of a Kirchhoff plate which is subject to thermal forces that contribute to plate deformation. At the end of the introduction the author specifies the problems which could not be studied in this book as anisotropic plates, composite structures, influence of external forces, etc.

The book is of interest for mathematicians and engineers, who are working in analysis and design of feedback stabilizers.

In Chapter 1 a specific example of the sort of problems to be studied is presented. In Chapter 2 five different models of thin plates are derived. Three are purely elastic: the fourth-order Kirchhoff model, the sixth- order Mindlin-Timoshenko (MT) model and that of Kármán (large deflection). The fourth model is a viscoelastic plate and the last a thermoelastic plate. In Chapter 3 the M-T-system is studied. The plate is assumed to be clamped along a portion of its edge, while forces and moments are applied on the remainder of the boundary. The influence of the shear modulus K of the M-T-system is discussed in the two limiting situations \(K\to 0\) and \(K\to \infty\) in Chapter 4. Uniform stabilization in nonlinear plate problems is the subject of Chapter 5. The author considers first the problem of stabilizing a linear Kirchhoff plate by using a nonlinear velocity feedback in the shear force at the boundary. The second topic is boundary stabilization of the Kármán model. In Chapter 6 a viscoelastic plate with long-range memory is studied. At last in Chapter 7 a uniform decay rate is established for the thermoelastic energy of a Kirchhoff plate which is subject to thermal forces that contribute to plate deformation. At the end of the introduction the author specifies the problems which could not be studied in this book as anisotropic plates, composite structures, influence of external forces, etc.

The book is of interest for mathematicians and engineers, who are working in analysis and design of feedback stabilizers.

Reviewer: W.Schnell

##### MSC:

74H45 | Vibrations in dynamical problems in solid mechanics |

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

74K20 | Plates |

93D20 | Asymptotic stability in control theory |