Estimation of higher-order damping terms in linear plate models
Timoshenko's model for computation of plates with internal damping involves a fifth order differential equation that can be written in the form (1) \(u_{tt}+Bu_ t+Au=f\) in \(\Omega\times [0,T]\), where A and B are fourth-order elliptic operators, familiar to researchers in linear elasticity. For example A can be the Kirchhoff operator \[ A\phi (x,y)=\nu \Delta (a(x,y)\Delta \phi)+(1- \nu)[(a\phi_{xx})_{xx}+2(a\phi_{xy})_{xy}+(a\phi_{yy})_{yy}], \] \[ and\quad B\phi =\Delta (b(x,y)\Delta \phi) \] (unfortunately letters a, b are abused in the article). The author sets the problem in the \(L^ 2\times H^ 2_ 0\) setting and introduces appropriate bilinear forms. Existence of solutions to (1) with zero initial conditions is easily established. Then, using a sequence of bilinear forms, convergence to a unique limit is proved in weak star \(L^{\infty}([0,T],H^ 2_ 0)\) for the Kevin-Voigt model with coefficients in the equation (1) not known, but prescribed within certain limits. Displacements at specifically assigned points are regarded as the (measured) data. Compact embedding of \(H^ 2(\Omega)\) in \(L^{\infty}(\Omega)\) is used to prove uniform convergence of the bilinear forms. The author determines optimality conditions and establishes criteria for positivity of Lagrangian multipliers. This leads to results concerning stability with respect to perturbations of the data. The article concludes with specific examples of computations. This is an excellent, carefully written article combining functional analysis, numerical methods and estimating techniques to derive results that should be of interest to engineers working on plates and shells. It is clear that authors' results should be easily extended to several types of shell models and problems in other areas of engineering.
optimal regularized output least squares estimators
regularity properties of optimal estimators
sequence of bilinear forms, convergence to a unique limit
scientific article; zbMATH DE number 4110231
scientific article