Inverse coefficient problems in mechanics

The main statements and methods for studying coefficient inverse problems are presented. The classification of coefficient inverse problems is given depending on the objects being reconstructed. Three classes of tasks are highlighted: finite dimensional tasks, problems on determining one-dimensional functions, problems of multidimensional functions restoring. The main approaches in the study of each type of inverse problems are discussed. Finite-dimensional inverse problems research methods including Prony's method that allows to simplify the solution scheme for a nonlinear inverse problem are described. As an example of a finite-dimensional inverse problem, a method for determining the linear laws of strip inhomogeneity is given. Basic technics for the study of the coefficient inverse problems by the definition of one or several functions in the analysis of steady-state oscillations in various settings are given. In the first setting, the components of the physical fields inside the body are specified as additional information. As an example, the problem of determining the variable Young's modulus of a beam in the analysis of bending vibrations is given. It reduces to a problem for a linear operator equation with a compact operator. As the second typical example of coefficient inverse problem in the first formulation, the problem of determining the variable Lame coefficients when analyzing rectangle oscillations is given. It is reduced to solving the Cauchy problem for a system of partial differential equations of the first order. In the second formulation, the displacement field at the boundary of the body is given in a certain frequency range. It leads to a significantly non-linear incorrect problem. Using the example of the problem of determining the Young's modulus, shear modulus, and density for a functionally gradient cantilever rod of the constant cross section, an iterative process is developed for the analysis of longitudinal, bending and torsional vibrations. At each step, a solution for a system of Fredholm integral equations of the first kind is constructed. The issues of the uniqueness of restoring the desired characteristics are discussed.