The identification of external force dynamics in the modeling of vibration

The linear homogeneous wave equations with initial and boundary conditions are considered. It is assumed that the non-uniform terms describing an external force, are expanded into Fourier series, and the objective is to determine its $N$ time depending coefficients. In order to determine these coefficients uniquely, $N$ non-local boundary conditions are introduced in accordance with the required averaged dynamics of oscillations. The sufficient conditions are given when the formulated problem enjoy unique classical solution, which can be found by solving the system of Volterra integral equations, explicitly built in this work. Kernels of such integral equations enable resolvent construction using Laplace transform. This problem statement and method can be generalized and applied for system of inhomogeneous wave equations. These results can be useful in the formulation and solution of some problems arising in the optimization of boundary controls of string vibrations.