Optimal Control of a Three-Component Electro Elastic Wave Energy Localization in a Piezoelectric Waveguide

A general formulation of the mathematical initial-boundary value problem of controlling the propagation of an electroelastic three-component wave in a waveguide made of a piezoelectric material of arbitrary anisotropy is presented. The control of the propagation of an electroacoustic wave is carried out by surface electromechanical effects in the case of non-acoustic contact. The control problem is reduced to a recurrent system of control problems for the eigenfunctions of the electromechanical oscillations of the waveguide, with the corresponding eigen harmonics of the electromechanical characteristics of the propagating wave. The control functions of the surface action are presented in the form of Fourier series of harmonics corresponding to the natural harmonics of the electromechanical oscillations of the waveguide. The problem of regulating the localization of wave energy by means of electromechanical surface effects is formulated as a problem of optimal control of the distributions of the characteristics of a three-component electroelastic wave across the thickness of a piezoelectric waveguide. Optimal control of surface effects is achieved by minimizing (or maximizing) the integral of the electromechanical wave energy along a selected thin strip across the thickness of the waveguide.