IDENTIFYING NONLINEAR OSCILLATORS BY AN ENERGETIC FUNCTIONAL IN THE LINEAR SPACE OF TEMPORAL BOUNDARY FUNCTIONS

Abstract

We resolve the inverse problems of a second-order nonlinear oscillator to recover time-dependent damping function and nonlinear restoring force, with the help of temporal boundary data measured at initial time and final time. By using these data, a sequence of temporal boundary functions of time is derived, which satisfy the measured temporal boundary conditions automatically, and are at least the fourth-order polynomials of time. All the temporal boundary functions and zero element constitute a linear space, and a new concept of energetic functional is introduced in the linear space, of which the energy is preserved for each energetic temporal boundary function. We employ the energetic temporal boundary functions as the bases of numerical solutions. Then, the linear systems are derived and the iterative algorithms used to recover the unknown nonlinear oscillators are developed from the energetic functional, which are convergent very fast. We can recover the damping functions and restoring forces of nonlinear oscillators, among them the nonlinear ship rolling oscillator and the Duffing nonlinear oscillator are of tested examples. The required data are parsimonious, merely the measured temporal boundary data of displacement and velocity, and the temporal boundary data of unknown function to be recovered.