The Optics and Optimal Control Theory Interpretation of the Parametric Resonance

Abstract

The aim of the article is the elaboration of parametric resonance theory at piecewise constant frequency modulation. The investigation is based on the analogy with optics and optimal control theory (OCT) application. The exact expressions of oscillation frequency, gain/damping coefficients, dependencies of these coefficients on the modulation depth, duty ratio and initial phase are derived. First of all, the results obtained on the basis of the energy behavior analysis (at the conjunction conditions execution) in frictionless systems are presented. The well-known parametric resonance triggering condition is revised and adjusted. The heuristic feedback introduction (based on the energy behavior analysis) in the oscillation equation permits one to prove that the frequency modulation satisfying the parametric resonance condition is not necessary and sufficient condition of the oscillations unlimited increase. Their damping/shaking up formally corresponds by the frequency and duty ratio to the condition of the equality of optical paths to the quarter-wavelength characteristic of the interference filter or mirror. The unity of space-time coordinates shows itself in this specific form of the optical-mechanical analogy due to the general Hill’s equation description. It is marked that this equation theory underlies most of metamaterials advantages because all transport phenomena imply different wave – electromagnetic, acoustic, spin etc. propagation one way or another. The question about control uniqueness arises that is modulating frequency, duty ratio and signature sign uniqueness. Another question of characteristic index extremum at different controls is tightly bound with the former. The answers to these questions are obtained on the basis of OCT. The similarity of the optimal control problem solution and the one obtained at the heuristic feedback introduction through fundamental solutions product permits one to introduced the new form named general or mixed Hamiltonian along with the ordinary and OCT Hamiltonians. Besides this mixed Hamiltonian equality to zero together with the Wronskian constancy (almost everywhere) is the useful analogous in form to the Liouville’s theorem equation. The nonlinearity accounting using the OCT formalism is described too.