Temporal homogenization for linear ordinary differential equations
Molei Tao, Houman Owhadi · arXiv (Cornell University) · 2013
Using a homogenization approach, we approximate the long-time solution of $\dot{x}=Ax+\epsilon P(t)x$ by $x(t)\approx \exp(At) \exp(\epsilon B t) x(0)$, where $B$ is an effective matrix and the solution of (the analogue of) a cell problem. We provide two ways for explicit computations of $B$, both at a cost independent of $\epsilon$. Although close in spirit to classical averaging theory, the algebraic nature of our analysis allows the following points: (i) the approximation error is shown to be $\mathcal{O}(\epsilon)$ up to time $\mathcal{O}(\epsilon^{-1})$ (which is sharper than the $o(1)$ result of classical general averaging theory); (ii) we provide a necessary and sufficient condition for accuracy, which permits the method to be used in settings where the validity of classical averaging theory has not been established. Furthermore, the algebraic identification of the effective matrix $B$ allows us to apply this method to (a) control (possibly damped) harmonic oscillators via parametric resonance; (b) identify specific coupling mechanisms between damped harmonic oscillators that lower the threshold on fluctuation amplitude required for energy harvesting via parametric resonance in presence of dissipative effects.