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Let $u^\varepsilon$ and $u$ be viscosity solutions of the oscillatoryHamilton-Jacobi equation and its corresponding effective equation. Givenbounded, Lipschitz initial data, we present a simple proof to obtain theoptimal rate of convergence $\mathcal{O}(\varepsilon)$ of $u^\varepsilon\rightarrow u$ as $\varepsilon \rightarrow 0^+$ for a large class of convexHamiltonians $H(x,y,p)$ in one dimension. This class includes the Hamiltoniansfrom classical mechanics with separable potential. The proof makes use ofoptimal control theory and a quantitative version of the ergodic theorem forperiodic functions in dimension $n = 1$.

Rate of convergence for periodic homogenization of convex Hamilton-Jacobi equations in one dimension