We study the small-time controllability problem on the Lie groups $SL_2(\mathbb{ R})$ and $SL_2(\mathbb{ R})\ltimes H_{ d}(\mathbb{ R})$ with Lie bracket methods (here $H_{ d}(\mathbb{ R})$ denotes the $(2d+1)$-dimensional real Heisenberg group). Then, using unitary representations of $SL_2(\mathbb{ R})\ltimes H_{ d}(\mathbb{ R})$ on $L^2(\mathbb{ R}^d,\mathbb{ C})$ and $L^p(T^*\mathbb{ R}^d,\mathbb{ R}), p\in[1,\infty)$, we recover small-time approximate reachability properties of the Schr\"odinger PDE for the quantum harmonic oscillator, and find new small-time approximate reachability properties of the Liouville PDE for the classical harmonic oscillator.
Categories: math.OC