Non-singular pre-big bang scenarios from all-order $\alpha'$ corrections

We reformulate in Hamiltonian language the recent proposal by Hohm and Zwiebach of an action yielding the most general $O(d,d)$-symmetric string cosmology equations, at tree-level in the string-loop expansion, but to all orders in the $\alpha'$ expansion. This allows us to give a simple characterization of a large class of non-singular, non-perturbative, pre-big bang scenarios smoothly interpolating between a low-energy initial accelerated (string frame) expansion and a phase of final (string and Einstein frame) decelerated expansion. Interestingly, these solutions must necessarily include, just around the bounce, a very short phase of (string-frame) contraction.