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We consider a bilinear control problem for the wave equation on a torus ofarbitrary dimension. We show that the system is globally approximatelycontrollable in arbitrarily small times from a dense family of initial states.The control strategy is explicit, and based on a small-time limit of conjugateddynamics to move along non-directly accessible directions (a.k.a. Lie bracketsof the generators).

Small-time global approximate controllability of bilinear wave equations