It is known that there are lattice models in which non-interacting particles get dynamically localized when periodic $delta$-function kicks are applied with a particular strength. We use both numerical and analytical methods to study the effects of interactions in three different models in one dimension. The systems we have considered include spinless fermions with interactions between nearest-neighbor sites, the Hubbard model of spin-1/2 fermions, and the Bose Hubbard model with on-site interactions. We derive effective Floquet Hamiltonians up to second order in the time period of kicking. Using these we show that interactions can give rise to a variety of interesting results such as two-body bound states in all three models and dispersionless many-body bound states for spinless fermions and bosons. We substantiate these results by exact diagonalization and stroboscopic time evolution of systems with a finite number of particles. We derive a low-energy pseudo-spin-1/2 limit of the Bose Hubbard system in the thermodynamic limit and show that a special case of this has an exponentially large number of ground states. Finally we study the effect of changing the strength of the $delta$-function kicks slightly away from perfect dynamical localization; we find that a single particle remains dynamically localized for a long time after which it moves ballistically.
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