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Much having learned about Floquet dynamics of pseudospin-$1/2$ system namely,graphene, we here address the stroboscopic properties of a periodically kicked{three-band fermionic system such as $\alpha$-T$_3$ lattice. This particularmodel provides an interpolation between graphene and dice lattice via thecontinuous tuning of the parameter $\alpha$ from 0 to 1.} In the case of dicelattice ($\alpha=1$), we reveal that one can, in principle, engineer varioustypes of low energy dispersions around some specific points in the Brillouinzone by tuning the kicking parameter in the Hamiltonian along a particulardirection. Our analytical analysis shows that one can experience differentquasienergy dispersions for example, Dirac type, semi-Dirac type, gapless line,absolute flat quasienergy bands, depending on the specific values of thekicking parameter. Moreover, we numerically study the dynamics of a wave packetin dice lattice. The quasienergy dispersion allows us to understand theinstantaneous structure of wave packet at stroboscopic times. We find asituation where absolute flat quasienergy bands lead to a complete dynamicallocalization of the wave packet. {Aditionally, we calculate the quasienergyspectrum numerically for $\alpha$-T$_3$ lattice. A periodic kick in aperpendicular (planar) direction breaks (preserves) the particle-hole symmetryfor $0<\alpha<1$. Furthermore, it is also revealed that the dynamicallocalization of wave packet does not occur at any intermediate $\alpha \ne0,\,1$.}

Floquet engineering of low-energy dispersions and dynamical localization in a periodically kicked three-band system