  We consider driving multi-orbital Mott insulators using laser radiation. We
derive general expressions for periodically driven spin-orbital models using
time-dependent perturbation theory in the strong interaction limit. We show
that the effective exchange interactions of the Floquet spin-orbital
Hamiltonian are highly tunable via variations of the frequency, amplitude, and
polarization of the laser. We also take the effect of finite bandwidth of
excitations into account and study possible heating effects. We further apply
our formalism to orthorhombic titanates YTiO$_3$ and LaTiO$_3$ based on
first-principles calculations, and find that the spin exchange interactions in
these compounds can be engineered to a large extent by tuning the frequency and
electric-field amplitude of the laser.
