  We propose a method for controlling the exchange interactions of Mott
insulators with strong spin-orbit coupling. We consider a multiorbital system
with strong spin-orbit coupling and a circularly polarized light field and
derive its effective Hamiltonian in the strong-interaction limit. Applying this
theory to a minimal model of $\alpha$-RuCl$_{3}$, we show that the magnitudes
and signs of three exchange interactions, $J$, $K$, and $\Gamma$, can be
changed simultaneously. Then, considering another case in which one of the
hopping integrals has a different value and the other parameters are the same
as those for $\alpha$-RuCl$_{3}$, we show that the Heisenberg interaction $J$
can be made much smaller than the anisotropic exchange interactions $K$ and
$\Gamma$.
