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 $Gamma$.
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