  In this paper, we first construct a differential graded Lie algebra that
controls deformations of a Lie-Yamaguti algebra. Furthermore, a relative
Rota-Baxter operator on a Lie-Yamaguti algebra is characterized as a
Maurer-Cartan element in an appropriate $L_\infty$-algebra that we build
through the graded Lie bracket of Lie-Yamaguti algebra's controlling algebra,
and gives rise to a twisted $L_\infty$-algebra that controls its deformation.
Next we establish the cohomology theory of relative Rota-Baxter operators on
Lie-Yamaguti algebras via the Yamaguti cohomology. Then we clarify the
relationship between the twisted $L_\infty$-algebra and the cohomology theory.
Finally as byproducts, we classify certain deformations on Lie-Yamaguti
algebras using the cohomology theory.
