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In this paper, we first construct a differential graded Lie algebra thatcontrols deformations of a Lie-Yamaguti algebra. Furthermore, a relativeRota-Baxter operator on a Lie-Yamaguti algebra is characterized as aMaurer-Cartan element in an appropriate $L_\infty$-algebra that we buildthrough 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 onLie-Yamaguti algebras via the Yamaguti cohomology. Then we clarify therelationship between the twisted $L_\infty$-algebra and the cohomology theory.Finally as byproducts, we classify certain deformations on Lie-Yamagutialgebras using the cohomology theory.

Maurer-Cartan characterization, $L_\infty$-algebras, and cohomology of relative Rota-Baxter operators on Lie-Yamaguti algebras