  A relative Rota-Baxter algebra is a triple $(A, M, T)$ consisting of an
algebra $A$, an $A$-bimodule $M$, and a relative Rota-Baxter operator $T$.
Using Voronov's derived bracket and a recent work of Lazarev et al., we
construct an $L_\infty [1]$-algebra whose Maurer-Cartan elements are precisely
relative Rota-Baxter algebras. By a standard twisting, we define a new
$L_\infty [1]$-algebra that controls Maurer-Cartan deformations of a relative
Rota-Baxter algebra $(A,M,T)$. We introduce the cohomology of a relative
Rota-Baxter algebra $(A, M, T)$ and study infinitesimal deformations in terms
of this cohomology (in low dimensions). As an application, we deduce cohomology
of coboundary skew-symmetric infinitesimal bialgebras and discuss their
infinitesimal deformations. Finally, we define homotopy relative Rota-Baxter
operators and find their relationship with homotopy dendriform algebras and
homotopy pre-Lie algebras.
