  The $L_\infty$-algebra is an algebraic structure suitable for describing
deformation problems. In this paper we construct one $L_\infty$-algebra, which
turns out to be a differential graded Lie algebra, to control the deformations
of Lie algebroids and a second one to control the deformations of Lie
subalgebroids. We also combine these two $L_\infty$-algebras into one to
control the simultaneous deformations of a Lie algebroid and its Lie
subalgebroids. The results generalize the deformation theory of Lie algebra and
Lie subalgebras. Applications of our results include deformations of
foliations, deformations of complex structures and deformations of
homomorphisms of Lie algebroids.
