We develop the deformation theory of $A_\infty$ algebras together with $\infty$-inner products and identify a differential graded Lie algebra that controls the theory. This generalizes the deformation theories of associative algebras, $A_\infty$ algebras, associative algebras with inner products, and $A_\infty$ algebras with inner products.
"Deformations of associative algebras with inner products." Homology Homotopy Appl. 8 (2) 115 - 131, 2006.