On convergence of infinite matrix products with alternating factors from two sets of matrices

We consider the problem of convergence to zero of matrix products $A_n{B}_n\cdots A_1{B}_1$ with factors from two sets of matrices, $A_i\in \mathcal{A}$ and $B_i\in \mathcal{B}$, due to a suitable choice of matrices $\{B_i\}$. It is assumed that for any sequence of matrices $\{A_i\}$ there is a sequence of matrices $\{B_i\}$ such that the corresponding matrix product $A_n{B}_n\cdots A_1{B}_1$ converges to zero. We show that in this case the convergence of the matrix products under consideration is uniformly exponential, that is, $\mathrm{\parallel }{A}_n{B}_n\cdots A_1{B}_1\mathrm{\parallel }\le C{\lambda }^n$, where the constants $C>0$ and $\lambda \in (0,1)$ do not depend on the sequence $\{A_i\}$ and the corresponding sequence $\{B_i\}$.