Given a constant $0 < \alpha < 1$, a matrix $A \in R^{n \times n}$ and a vector $b \in \mathbb{R}^n$, it is well-known that a sufficient condition for the iteration $x_{t+1} = \alpha A x_t + b$ to converge is that the spectral radius of A is bounded by one.
This question concerns a generalization of this problem to the case where we have a finite set of pairs $(A^j,b^j)$. Consider the set $A^R$ of matrices obtained by taking rows from the matrices $A^j$, i.e. $A^R = \{ A: \forall r. \exists j. A^j_{r,:} = A_{r,:} \}$. Here by $A_{r,:}$ we denote the rth row of $A$. We assume that the joint spectral radius of the set $A^R$ is bounded by one, i.e. the spectral radius of any finite product of the elements of $A^R$ is at most one.
Now, consider the following iteration.
$ \forall i. e_i^\top x_{t+1} = \max_j e_i^{\top} (\alpha A^j x_{t} + b^j)$
Here by $e_i$ we denote the canonical basis vector.
My questions are:
- Given the condition that the joint spectral radius of the set $A^R$ is bounded by one, does this converge to a single value?
- Does this converge to the same thing regardless of the starting value?
- If it doesn't converge to a single value, does it converge to a limiting cycle (i.e. a periodic orbit)?
Note: This is a generalized version of this question, where the difference is in a weaker condition on the $A_j$s.
I would also be very grateful for hints or incomplete solutions.