Complete controllability and contractibility in multimodal systems

Mather sets for sequences of matrices and applications to the study of joint spectral radii

“…is satisfied for all i ∈ I. Note that if this is the case, then each of the maps i → A (1) i , i → A (2) , i → B i is continuous. The first of the two main results of this section is the following proposition, which guarantees the existence of upper triangularisations with favourable properties.…”

“…By Lemma 3.3, it follows that A (1) must be relatively product bounded. Now let us suppose, for a contradiction, that (A (2) ) < (A). We make estimates similar to those in the proof of Lemma 3.3.…”

“…We claim that there does not exist a subspace V of C d−r such that A (2) i V ⊆ V for all i ∈ I, and 0 < dim V < d − r. To see this, it suffices to note that if such a subspace V were to exist, then U ⊕ MV would be an invariant subspace for A with dimension strictly between r and d, which is impossible by the definition of r. We conclude via Lemma 3.2 that A (2) is relatively product bounded.…”

“…For a contradiction, let us assume that (A (1) ) < (A). Since A (2) is relatively product bounded with joint spectral radius not greater than (A), and since (A (1) ) < (A), we may choose constants C 1 , C 2 > 1 and θ ∈ (0, (A)) such that, for all n 1, we have |A (1) i1 · · · A (1) in | Cθ n and |A (2) i1 · · · A (2) in | C 2 (A) n for every (i 1 , . .…”

“…Let (A (1) , A (2) , B, M) be an upper triangularisation of A such that (A (1) ) = (A), which exists by Lemma 3.3, and suppose that the matrices comprising A (1) have the smallest possible dimension r for which this relationship can hold. It follows, in particular, that there cannot exist an upper triangularisation (C (1) , C (2) , D,M ) of A (1) such that (A (1) ) = (C (1) ), since this would lead to a new upper triangularisation of A in which the upper-left matrices have smaller than the minimum possible dimension. By Lemma 3.3, it follows that A (1) must be relatively product bounded.…”