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We consider the problem of convergence to zero of matrix products

Denote by

As an example of a problem in which such a question arises, let us consider one of the varieties of the stabilizability problem for discrete-time switching linear systems [

As was noted, for example, in [

If, in considering the switching system, it is assumed that there are actually no control actions, that is,

The presence of alternating factors in the products of matrices (

Every product (

The matrix products (

As an example, consider the case where sets

Let the set of matrices

Our goal is to prove that an analogue of Theorem A (on exponential convergence) is valid for the path-dependent stabilizable matrix products (

Let

To prove the theorem, we need the following auxiliary assertion.

Let the conditions of Theorem

By Definition

Given a sequence

Assuming that inequality (

Let us denote by

We now note that each of the sets

Thus, we have proven the existence of a number

We now proceed directly to the proof of Theorem

Given an arbitrary sequence

Next, consider the sequence of matrices

We continue in the same way constructing for each

Let us show that, for the obtained sequence of matrices

Let us now consider another variant of the stabilizability of matrix products (

The matrix products (

It is clear that path-independent periodically stabilized products (

Let

Denote by

Since the set of matrices

Further, repeating the proof of the corresponding part of Theorem

First of all, we would like to make the following remarks.

In the proof of Lemma

Let the sets of matrices

All the above statements remain valid for the sets of matrices

Throughout the paper, in order to avoid inessential technicalities in proofs, it was assumed that the sets of matrices

Comparing the notions of path-dependent stabilizability and path-independent periodic stabilizability, one can note that in the second of them the requirement of periodicity of the sequence

The matrix products (

It is not difficult to construct an example of the sets of square matrices in which the matrix products

Let the matrix products (

Let us consider one more issue, which is adjacent to the topic under discussion. In the theory of matrix products, the following assertion is known [

Let finite sets of matrices

The author declares that he has no conflicts of interest.

The work was carried out at the Kharkevich Institute for Information Transmission Problems, Russian Academy of Sciences, and was funded by the Russian Science Foundation (Project no. 14-50-00150).