On Convergence of Infinite Matrix Products with Alternating Factors from Two Sets of Matrices

Abstract

We consider the problem of convergence to zero of matrix products with factors from two sets of matrices, and , due to a suitable choice of matrices . It is assumed that for any sequence of matrices there is a sequence of matrices such that the corresponding matrix products converge to zero. We show that, in this case, the convergence of the matrix products under consideration is uniformly exponential; that is, , where the constants and do not depend on the sequence and the corresponding sequence . Other problems of this kind are discussed and open questions are formulated.

1. Introduction

Denote by the space of matrices of dimension with real elements and the topology of elementwise convergence. Let and be finite sets of matrices. We will be interested in the question of whether it is possible to ensure the convergence to zero of matrix products, for all possible sequences of matrices , due to a suitable choice of sequences of matrices .

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 [1–5]. Consider a system whose dynamics is described by the equations where the first of them describes the functioning of a plant, whose properties are uncontrollably affected by perturbations from class , while the second equation describes the behavior of a controller. Then, by choosing a suitable sequence of controls , one can try to achieve the desired behavior of system (2), for example, the convergence to zero of its solutions:

As was noted, for example, in [6, 7], the question of the stabilizability of matrix products with alternating factors from two sets, due to a special choice of factors from one of these sets, can also be treated in the game-theoretic sense.

If, in considering the switching system, it is assumed that there are actually no control actions, that is, , then (2) take the form In this case, the problem of the stabilizability of the corresponding switching system turns into the problem of its stability for all possible perturbations of the plant in class , that is, into the problem of convergence to zero of the solutions of (4) for all possible sequences of matrices . Convergence to zero of the matrix products , arising in this case, has been investigated by many authors (see, e.g., [2, 8–11], as well as the bibliography in [12]).

The presence of alternating factors in the products of matrices (1) substantially complicates the problem of convergence of the corresponding matrix products for all possible sequences of matrices due to a suitable choice of sequences of matrices in comparison with the problem of convergence of matrix products for all possible sequences of matrices . A discussion of the arising difficulties can be found, for example, in [13]. One of the applications of the results obtained in this paper for analyzing the new concept of the so-called minimax joint spectral radius is also described there.

2. Path-Dependent Stabilizability

Every product (1) is a matrix of dimension ; that is, it is an element of the space . As is known, the space with the topology of elementwise convergence is normable; therefore we assume that is some norm in it. We note here that since all norms in the space are equivalent, the choice of a particular norm when considering the convergence of products (1) is inessential. Nevertheless, in what follows, it will be convenient for us to assume that the norm in is submultiplicative; that is, for any two matrices , the inequality holds. In particular, a norm on is submultiplicative if it is generated by some vector norm on ; that is, its value on matrix is defined by the equality , where and are the norms of the corresponding vectors in .

Definition 1. The matrix products (1) are said to be path-dependent stabilizable by choosing the factors if for any sequence of matrices there exists a sequence of matrices for which

As an example, consider the case where sets and consist of square matrices of dimension , and , where is the identical matrix. In this case, Definition 1 of the path-dependent stabilizability of the matrix products (1) reduces to the following condition: for each sequence . As is known, in this case, convergence (7) is uniformly exponential. Namely, the following statement, which was repeatedly “discovered” by many authors, is true (see, e.g., [2, 8–11]).

Theorem A (on exponential convergence). Let the set of matrices be such that for each sequence convergence (7) holds. Then there exist constants and such that for each sequence .

Our goal is to prove that an analogue of Theorem A (on exponential convergence) is valid for the path-dependent stabilizable matrix products (1).

Theorem 2. Let and be the sets of matrices for which the matrix products (1) are path-dependent stabilizable. Then there exist constants and such that for any sequence of matrices there is a sequence of matrices for which

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

Lemma 3. Let the conditions of Theorem 2 be satisfied. Then there exist constants and such that for any sequence of matrices there is a positive integer and a set of matrices for which .

Proof. By Definition 1 of the path-dependent stabilizability of the matrix products (1) for each matrix sequence there exists a natural such that for some sequence of matrices .
Given a sequence , let us denote by the smallest under which inequality (10) holds. To prove the lemma, it suffices to show that the quantities are uniformly bounded; that is, there is a such that Assuming that inequality (11) is not true, for each positive integer , we can find a sequence such that . In this case, by the definition of the number , for each positive integer .
Let us denote by the set of all sequences , for each of which inequalities (12) hold. Then and, therefore, . Moreover, and each set is closed since inequalities (12) hold for all its elements; sequences , for each positive integer .
We now note that each of the sets is a subset of the topological space of all sequences with the topology of infinite direct product of the finite set of matrices . By the Tikhonov theorem in this case is a compact. Then, each of the sets is also a compact. In this case, it follows from (13) that and, therefore, there is a sequence such that By the definition of the sets , for the sequence , the inequalities hold for each and any which contradicts the assumption of the path-dependent stabilizability of the matrix products (1). This contradiction completes the proof of the existence of a number for which inequalities (11) are valid.
Thus, we have proven the existence of a number such that, for each sequence and some corresponding sequence , strict inequalities (10) are satisfied with . Moreover, since the number of all products participating in inequalities (10) is finite, then the corresponding inequalities (10) can be strengthened: there is a such that for any sequence of matrices there exist a natural and a set of matrices for which .

We now proceed directly to the proof of Theorem 2.

Proof of Theorem 2. Given an arbitrary sequence , by Lemma 3, there exist a number and a set of matrices such that Next, consider the sequence of matrices (the “tail” of the sequence starting with the index ). Again, by virtue of Lemma 3, there exist a and a set of matrices such that We continue in the same way constructing for each numbers and sets of matrices for which Let us show that, for the obtained sequence of matrices for some and , which do not depend on the sequences and , inequalities (9) are valid. Fix a positive integer and specify for it a number such that Such exists, since the sequence strictly increases by construction. We now represent the product in the form where Then (since the sets and are finite, ). Further, by the definition of the matrices and inequalities (19), Taking into account the fact that, by virtue of (18), for each , the estimate is fulfilled, from here and from (20) we obtain for the number a lower estimate: . And then from the estimates established earlier for , , we deduce that Hence, putting and , we obtain inequalities (9).

3. Path-Independent Stabilizability

Let us now consider another variant of the stabilizability of matrix products (1) due to a suitable choice of matrices .

Definition 4. The matrix products (1) are said to be path-independent periodically stabilizable by choosing the factors if there exists a periodic sequence of matrices such that for any sequence of matrices .

It is clear that path-independent periodically stabilized products (1) are path-dependent stabilized.

Theorem 5. Let and be the sets of matrices for which the matrix products (1) are path-independent periodically stabilizable by a sequence of matrices . Then there exist constants and such that for any sequence of matrices .

Proof. Denote by the period of the sequence . Consider the set of -matrices: Since the set of matrices is finite, set is also finite. Moreover, by Definition 4 of path-independent periodic stabilization, for each sequence . Hence, for each sequence , there is also In this case, by Theorem A (on exponential convergence), there are and such that or, equivalently, Further, repeating the proof of the corresponding part of Theorem 2 word for word, we derive from inequalities (32) the existence of constants and such that for any sequence of matrices inequalities (27) hold.

4. Remarks and Open Questions

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

Remark 6. In the proof of Lemma 3, in fact, we used not the condition of path-dependent stabilizability of the matrix products (1) but the weaker condition that for each matrix sequence there exist a natural and a collection of matrices for which equality (10) holds. Correspondingly, the statement of Theorem 2 is valid under weaker assumptions.

Theorem 7. Let the sets of matrices and be such that for each matrix sequence there are a natural and a collection of matrices for which Then there exist constants and such that for any sequence of matrices there is a sequence of matrices for which

Remark 8. All the above statements remain valid for the sets of matrices and with complex elements.

Remark 9. Throughout the paper, in order to avoid inessential technicalities in proofs, it was assumed that the sets of matrices and are finite. In fact, all the above statements remain valid in the case when the sets of matrices and are compacts, not necessarily finite, that is, are closed and precompact.

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 stabilizing the matrix products (1) appeared. Therefore, the following less restrictive concept of path-independent stabilizability seems rather natural.

Definition 10. The matrix products (1) are said to be path-independent stabilizable by choosing the factors if there is a sequence of matrices such that convergence (26) holds for any sequence of matrices .

It is not difficult to construct an example of the sets of square matrices in which the matrix products converge slowly enough, slower than any geometric progression. For this, it is enough to put and , where , and define sequence so that the matrix appears in it “fairly rare,” at positions with numbers ,

Question 11. Let the matrix products (1) be path-independent stabilizable by choosing a certain sequence of matrices . Is it possible in this case to specify a sequence of matrices (possibly different from ) and constants and such that, for any sequence of matrices for all , the inequalities will be valid?

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 [2, 8–11]: let be a finite set such that for each sequence of matrices the sequence of norms is bounded. Then all such sequences of norms for the matrices are uniformly bounded; that is, there exists a constant such that for each sequence of matrices .

Question 12. Let finite sets of matrices and be such that for each sequence of matrices there is a sequence of matrices for which the sequence of norms is bounded. Does there exist in this case a constant such that for every matrix sequence there is a sequence of matrices , for which the sequence of norms is uniformly bounded, that is, for all , the inequalities hold?

Conflicts of Interest

The author declares that he has no conflicts of interest.

Acknowledgments

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).