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, $A_{i}\in\mathscr{A}$ and $B_{i}\in\mathscr{B}$, 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, $\|A_{n}B_{n}\cdots A_{1}B_{1}\|\le C\lambda^{n}$, where the constants $C>0$ and $\lambda\in(0,1)$ do not depend on the sequence $\{A_{i}\}$ and the corresponding sequence $\{B_{i}\}$.


Introduction
Denote by M(p, q) the space of matrices of dimension p × q with real elements and the topology of elementwise convergence. Let A ⊂ M(N, M ) and B ⊂ M(M, N ) 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 A n B n · · · A 1 B 1 , for all possible sequences of matrices {A i }, due to a suitable choice of sequences of matrices {B i }.
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][2][3][4][5]. Consider a system whose dynamics is described by the equations where the first of them describes the functioning of a plant, whose properties uncontrollably affected by perturbations from the class A, while the second equation describes the behavior of a controller. Then, by choosing a suitable sequence of controls {B n ∈ B}, one can try to achieve the desired behavior of the system (2), for example, the convergence to zero of its solutions x(n) = A n B n · · · A 1 B 1 x(0).
As was noted, e.g., 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, i.e. B n ≡ I, then equations (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 the class A, that is, into the problem of convergence to zero of the solutions of equation (3) for all possible sequences of matrices {A i ∈ A}. Convergence to zero of the matrix products A n · · · A 1 , arising in this case, has been investigated by many authors, see, e.g., [2,[8][9][10][11] as well as the bibliography in [12]. The presence of alternating factors in the products of the matrices (1) substantially complicates the problem of convergence of the corresponding matrix products for all possible sequences of matrices {A i ∈ A} due to a suitable choice of sequences of matrices {B i ∈ B} in comparison with the problem of convergence of matrix products A n · · · A 1 for all possible sequences of matrices {A i ∈ A}. A discussion of the arising difficulties can be found, e.g., 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.

Path-Dependent Stabilizability
Every product (1) is a matrix of dimension N × N ; i.e. it is an element of the space M(N, N ). As is known, the space M(N, N ) 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 M(N, N ) 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 M(N, N ) is submultiplicative, that is, such that for any two matrices X, Y , the inequality XY ≤ X · Y holds. In particular, a norm on M(N, N ) is submultiplicative if it is generated by some vector norm on R N , that is, its value on the matrix A is defined by the equality A = sup x =0 Ax x , where x and Ax are the norms of the corresponding vectors in R N . Definition 1. The matrix products (1) are said to be path-dependent stabilizable by choosing the factors {B n } if for any sequence of matrices {A n ∈ A} there exists a sequence of matrices {B n ∈ B} for which As an example, consider the case where the sets A and B consist of square matrices of dimension N × N , and B = {I}, where I 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 {A n ∈ A}. As is known, in this case the convergence (5) is uniformly exponential.
Theorem on Exponential Convergence. Let the set of matrices A be such that for each sequence {A n ∈ A} the convergence (5) holds. Then there exist constants C > 0 and λ ∈ (0, 1) such that A n · · · A 1 ≤ Cλ n , n = 1, 2, . . . , Our goal is to prove that an analogue of Theorem on Exponential Convergence is valid for the path-dependent stabilizable matrix products (1). Theorem 1. Let A and B be the sets of matrices for which the matrix products (1) are pathdependent stabilizable. Then there exist constants C > 0 and λ ∈ (0, 1) such that for any sequence of matrices {A n ∈ A} there is a sequence of matrices {B n ∈ B} for which To prove the theorem we need the following auxiliary assertion.
Lemma 1. Let the conditions of Theorem 1 be satisfied. Then there exist constants k * > 0 and µ ∈ (0, 1) such that for any sequence of matrices {A n ∈ A} there is a positive integer k ≤ k * and a set of matrices Proof. By Definition 1 of the path-dependent stabilizability of the matrix products (1) for each matrix sequence {A n ∈ A} there exists a natural k such that for some sequence of matrices {B n ∈ B}.
Given a sequence {A n } let us denote by k({A n }) the smallest k under which the inequality (7) holds. To prove the lemma it suffices to show that the quantities k({A n }) are uniformly bounded, that is, there is a k * such that Assuming that the inequality (8) is not true, for each positive integer k we can find a sequence {A for each positive integer n }) − 1. Let us denote by A k the set of all sequences {A n ∈ A}, for each of which the inequalities (9) hold. Then {A (k) n } ∈ A k and, therefore, A k = ∅. Moreover, and each set A k is closed since inequalities (9) hold for all its elements, sequences {A n } ∈ A k , for each positive integer m ≤ k − 1.
We now note that each of the sets A k is a subset of the topological space A ∞ of all sequences {A n ∈ A} with the topology of infinite direct product of the finite set of matrices A. By the Tikhonov theorem in this case A ∞ is a compact. Then, each of the sets A k is also a compact.
In this case it follows from (10) that ∞ k=1 A k = ∅ and, therefore, there is a sequence {Ā n ∈ A} such that By the definition of the sets A k , for the sequence {Ā n ∈ A} the inequalities Ā m B m · · ·Ā 1 B 1 ≥ 1 hold for each m ≥ 1 and any B 1 , . . . , B m ∈ B which contradicts the assumption of the pathdependent stabilzability of the matrix products (1). This contradiction completes the proof of the existence of a number k * for which the inequalities (8) are valid.
Thus, we have proved the existence of a number k * such that, for each sequence{A n ∈ A} and some corresponding sequence {B n ∈ B}, strict inequalities (7) are satisfied with k = k({A n }) ≤ k * . Moreover, since the number of all products A k B k · · · A 1 B 1 participating in the inequalities (7) is finite, then the corresponding inequalities (7) can be strengthened: there is a µ ∈ (0, 1) such that for any sequence of matrices {A n ∈ A} there exist a natural k ≤ k * and a set of matrices B 1 , . . . , B k ∈ B for which A k B k · · · A 1 B 1 ≤ µ < 1.
We now proceed directly to the proof of Theorem 1.
Proof of Theorem 1. Given an arbitrary sequence {A n ∈ A}, then by Lemma 1 there is a number k 1 ≤ k * and a set of matrices B 1 , . . . , B k1 such that Next, consider the sequence of matrices {A n ∈ A, n ≥ k 1 + 1} (the 'tail' of the sequence {A n ∈ A} starting with the index k 1 + 1). Again, by virtue of Lemma 1, there is a k 2 ≤ k 1 + k * and a set of matrices B k1+1 , . . . , B k2 such that We continue in the same way constructing for each m = 3, 4, . . . numbers and sets of matrices B km−1+1 , . . . , B km for which Let us show that for the obtained sequence of matrices {B n } for some C > 0 and λ ∈ (0, 1), that do not depend on the sequences {A n } and {B n }, the inequalities (6) are valid. Fix a positive integer n and specify for it a number p = p(n) such that Such p exists, since the sequence {k m } strictly increases by construction. We now represent the product A n B n · · · A 1 B 1 in the form Then (since the sets A and B are finite then κ < ∞). Further, by the definition of the matrices D i and the inequalities (12), Taking into account that by virtue of (11), for each m, the estimate k m ≤ k * m is fulfilled, from here and from (13) we obtain for the number p a lower estimate: p ≥ n k * − 1. And then from the estimates established earlier for D * , D 1 , . . . , D m we deduce that Hence, putting C = κ k * µ , λ = µ 1 k * , we obtain the inequalities (6).

Path-Independent Stabilizability
Let us now consider another variant of the stabilizability of matrix products (1) due to a suitable choice of matrices {B i }.
Definition 2. The matrix products (1) are said to be path-independent periodically stabilizable by choosing the factors {B n } if there exists a periodic sequence of matrices {B n ∈ B} such that for any sequence of matrices {A n ∈ A}.
It is clear that path-independent periodically stabilized products (1) are path-dependent stabilized.
Theorem 2. Let A and B be the sets of matrices for which the matrix products (1) are pathindependent periodically stabilizable by a sequence of matrices {B n ∈ B}. Then there exist constants C > 0 and λ ∈ (0, 1) such that A nBn · · · A 1B1 ≤ Cλ n , n = 1, 2, . . . , for any sequence of matrices {A n ∈ A}.
Proof. Denote by p the period of the sequence {B n }. Consider the set of (N × N )-matrices Since the set of matrices A is finite, the set D is also finite. Moreover, by Definition 2 of path-independent periodic stabilization, for each sequence {A n ∈ A}. Hence for each sequence {D n ∈ D} there is also D n · · · D 1 → 0 as n → ∞.
In this case, by Theorem on Exponential Convergence, there are k * > 0 and µ ∈ (0, 1) such that or, equivalently, Further, repeating the proof of the corresponding part of Theorem 1 word for word, we derive from the inequalities (16) the existence of constants C > 0 and λ ∈ (0, 1) such that for any sequence of matrices {A n ∈ A} the inequalities (15) hold.

Remarks and Open Questions
First of all, we would like to make the following remarks. Remark 1. In the proof of Lemma 1, 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 {A n ∈ A} there exist a natural k = k({A n }) and a collection of matrices B 1 , . . . , B k ∈ B for which equality (7) holds. Correspondingly, the statement of Theorem 1 is valid under weaker assumptions.
Theorem 3. Let the sets of matrices A and B be such that for each matrix sequence {A n ∈ A} there are a natural k and a collection of matrices B 1 , . . . , B k ∈ B for which Then there exist constants C > 0 and λ ∈ (0, 1) such that for any sequence of matrices {A n ∈ A} there is a sequence of matrices {B n ∈ B} for which A n B n · · · A 1 B 1 ≤ Cλ n , n = 1, 2, . . . .

Remark 2.
All the above statements remain valid for the sets of matrices A and B with complex elements.
Remark 3. Throughout the paper, in order to avoid inessential technicalities in proofs, it was assumed that the sets of matrices A and B are finite. In fact, all the above statements remain valid in the case when the sets of matrices A and B 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 {B n } stabilizing the matrix products (1) appeared. Therefore, the following less restrictive concept of path-independent stabilizability seems rather natural.
Definition 3. The matrix products (1) are said to be path-independent stabilizable by choosing the factors {B n } if there is a sequence of matrices {B n ∈ B} such that the convergence (14) holds for any sequence of matrices {A n ∈ A}.
It is not difficult to construct an example of the sets of square matrices in which the matrix products A nBn · · · A 1B1 converge slowly enough, slower than any geometric progression. For this it is enough to put A = {I}, B = {I, λI}, where λ ∈ (0, 1), and define the sequence {B} so that the matrix λI appears in it "fairly rare", at positions with numbers k 2 , k = 1, 2, . . . . Question 1. Let the matrix products (1) be path-independent stabilizable by choosing a certain sequence of matrices {B n ∈ B}. Is it possible in this case to specify a sequence of matrices {B n ∈ B} (possibly different from {B n ∈ B}) and constants C > 0 and λ ∈ (0, 1) such that for any sequence of matrices {A n ∈ A} for all n = 1, 2, . . . the inequalities A nBn · · · A 1B1 ≤ Cλ n 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][9][10][11]: let A be a finite set such that for each sequence of matrices {A n ∈ A} the sequence of norms { A n · · · A 1 , n = 1, 2, . . .} is bounded.
Then all such sequences of norms for the matrices are uniformly bounded, that is, there exists a constant C > 0 such that A n · · · A 1 ≤ C, n = 1, 2, . . . , for each sequence of matrices {A n ∈ A}.
Question 2. Let finite sets of matrices A and B be such that for each sequence of matrices {A n ∈ A} there is a sequence of matrices {B n ∈ A} for which the sequence of norms { A n B n · · · A 1 B 1 , n = 1, 2, . . .} is bounded. Does there exist in this case a constant C > 0 such that for every matrix sequence {A n ∈ A} there is a sequence of matrices {B n ∈ A}, for which the sequence of norms { A n B n · · · A 1 B 1 , n = 1, 2, . . .} is uniformly bounded, that is, for all n = 1, 2, . . . the inequalities A n B n · · · A 1 B 1 ≤ C hold?