Sufficient conditions for assignability of nonuniform dichotomy spectrum of discrete time-varying linear systems

We consider a version of the pole placement problem for tempered one-sided linear discrete-time time-varying linear systems. We prove a sufficient condition for assignability of the nonuniform dichotomy spectrum by linear feedback. The main result is that the nonuniform dichotomy spectrum is assignable if the system is completely controllable and certain lower asymptotic bound for the controllability Gramian holds.


Introduction
The concept of exponential dichotomy is closely related to the problem of inferring the stability of a nonlinear system based on its linear approximation. This problem was first formulated by Lyapunov (see [1]), who showed that for the class of regular systems introduced by him, the stability of the linear approximation implies the local stability of the nonlinear system. In [2], Perron showed that this implication is also true for nonlinear systems, the linear approximation of which has the property that its inhomogeneous version has a bounded solution to any bounded inhomogeneity. Perron thus introduced for the first time a class of systems that we now call systems with an exponential dichotomy.
On the other hand, Maizel [3] extended the Persidskii criterion [4] to the case of conditional stability and Malkin's criterion [5] on the stability of the first approximation, and thus defined two classes of linear differential systems for which the Lyapunov problem also has a positive answer. It turned out [3] that these two classes coincide with each other and with the class of systems defined by Perron-thus, the class of exponentially dichotomous systems was characterized from three essentially different points of view.
The works of Perron and Maisel were preceded by the works of Hadamard [6] and Bohl [7], which contained essentially the same key ideas that later led to the concept of exponential dichotomy.
The class of exponentially dichotomic systems was singled out as an independent subject of research with its own problems in a series of works [8,9] by Massera and Schaeffer, who also coined the name "exponential dichotomy" and formulated the canonical definition of exponentially dichotomous systems.
The final formulation of fundamentals of the theory of exponentially dichotomous systems was completed thanks to the monographs of Massera and Schaeffer [10] and Daleckii and Krein [11], who summarized the earlier obtained results on exponentially dichotomous systems and outlined the development paths of this theory by setting many new problems and by formulating interesting and deep open questions.
Later, an equally important role in the development of the theory of dichotomy was played by Coppel's monograph [12], in which the results on finite-dimensional exponentially dichotomous systems obtained before 1978 were collected and strengthened.
The effectiveness of the notion of exponential dichotomy both in the study of the asymptotic behavior of solutions of nonlinear systems, the first approximation of which is exponentially dichotomous, and in its applications to dynamical systems, has caused various generalizations of this notion both within the theory of linear differential systems itself and beyond-in the theory of evolution operators and the theory of linear extensions of dynamical systems. One of them is the concept of nonuniform exponential dichotomy, which is defined in the literature in many nonequivalent ways (see [13][14][15][16]).
The concept of the dichotomy is closely related to the concept of the dichotomy spectrum, introduced by Sacker and Sell in [17]. In [17], the authors developed the Sacker-Sell spectral theory, which is now also called dichotomy spectrum for nonautonomous differential equations [18]. Nowadays, the dichotomy spectrum is an important tool in the qualitative theory of dynamical systems. This is due to the following reasons. The dichotomy spectrum, together with the associated spectral manifolds, completely describes the dynamical skeleton of a linear system. This spectrum depicts uniform exponential stability as follows: if the dichotomy spectrum lies left of zero, then the uniform exponential stability of nonlinearly perturbed systems is guaranteed [19]. More generally, this concept may be used to discuss the existence and the smoothness of invariant manifolds for nonautonomous differential equations, to obtain a version of the Grobman-Hartman theorem for nonautonomous systems (in this context, the hyperbolicity is formulated as zero does not belong to the dichotomy spectrum) [20], to characterize the existence of center manifolds [21] and in the theory of the Lyapunov regularity [22]. Using a resonance condition of the dichotomy spectrum to study the normal forms of nonautonomous system, in [18], a finite order normal form was obtained, and in [23], analytic normal forms of a class of analytic nonautonomous differential systems were presented. Finally, information on the fine structure of the dichotomy spectrum allows classifying various types of nonautonomous bifurcations [24]. In this paper, we consider nonuniform dichotomy spectrum as it was defined in [25] (see also [26]).
In control theory, one of the most effective methods of designing control systems for stationary systems is the pole placement method (see [27]). The most important result here is the fact that the controllability of a linear time-invariant system is equivalent to the fact that for each set of complex numbers whose cardinality is equal to the size of the state vector and is symmetric about the real axis, there exists a stationary feedback such that the poles of the closed-loop system form this set [28].
The generalization of this methodology to systems with time-varying coefficients encounters two main difficulties: there are many nonequivalent controllability concepts, and for systems with variable coefficients, there are many numerical characteristics that play, to a certain extent, the role of the poles of stationary systems (e.g., the Lyapunov, Bohl and Perron exponents or dichotomy spectra). The methods of assignability of the so-called Lyapunov invariants through linear feedback for continuous systems are fully described in the monographs of Makarov and Popova [29], and for discrete systems, the assignability of the Lyapunov exponents is discussed in the series of papers [30,31], and [32]. The problem of assignability of the spectrum of uniform exponential dichotomy was discussed in the works [33,34], and [35], where it was shown that both for discrete and continuous systems with bounded time-varying coefficients, considered both on the half-line and on the whole line, the spectrum of uniform exponential dichotomy is assignable if and only if the system is uniformly completely controllable.
Although a uniform dichotomy is a common phenomenon (see [10]), it may not be satisfied in many important situations, for example, when the linear equation is a result of linearization of certain nonlinear dynamics [36]. Moreover, as it was shown in this paper, analysis of such dynamics can be successfully performed using the weaker concept of nonuniform dichotomy. The corresponding spectrum of the nonuniform dichotomy is a subset of the spectrum of the uniform dichotomy, but it contains all the Lyapunov exponents as subsets, and therefore, on its basis, it is possible to infer exponential stability. For this reason, among other things, in control systems, as the aim of control can be considered, placement of the spectrum of nonuniform dichotomies in a given position, in particular by selecting the position as a set of points, leads to the task of locating Lyapunov exponents.
In this paper, we investigate the problem of the relationship between the assignability of the nonuniform dichotomy spectrum and complete controllability for discrete linear systems with not necessarily bounded time-varying coefficients.
In all previous works on the assignability of the Lyapunov exponents ( [29][30][31], and [32]) and the dichotomy spectrum ( [33,34], and [35]), there is an assumption that the coefficients are bounded, and this assumption plays an important role there. In the present work, it was possible to significantly weaken this assumption and obtain results on the assignability of the spectrum, assuming that the coefficients are tempered. Related to the concept of a tempered sequence is the concept of tempered exponential dichotomy. In [37], the authors characterize the concept of a tempered exponential dichotomy on a Banach space in terms of an admissibility property. Additionally, they show a new proof of the robustness property of the notion of a tempered exponential dichotomy under sufficiently small linear perturbations for continuous-time dynamical systems. For discrete-time dynamical systems, the characterization of tempered exponential dichotomy on a Banach space is presented in [38]. As a result, the authors show that the concept of an exponential dichotomy under sufficiently small parameterized perturbations perseveres and that their stable and unstable spaces are as regular as the perturbation.
For the consideration of the systems with not necessarily bounded coefficients, it was compulsory to create new methods that allowed developing the key for the main result of Theorem 12 about transforming the system through linear feedback to an upper triangular form with predetermined elements on the main diagonal. The diagonal significance of nonuniform dichotomy, i.e., the property that the spectrum of an upper triangular system is the union of spectra of one dimensional systems corresponding to the elements of the main diagonal, shown in Theorems 6 and 7, also plays an important role in proving the main result and is interesting for the theory of dichotomy itself. For the purpose of this result, we adapted to discrete systems the idea of the linking operator proposed in [39] for continuous systems. Finally, it should be noted that as shown in [34] and [35], the assignability of the spectrum of a uniform dichotomy requires the assumption of one of the strongest types of controllability, i.e., uniform complete controllability, while in our work, we consider a significantly weaker assumption about complete controllability.
The paper is organized as follows. In the next section, we present the basic definition, formulation of the problem, and the main result. In the third section, we discuss properties of nonuniform dichotomy for upper triangular systems. The fourth section is devoted to properties of complete controllability. The proof of the main results is presented in the fifth section. Section six contains an example. We formulate conclusions in the last section.
Denote by N the set of natural numbers. We denote the s-dimensional Euclidean space with Euclidean norm · and the set of matrices of size s by t with real entries by R s and R s×t , respectively. For a matrix X ∈ R s×t , A denotes the operator norm generated by the Euclidean norm. GL s (R) is the subset of R s×s consisting of invertible matrices. If X, Y ∈ R s×s are symmetric, then X ≥ Y , (X > Y ) means that the matrix X -Y is nonnegative definite (positive definite). For a sequence A : N → GL s (R), A = (A(k)) k∈N , we denote by A -1 the sequence (A(k) -1 ) k∈N , which we will also write as (A -1 (k)) k∈N . The identity matrix of size s by s is denoted by I s . The set of all sequences X : N → R s×t , X = (X(k)) k∈N such that lim sup is denoted by L tem (N, R s×t ), and its elements are called tempered sequences.

Preliminaries and statements of main results
For a sequence A : N → GL s (R), A = (A(k)) k∈N , let A (·, ·) denote the evolution operator generated by A, i.e., Consider a discrete time-varying linear system where A := (A(k)) k∈N , A(k) ∈ GL s (R). If (x A (k, k 0 , x 0 )) k∈N denotes the solution of (1), satisfying the condition x A (k 0 , k 0 , x 0 ) = x 0 , then A matrix P ∈ R s×s is called projector if P 2 = P. An invariant projector of (1) is defined to be a function P : N → R s×s of projections P(k), k ∈ N, such that P(k) A (k, l) = A (k, l)P(l), k, l ∈ N.
In this paper, we consider the following definition of nonuniform exponential dichotomy (NED) and nonuniform exponential dichotomy spectrum (NEDS) introduced in [25].
Definition 1 (Nonuniform exponential dichotomy) We say that (1) admits a NED if there exists an invariant projector (P(m)) m∈N of (1), a constant α > 0, and for each ε > 0 a constant D(ε) > 0 such that for k, l ∈ N Definition 2 (Nonuniform exponential dichotomy spectrum) The nonuniform exponential dichotomy spectrum (NEDS) of (1) is defined by The structure of the NEDS is given by the following theorem from [25]. where Consider now a discrete linear time-varying system with control where A = (A(k)) k∈N , A(k) ∈ GL s (R), B : N → R s×t , B = (B(k)) k∈N , and the control sequence u = (u(k)) k∈N is t-dimensional. The (forward) solution of (5) corresponding to the control u and initial condition x(k 0 ) = x 0 , where k 0 ∈ N and x 0 ∈ R s , is denoted by x(k, k 0 , x 0 , u) k≥k 0 and is given by the following formula If we have a sequence U : N → R t×s , U = (U(k)) k∈N , then we may define a so-called feedback control u = (u(k)) k∈N by With this control, system (5) takes the following closed-loop form Our main result will be formulated under the assumption that A, A -1 ∈ L tem (N, R s×s ) and B ∈ L tem (N, R s×t ); therefore the following class of feedback is important.

Definition 3 (Admissible feedback) Suppose that
is called an admissible feedback for (5) Remark 1 (Tempered sequences are not exponentially growing) For a tempered sequence X = (X(k)) k∈N ∈ L tem (N, R s×t ), there exists for all ε > 0 a D(ε) > 0 such that The next definition contains the precise statement of the main objective of this paper.
Definition 4 (Assignability of spectrum) The NEDS of (7) is called assignable if for arbitrary 1 ≤ p ≤ s and an arbitrary set = ∅, R or of the form (4), there exists an admissible feedback U such that (A + BU) = .
The main result of the paper, which contains sufficient conditions for assignability of NEDS, is formulated in terms of complete controllability and the controllability Gramian. The formal definitions are as follows.
In the investigation of controllability, a crucial role is played by the following Kalman controllability matrix The next theorem (see [40,41]) gives, in terms of the Kalman controllability matrix, necessary and sufficient conditions for complete controllability. For k, l ∈ N and k < l, denote by α A,B (k, l) the smallest eigenvalue of W A,B (k, l). The next theorem is the main result of the paper.

Theorem 3 (Assignability theorem)
Suppose that A, A -1 ∈ L tem (N, R s×s ) and B ∈ L tem (N, R s×t ), system (5) is K -completely controllable and Then the NEDS of (5) is assignable.

Upper triangular sequences and dichotomy
In this section, we present results about the NEDS for upper triangular systems. The main role is played by the linking operator introduced by Batteli and Palmer in [39]. Our main Theorem 7 shows that the NEDS of a system with upper triangular coefficients is the union of the NEDS of the scalar systems formed from the diagonal entries. Suppose that A = (A(k)) k∈N is an upper triangular sequence . Then for k, l ∈ N, the matrices A (k, l) are upper triangular and where Let us denote It is clear that V 1 is nonempty (0 ∈ V 1 ) and is a linear subspace of R¯s; therefore, V 2 is well-defined. For a fixed x 0 ∈ R s-s , consider the equation and denote its solution, satisfyingx(0) =x 0 ∈ R¯s by (x(k, 0,x 0 )) k∈N . By the variation of constants, formula (see [42, pp. 83]), (x(k, 0,x 0 )) k∈N is given bȳ Let us denote Observe that 0 ∈ W 1 , and from the superposition principle (see [42, pp. 49]), it follows that W 1 is a subspace of R s-s ; therefore, W 2 is well-defined.
This simple observation leads to the following nontrivial fact.
Using this Lemma, we may associate with system (10) a linear operator L : W 1 → V 2 , the so-called linking operator (see [39] and [43]), by the formula Lx 0 =x 0 , wherex 0 is the unique element in V 2 such that the solution (x(k, 0,x 0 )) k∈N of (12) is bounded. By Lemma 1, this operator is well-defined, and its linearity is obvious.
LetPĀ 1 : R¯s → R¯s and Q : R s-s → R s-s be the projections onto V 1 and W 1 , respectively. We consider the projection P : R s → R s given by and define P(m) : R s → R s for m ∈ N by Observe that (P(m)) m∈N , (PĀ 1 (m)) m∈N and (Q(m)) m∈N satisfy the invariance properties The following remark follows directly from Definition 1 of nonuniform exponential dichotomy.
Remark 3 (Alternative characterizations of NED) The following three statements are equivalent: (i) System (1) admits a NED.
(ii) There exists an invariant projector (P(m)) m∈N , a constant α > 0, and for each ε > 0 a constant D(ε) > 0 such that for k, l ∈ N and x ∈ R s (iii) There exists a projection P, a constant α > 0, and for each ε > 0 a constant D(ε) > 0 such that for k, l ∈ N Remark 4 It is clear that if the inequalities (16) and (17) hold for all ε ∈ (0, ε ) for a certain ε > 0, then they hold for all ε > 0. Therefore, to show that a system has a NED it is enough to show that (16) and (17) hold for all sufficiently small ε > 0.
Lemma 2 (Uniqueness of the image of the projector) Suppose that system (1) admits a NED. Then for each k ∈ N, and where P(k) are any projections from Definition 1, and P is any projection from Remark 3(iii).
In particular, the images of the projections P(k), k ∈ N, and P, satisfying Definition 1 and Remark 3(iii), respectively, are unique.
Proof The equality is proved in [26, Proposition 1]. If P is any projection from Remark 3(iii), it is easy to show that the projections satisfy Definition 1, and therefore, Theorem 4 (NED for blocks of the upper triangular system) Suppose that (10) admits a NED on N with invariant projections (P(m)) m∈N and constant α > 0. Then both systems and have a NED with constant α > 0. Moreover, the invariant projector (P(m)) m∈N of the dichotomy for (10) can be taken in the block upper triangular form Proof Observe that for anyx ∈ R¯s and m ∈ N, we have As a consequence and forx ∈ R¯s and m ≥ k, we get for a fixed ε > 0 and similarly for anyx ∈ R¯s and m ≤ k Using Remark 3, this completes the proof of NED of (18). The fact that (19) has a NED can be proven by considering the system exactly as in step 2 of the proof of Theorem 1 in [39].
From the above theorem and its proof, the following remark follows.
Remark 6 (NED projector for the upper triangular systems) Suppose that system (5) with the coefficient of the upper triangular form (10) admits a NED. By Theorem 4, the projector can be taken in the form (20). Using the definitions of P(m),PĀ 1 (m), Q(m), and (11), it follows that where and In fact, we have By induction, it can be proven that the explicit solution of (21) is given by Now we will show that the opposite implication to this from Theorem 4 holds for tempered sequences C.
Theorem 5 (NED for the upper triangular tempered system) Suppose that (C(k)) k∈N ∈ L tem (N, R s×(s-s) ) and that the systems (18) and (19) admit a NED. Then also system (5) with coefficient of the upper triangular form (10) admits a NED.
Summarizing, we have proved that for each ε ∈ (0, α/6), there exists D(ε ) > 0 such that This, together with the fact that the systems (18) and (19) admit a NED and (22), implies (16). Suppose now that k > m. Using the identities and we can show analogically, as in the case k < m, that (17) holds.
Now, we will consider system (1) with A being in upper triangular form, i.e., Denote and F(k) = a ss (k).
Theorem 6 (Characterization of NED for the upper triangular tempered system) Suppose that (A(k)) k∈N , (A -1 (k)) k∈N ∈ L tem (N, R s×s ) and that (A(k)) k∈N is in the upper triangular form (23). Then (1) admits a NED if and only if every scalar equation admits a NED.
Proof We prove the theorem by induction over j ∈ {1, . . . , s}. When j = 1, the conclusion is obvious. When j = 2, the conclusion follows from Theorems 4 and 5. Suppose that the result is true when the system is (s -1)-dimensional. Since system (1) with A in the form (23) has the form (10) with the conclusion of the theorem for s-dimensional systems follows from Theorems 4 and 5, and the fact that if (A(k)) k∈N is tempered, then, so are (D(k)) k∈N , (E(k)) k∈N , and (F(k)) k∈N .
From Theorem 6, the following pivotal result follows.
Theorem 7 (NEDS for the upper triangular tempered systems) Suppose that the sequence (A(k)) k∈N in form (23) is tempered. Then where (a jj ) is the NEDS of (24).

Complete controllability
This paragraph contains some considerations about complete controllability, which led to the main result formulated in Theorem 12. This theorem is a generalization of Theorem 4.6 in [30] to systems with unbounded coefficients. The proof of the following lemma is contained in the proof of Proposition 7 in [44]. (5) is K -completely controllable, then for any U : N → R t×s , U = (U(k)) k∈N , the system

Lemma 3 (Complete controllability persists under feedback) If system
is K -completely controllable.
If D = (D(k)) k∈N is a sequence of invertible s by s matrices, then with we get In such a situation, we will say that D transforms system (5) into the system wherē In our investigation, we will use the concept of weak equivalence. In the literature, one can find various definitions of equivalence for different types of dynamical systems (e.g., see [45][46][47][48]). Definition 6 (Weak equivalence) If there exists a sequence D such that D, D -1 ∈ L tem (N, R s×s ), which transforms system (5) into (27), then systems (5) and (27) are called weakly equivalent. (5) is completely controllable and a transformation D = (D(k)) k∈N , D, D -1 ∈ L tem (N, R sxs ) transforms system (5) into system (27). Then (27) is completely controllable. Moreover, if

Lemma 4 (Weak equivalence preserves complete controllability) Suppose that
then Proof Formula (28) implies that and therefore, by Theorem 2, complete controllability (5) is equivalent to complete controllability of (27). Since for any symmetric matrix V ∈ R s×s and any matrix D ∈ R s×s , we have and (see [49]), where λ min (V ) and λ max (V ) are the smallest and the greatest eigenvalue of V , respectively. Then using (32) with V = W A,B (k 0 , k 0 + K), we get Denoting y = D -T (k 0 )x and having in mind that D -T (k 0 ) is invertible, we have for any y ∈ R s , y = 0. Using (32) with V = D(k 0 )D T (k 0 ) and (33) with D = D(k 0 ), we get Using the last inequality in (34), we obtain α A,B (k 0 , k 0 + K) ≤ y T W A,B (k 0 , k 0 + K)y y T y D -1 (k 0 ) 2 and taking into account (29), we have Taking the infimum over y ∈ R s , y = 0 and using (32), we get The last inequality implies (30).
Theorem 8 For any K , k 0 ∈ N and ξ ∈ R s , ξ = 1, there exist ν ∈ R t , ν = 1 and k v ∈ N, Proof First, we show that for any k 0 , K ∈ N and ξ ∈ R s , ξ = 1, there exists ν ∈ R t , ν = 1 such that Since, W A,B (k 0 , k 0 + K) ≥ α A,B (k 0 , k 0 + K)I s , using the notation (35), we get for any k 0 , K ∈ N and any ξ ∈ R s , ξ = 1. Let us fix ξ ∈ R s , ξ = 1, and define as ν as one of the vectors of the standard basis e i , i ∈ {1, . . . , t}, of R t for which the expression achieves its maximum. Then we have The last inequality implies (37). From (37), the inequality (36)  For 0 < δ ≤ 1 and r ≥ 0, denote H(r, δ) = H ∈ R s×s : H -I s < r, det(H) j ≥ δ, j = 1, . . . , s , and by H the set of all s by s matrices with leading minors with positive determinants. The proof of the next lemma may be found in [29].

Lemma 5 A matrix H belongs to H if and only if H = LG, where L and G are lower and upper triangular matrices with positive diagonal elements, respectively.
Let us introduce the following notation a(k 0 , K) = max max A -1 (l) , 1 : l = k 0 , . . . , k 0 + K -1 and b(k 0 , K) = max B(l) : l = k 0 , . . . , k 0 + K -1 .

Lemma 6 Suppose that system
Proof According to Theorem 8, for any k 0 ∈ N and ξ ∈ R s , ξ = 1, there exist ν ∈ R t , ν = 1 and k v ∈ N, k 0 ≤ k v ≤ k 0 + K -1 such that (36) holds. Let us fix k 0 ∈ N. The construction will be done by induction. In the first step, consider any ξ 1 ∈ R s , ξ 1 = 1. Then we find ν 1 ∈ R t , ν 1 = 1 and k 1 ∈ N, k 0 ≤ k 1 ≤ k 0 + K -1, such that Suppose that after the dth step, we have ξ i ∈ R s , ξ i = 1, ν i ∈ R t , ν i = 1 and k i ∈ N, k 0 ≤ k i ≤ k 0 + K -1, i = 1, 2, . . . , d such that Denote by M d the orthogonal complement of In the (d + 1)-st step, we take any ξ d+1 ∈ M d , ξ d+1 = 1, and for it, we define ν d+1 ∈ R t , ν d+1 = 1 and k d+1 ∈ N, k 0 ≤ k d+1 ≤ k 0 + K -1 such that After s steps, we will have s vectors ξ i , i = 1, . . . , s, such that the inequality (41)  for any matrix ∈ R s×s , and we have Let us rearrange the sequence k 1 , k 2 , . . . , k s such that it forms a nondecreasing sequence. The elements of the new sequence will be denoted by m i . So, we have In the same way, we rearrange the vectors ν 1 , . . . , ν s to obtain vectors η 1 , . . . , η s . Then the matrix which is obtained from F(k 0 ) by swapping columns accordingly, satisfies (39) and (40).
The next theorem shows that by a special choice of control, we may connect the transition matrices of (1) and (7) by a particular relation.
Theorem 9 Suppose that system (5) is K -completely controllable. Then for any k 0 ∈ N, there exists an invertible s by s matrix (k 0 ) with the following property: for any 0 < δ ≤ 1 and r ≥ 0, there exist β 1 (r, δ, k 0 ) > 0 and β 2 (r, δ, k 0 ) > 0 such that for any H ∈ H(r, δ), there exist a feedback control U = U(i) i=k 0 ,...,k 0 +K-1 such that where A+BU is the transition matrix of the closed loop system (7) and max i=k 0 ,...,k 0 +K-1 where Proof Let us fix k 0 ∈ N, and let η i ∈ R t , m i ∈ N, i = 1, . . . , s, and (k 0 ) = F(k 0 ) be as in Lemma 6. Consider any 0 < δ ≤ 1, r ≥ 0, H ∈ H(r, δ) and the matrix equations where Y (k) is a s by s, and V (k) is a t by s matrix. We will consider this equation with initial condition Y (k 0 ) = I s , and we construct a sequence V (k), k = k 0 , k 0 + 1, . . . , k 0 + K -1, such that The solution of equation (44), with the considered initial condition, is given by and the condition (45) will be satisfied if and only if In the further calculation, we will consider sums of the form i∈I j and i∈I j . In the case when I j or I j is empty, we define the sum as equal to zero. Set where e i , i = 1, . . . , s, is the standard basis of R s , and η i are defined in Lemma 6, equation (38). We have and Because of Lemma 6, we know that and therefore, We have constructed matrices V (k), k = k 0 , . . . , k 0 + K -1, such that the matrix is invertible for any k = k 0 , . . . , k 0 + K -1, and such that From the construction of the control sequence, it follows that the matrix is invertible for any k = k 0 , . . . , k 0 + K -1, and Taking we get and because Y (k 0 ) = I s , the equality (42) holds. Moreover, where Observe that for any k ∈ {k 0 , k 0 + 1, . . . , k 0 + K -1}, we have

This implies
Theorem 10 Suppose that system (5) is K -completely controllable, then there exists a sequence (T(k)) k∈N , T(k) ∈ GL s (R), such that and for each upper triangular sequence (L(k)) k∈N , L(k) ∈ GL s (R) and each lower triangular sequence (G(k)) k∈N , G(k) ∈ GL s (R), of matrices with positive diagonal elements (both L and G), there exists a feedback control U = (U(k)) k∈N such that and where Proof Consider sequences (L(k)) k∈N and (G(k)) k∈N as in the theorem. Then for the diagonal elements l ii (k), i = 1, . . . , s, of the matrix L(k), we have for all k ∈ N. From the last two inequalities, we obtain The same estimates can be obtained for the diagonal elements g ii (k), i = 1, . . . , s, of the matrix G(k). Therefore, for all k ∈ N and j = 1, . . . , s, where (H) j is the jth leading minor of the matrix H.
It follows from the proof of Theorem 9 that ((k -1)K) = F((k -1)K), where F is defined by (38). Set From (39) and (40), it is clear that (T(k)) k∈N satisfies Let us define then it is clear that U(k) ≤ γ 2 (k) + 1 β 1 γ 2 (k) + 1, γ -2s (k), (k -1)K and (49) holds. Moreover, by (43), it follows that Condition (51) in the following corollary to Theorem 10 ensures the existence of a tempered transformation and a control in a specific situation. It will be useful to prove our main result.
Proof Let T = (T(k)) k∈N be the sequence from Theorem 10. First, we will show that T and T -1 are tempered. Since A, A -1 and B are tempered for each ε > 0, there exists C(ε) > 0 such that and by (51) for all k ∈ N. This implies that From the last three inequalities, (46) and (47), we get and where From (55) and (56), it follows that T and T -1 are tempered since ε 1 and ε 2 tend to zero when ε tends to zero. Let us fix L and G as in the assumptions of the theorem, and let U = (U(k)) k∈N be from Theorem 10. Then from Theorem 10, it follows that (52) holds, and therefore, A(k) + B(k)U(k) ∈ GL s (R). Let us estimate ( U(k) ) k∈N . According to (49), we have U(k) ≤ γ 2 (k) + 1 β 1 γ 2 (k) + 1, γ -2s (k), (k -1)K .
Since L, L -1 , G and G -1 are tempered for each ε > 0, there exists C(ε) > 0 such that for all k ∈ N, and therefore, by (53) and (54), we get It is clear that there exists a constant C 4 (ε) > 0 such that for all k ∈ N. Using the last inequality in (59), we get where and ε 5 = ε(9s + 2sK -3).
The last inequality implies lim sup since ε 5 tends to zero when ε does so.
Definition 7 (Global positive scalarizability) We say that the system (7) is globally positively scalarizable if for any sequence p = (p(k)) k∈N , p -1 ∈ L tem (N, R), of positive real numbers, there exists an admissible control U = (U(k)) k∈N such that (7) is weakly equivalent to the system (1) with A(k) = p(k)I s .
Theorem 11 (Sufficient condition for global positive scalarizability) Suppose that A, A -1 ∈ L tem (N, R s×s ), B ∈ L tem (N, R s×t ) and system (5) is K -completely controllable and Then system (7) is globally positively scalarizable. Proof Assume that system (5) is K -completely controllable. Let T = (T(k)) k∈N be a sequence according to Corollary 1. By the QR factorization theorem (see [49, p. 112]) for the invertible matrix T -1 (k + 1) A (kK, (k -1)K)T(k), there is an orthogonal matrix Q(k) and an upper triangular matrix R(k) with positive diagonal elements such that and consequently, Since it follows that R = (R(k)) k∈N is a tempered sequence. Consider any positive sequence p = (p(k)) k∈N such that p, p -1 ∈ L tem (N, R) and denote Note that for all k, m ∈ N. Let us fix ε > 0. Since A, A -1 , B, T, T -1 , R, R -1 , p and p -1 are tempered sequences, there exists a constant C(ε) > 0 such that For an upper triangular matrix therefore, (H(k)) k∈N is a tempered sequence. According to the definition of T(k) and Corollary 1, we know that there exists a feedback control U = (U(i)) i∈N such that Multiplying these equalities, we get Note that Q(k) is orthogonal as a product of orthogonal matrices. As it follows from Corollary 1, the sequences (A(k) + B(k)U(k)) k∈N and ((A(k) + B(k)U(k)) -1 ) k∈N are tempered and (61) holds. Let us define a sequence D = (D(i)) i∈N as follows We will show that D is a tempered sequence. Let us fix i ∈ N. Then there exists exactly one k ∈ N such that i ∈ [kK, (k + 1)K). By the properties of the transition matrix and the function ϕ(m, k), we get Since i -kK < K , it follows that and similarly Observe that the sequence D establishes weak equivalence of the system (7) with the defined control U and system (1) with In fact, From Theorem 6, we obtain the following corollary.

Corollary 2
Under the assumption of Theorem 11, there exists an admissible feedback control U 1 = (U 1 (k)) k∈N such that (7) is weakly equivalent to In our further considerations, we will use the following result.

Lemma 7
Suppose that A = (A(k)) k∈N , C = (C(k)) k∈N and A -1 , C -1 are tempered sequences and assume that for all k ∈ N, where (k k ) k∈N is a sequence of natural numbers such that 0 < k k+1k k ≤ c < ∞ for all k ∈ N. Then the systems (1) and are weakly equivalent.
Proof Let us fix ε > 0 and consider M(ε) > 1 such that Let us fix k ∈ N. Then there exists a unique k ∈ N such that k k ≤ k < k k+1 . Observe that 0 ≤ kk k ≤ c. Moreover, and From the last two equalities we get This implies that the sequence (D(k)) k∈N is a tempered sequence. Finally, The following theorem plays a key role in obtaining our main results.

Theorem 12
Suppose that A, A -1 ∈ L tem (N, R s×s ), B ∈ L tem (N, R s×t ) and system (5) is Kcompletely controllable and Then for any positive sequences p i = (p i (k)) k∈N , p -1 i ∈ L tem (N, R), i = 1, . . . , s, there exists an admissible feedback control U = (U(k)) k∈N such that system (7) is weakly dynamically equivalent to where (C(k)) k∈N is an upper triangular tempered sequence with p i (k), i = 1, . . . , s, on the main diagonal for all k ∈ N.
Proof Consider positive sequences p i = (p i (k)) k∈N , p -1 i ∈ L tem (N, R), i = 1, . . . , s, and denote According to Corollary 2, there exists a feedback control U 1 = (U 1 (k)) k∈N such that lim sup and the system is weakly dynamically equivalent to y(k + 1) = y(k), k ∈ N. Let be the transformation that establishes this equivalence. Then we have and consequently, Note that this sequence establishes also a weak dynamic equivalence of and y(k + 1) = y(k) + D(k + 1)B(k)u(k).
By Lemma 3, we know that (66) is K -completely controllable. Since by Lemma 4, dynamic equivalence preserves K -complete controllability, it follows that system (66) is Kcompletely controllable and satisfies (63). According to Corollary 1, there exists a sequence T = (T(k)) k∈N , T, T -1 ∈ L tem (N, R s×s ) such that for each upper triangular sequence (L(k)) k∈N , (L -1 (k)) k∈N ∈ L tem (N, R s×s ), L(k) ∈ GL s (R) and each lower triangular sequence (G(k)) k∈N , (G -1 (k)) k∈N ∈ L tem (N, R s×s ) of matrices with positive diagonal elements (both L and G), there exists a control V = (V (k)) k∈N such that lim sup and where Y V is the transition matrix of Let us apply the QR factorization theorem to T(k) and let where R(k) and Q(k) are the upper triangular and orthogonal matrices, respectively. We have and Applying the QR factorization theorem to J(k)Q(k), we have where R(k) is an upper triangular matrix, and Q(k) is an orthogonal matrix. Moreover, and similarly, Corollary 1 implies that for the control V = (V (k)) k∈N , we have Consider the system (64) with The matrices C(k) are upper triangular with p i (k), i = 1, . . . , s, on the main diagonal and This implies that (C(k)) k∈N and (C -1 (k)) k∈N are tempered sequences. For the transition matrix C of (64), we have C lK, (l -1)K = C(lK -1) · · · C (l -1)K = R(l) diag According to Lemma 7, the system (69) with the defined control V and the system (64) are dynamically equivalent. Applying to (69) the inverse Lyapunov transformation we get x(k + 1) = D -1 (k + 1)y(k + 1) We see that the system (7) with the defined control U is dynamically equivalent to (64). The fact that (A(k) + B(k)U(k)) k∈N and ((A(k) + B(k)U(k)) -1 ) k∈N are tempered follows from the fact that (C(k)) k∈N and (C -1 (k)) k∈N are tempered sequences and where (D 1 (k)) k∈N is the tempered sequence that establishes the dynamic equivalence of (7) and (64). Finally, (8) follows from (65), (68), (70), and the fact that (D(k)) k∈N and (D -1 (k)) k∈N are tempered.

Proof of Theorem 3
We will start with some facts about NEDS of one-dimensional systems.
Remark 7 Observe that for one-dimensional systems, the definition of NED means that there exist constants α, η 0 > 0 such that for all ε ∈ (0, η 0 ), there exists K(ε) > 0 such that either Remark 8 In particular, if a one-dimensional sequence (c(n)) n∈N has a NED, then either the Lyapunov exponent The next lemma shows that for one-dimensional systems the NEDS may have all the forms described in Theorem 1. We will show that (a) = [x, y]. We will do this in three steps.
Step 1. For each γ > y, the system c = (e -γ a(n)) k∈N  . . , q -1, and I q = [a q , b q ] or I q = [a q , +∞) for some numbers a 1 ≤ b 1 < a 2 ≤ b 2 < · · · < a q ≤ b q and q ≤ s. Consider now the case when = q i=1 I i . For 1 ≤ i ≤ q, we construct a sequence p i = (p i (k)) k∈N , p -1 i ∈ L tem (N, R) of positive real numbers such that (p i ) = I i . This is possible by Lemma 8. For q + 1 ≤ i ≤ s, let p i (k) = p 1 (k). According to Theorem 12, there exists an admissible feedback control (U(k)) k∈N and a sequence of upper triangular matrices (C(k)) k∈N , (C -1 (k)) k∈N ∈ L tem (N, R n×n ), where C(k) = (c ij (k)) 1≤i,j≤n with c ii (k) = p i (k) such that x(k + 1) = A(k) + B(k)U(k) x(k), y(k + 1) = C(k)y(k) are weakly equivalent. From the definition of weak equivalence, it is clear that weakly equivalent systems have the same NEDS; therefore, using Theorem 7, we get In the case of being ∅ or R, the proof is analogical. We put p i (k) = p(k) for all 1 ≤ i ≤ s, where (p(k)) k∈N , p -1 ∈ L tem (N, R) is any sequence of positive real numbers such that (p) = ∅ or (p) = R.

Example
Example 1 Let us consider the sequence a = (a(n)) n∈N from the proof of Lemma 8 for x = 2 and y = 3. Consider the uncontrolled system (1) with A(k) = 1 2 1 + a(k) 1-a(k) 1a(k) 1+a(k) for k ∈ N.
For this system we have (A) = {1, [2,3]} and in particular, the system is not stable. It is also clear that A, A -1 ∈ L tem (N, R 2×2 ). Consider now the controlled system (5) with It is clear that B ∈ L tem (N, R 2×2 ) and for l > k, we have Therefore, the assumption of Theorem 3 is satisfied, and the considered system has the NEDS assignable. If we want to stabilize the system, we may try to construct an admissible feedback U such that Observe also that the coefficients of the controlled system (5) considered in this example are unbounded, and therefore, the result of papers [30][31][32][33][34] can not be applied here to stabilize this system.

Conclusion
In this paper, we investigated a problem of assignability of nonuniform dichotomy spectrum by time-varying linear feedback for linear discrete time-varying systems with tempered coefficients. The main result is that the spectrum is assignable if the system is completely controllable and certain lower asymptotic bound for the controllability Gramian holds (see (9)). To obtain this result, we generalize to the case of tempered and completely controllable systems the Theorem 4.6 from [30], which makes it possible to bring the system into an upper triangular form through linear feedback. The original theorem was proved for uniformly completely controllable systems with bounded coefficients. To the upper triangular system, we apply the idea of linking operator proposed by Batelli and Palmer in [39], and we obtain the result that the nonuniform dichotomy spectrum of an upper triangular system is the union of spectra of the one-dimensional systems from its main diagonal. It has been recently shown (see [33,34], and [35]) that for systems with bounded coefficients, the assignability of uniform exponential dichotomy spectrum is equivalent to uniform complete controllability. It is an open question whether our sufficient conditions for assignability of the nonuniform dichotomy spectrum of systems with tempered coefficients are also necessary.