Linear Algebra and its Applications

Linear Algebra and its Applications 433 (200) 867 875 Contents lists available at ScienceDirect Linear Algebra and its Applications journal homepage: www.elsevier.com/locate/laa On the exponential exponents of discrete linear systems Adam Czornik, Piotr Mokry, Aleksander Nawrat Department of Automatic Control, Silesian Technical University, ul. Akademicka 6, 44-0 Gliwice, Poland A R T I C L E I N F O A B S T R A C T Article history: Received 0 September 2009 Accepted 6 April 200 Available online 0 May 200 Submitted by H. Schneider Keywords: Time varying discrete linear systems Lyapunov exponents Perturbation theory Characteristic exponents In this paper we introduce the concepts of exponential exponents of discrete linear time varying systems. It is shown that these exponents describe the possible changes in the Lyapunov exponents under perturbation decreasing at infinity at exponential rate. Finally we present formulas for the exponential exponents in terms of the transition matrix of the system. 200 Elsevier Inc. All rights reserved.. Introduction Consider the linear discrete time-varying system x(n + ) = A(n)x(n), n 0 () where (A(n)) n N is a bounded sequence of invertible s-by-s real matrices. By we denote the Euclidean norm in R s and the induced operator norm. The transition matrix is defined as A(m, k) = A(m ) A(k) for m > k and A(m, m) = I, where I is the identity matrix. For an initial condition x 0, the solution of () is denoted by x(n, x 0 ) so x(n, x 0 ) = A(n, 0)x 0. Let a = (a(n)) n N be a sequence of real numbers. The number (or the symbol ± ) defined as λ(a) = lim sup n ln a(n) Corresponding author. E-mail address: Adam.Czornik@polsl.pl 0024-3795/$ - see front matter 200 Elsevier Inc. All rights reserved. doi:0.06/j.laa.200.04.09

868 A. Czornik et al. / Linear Algebra and its Applications 433 (200) 867 875 is called the characteristic exponent of the sequence (a(n)) n N.Forx 0 R s with x 0 /= 0, the Lyapunov exponent λ(x 0 ) of () is defined as the characteristic exponent of ( x(n, x 0 ) ) n N, that is λ(x 0 ) = lim sup n ln x(n, x 0 ). In the case of a sequence of matrices = ( (n)) n N we define the characteristic exponent λ( ) as the characteristic exponent of ( (n) ) n N. It is well known 3] that the set of all Lyapunov exponents of system () contains at most s elements, say λ (A) <λ 2 (A) < <λ r (A) < and the set {λ, λ 2,..., λ r } is called the spectrum of (). The greatest and the smallest exponent of () we will denote λ g (A) and λ s (A), respectively. Together with () we consider the following perturbed system z(n + ) = (A(n) + (n)) z(n), (2) where ( (n)) n N is a sequence of s-by-s real matrices from a certain class M. Under the influence of the perturbation ( (n)) n N, the characteristic exponents of () vary, in general, discontinuously. It is possible that a finite shift of the characteristic exponents of the original system () corresponds to an arbitrarily small sup (n). In particular, it is possible for an exponentially stable system to be perturbed by an exponentially decreasing perturbation and the resulting system is not stable. An example which illustrates this situation is given in 6]. The quantity Λ (M) = sup { λ g (A + ) : M } is referred to as the maximal upper movability boundary of the higher exponent of (2) with perturbation in the class M. In the paper we will consider exponentially small perturbations. The determination of the movability boundary of the higher exponent under various perturbations is one of important problems of the modern theory of characteristic exponents. This problem has been solved for continuous-time systems for many classes M. For example, upper bound for the higher exponent of (2) under small perturbations, the so-called central exponent (A), was constructed in 5, p. 4]. The attainability of this estimate was proved in ] with the use of the classical rotation method. This problem was solved in 7,8] for linear systems with perturbations decreasing at infinity at various rates and in 2] for linear systems with perturbations determined by integral conditions. Later in 9,0] perturbations infinitesimal in mean with a weight function have been investigated. Similar problems for discrete-time systems have been investigated in 6]. In this paper we introduce the notion and properties of exponential exponents. We will study the properties of the spectrum and the relations with the exponential exponents, and the influence of these objects on how the exponents vary when passing from ()to(2). According to the authors knowledge, exponential exponents of difference equations has not yet been investigated. 2. Exponential exponents In this paper, M is defined to be the class of perturbations M = {( (n)) n N : λ( ) <0}. Definition. Bounded sequences (r(n)) n N and (R(n)) n N are said to be lower and upper sequences for (), respectively, if for every ε>0 there are constants d r,ε and D R,ε such that for all natural m, k, m k we have m m d r,ε exp r(i) εk A(m, k) D R,ε exp R(i) + εk. (3) The numbers (A) = inf R i=k lim sup R(i), n (A) = sup r i=k lim inf r(i) n

A. Czornik et al. / Linear Algebra and its Applications 433 (200) 867 875 869 are called the upper exponential exponent and the lower exponential exponent of system (). Here the infimum is taken over the set of all upper sequences whereas the supremum is taken over the set of all lower sequences. We will denote by L(A) and U(A) the set of all lower and upper sequences of (), respectively. Because system () is uniquely defined by the sequence (A(n)) n N we will also talk about lower and upper sequences and lower exponential exponent and upper exponential exponent of sequence (A(n)) n N. The following relations between the exponential and Lyapunov exponents immediately follow from the definition: (A) λ (A) λ r (A) (A). The dual or adjoint equation to () is defined to be y(n + ) = B(n)y(n), n 0, where B(n) = (A T (n)) and A T denotes the transpose of A. The transition matrix of the dual system is given by B(m, k) = B(m ) B(k) for m > k and B(m, m) = I. It appears that the lower exponential exponent does not require special consideration since the problem can be reduced to the investigation of upper exponential exponent for the adjoint system. This is the content of the next theorem. Theorem. The lower exponential exponent of () is equal to the upper exponential exponent of the adjoint system (4) taken with the opposite sign. Proof. By the definition of adjoint system we have A(m, k) = ( B(m, k) ) T. Therefore the estimate m d R,ε exp r(i) εk A(m, k) i=k implies B(m, k) m exp r(i) + εk d R,ε i=k and the latter implies our statement. Now we show that under the condition that λ( ) <0, the exponential exponents of () and (2) coincides. Theorem 2. If λ( ) <0 then L(A) = L(A + ), U(A) = U(A + ) and in particular (A) = (A + ) and (A) = (A + ). In the proof of this theorem we will use the following version of the discrete Gronwall s inequality ]. Theorem 3. Suppose that for two sequences u(n) and f (n),n= m, m +,...of nonnegative real numbers the following inequality u(n) p + q u(i)f (i) (4)

870 A. Czornik et al. / Linear Algebra and its Applications 433 (200) 867 875 holds for certain p, q R and all n = m, m +,..., then u(n) p ( + qf (i)) for all n = m, m +,... (5) Proof. Denote by C(n, m) the transition matrix of (2) and consider an upper sequence R(n) for (). We show that it is also an upper sequence for (2). For the transition matrix C(n, m) we have C(n, m) = A(n, m) + A(n, i + ) (i)c(i, m) and therefore C(n, m) A(n, m) + A(n, i + ) (i) C(i, m). Fix an ε>0 such that ε< λ( ) and estimate A(n, i) according to (3). Then C(n, m) D R,ε exp R(i) + mε + D R,ε exp R(j) + (i + ) ε (i) C(i, m) j=i+ (6) or C(n, m) exp R(i) mε i D R,ε + D R,ε C(i, m) exp R(j) mε (i) exp ( R(i) + (i + ) ε). j=m By the Gronwall s inequality (5) with i u(i) = C(i, m) exp (R(j) + mε), j=m p = D R,ε, q = D R,ε, f(i) = (i) exp ( R(i) + (i + ) ε) we get ( C(n, m) exp R(i) mε D R,ε + DR,ε (i) exp ( R(i) + (i + ) ε) ) and C(n, m) D R,ε exp (mε) ( exp (R(i)) + DR,ε (i) exp ((i + ) ε) ). (7)

A. Czornik et al. / Linear Algebra and its Applications 433 (200) 867 875 87 By the boundedness of R(n) there exist ε > 0 and C > 0 such that ( ) exp (R(i)) + DR,ε/2 ε C exp R(i) + δ, for all natural numbers n > m and all δ>0. By (6) there exists a positive integer N such that for all n > N (i) exp ((i + ) ε) <ε. By the last two inequalities we obtain from (7) that for all n > m > N the following holds C(n, m) D R,ε C exp R(i) + mε + δ. Increasing if necessary the constant C, we conclude that R(n) is the upper function for (2) because δ>0is arbitrary. Changing the roles of systems () and (2) in the above arguments, we come to the conclusion that an upper function for (2) is also an upper function for (). Thus, the sets of upper functions of these systems coincide; hence the upper exponential exponents also coincide. For the lower functions and lower exponential exponents, the arguments are carried out by passing to the adjoin system. For a sequence of perturbation matrices = ( (n)) n N, we define λ g (A + ) and λ s (A + ) as the greatest and smallest Lyapunov exponent of (2), respectively. Moreover for δ>0define Λ δ = sup λ g (A + ) and λ δ = inf λ s (A + ). λ( )<δ λ( )<δ Because the functions Λ δ and λ δ (considered as functions of δ>0) are nondecreasing and nonincreasing, respectively, the following limits lim Λ δ 0 δ and lim λ δ 0 δ exist and are equal to sup λ g (A + ) and inf λ s (A + ), λ( )<0 λ( )<0 respectively. Theorem 4. The following inequalities hold sup λ g (A + ) (A) and λ( )<0 inf λ s (A + ) (A). λ( )<0 Proof. By Theorem it is enough to show the case of the upper exponential exponent. In that end, we show that for every ε>0, there exists δ>0such that λ g (A + ) (A) + ε for all perturbation matrices = ( (n)) n N satisfying λ( ) <δ. Fix an upper sequence R(n) such that lim sup R(i) (A) + ε n 2, (8) and denote C(n, m) the transition matrix of (2). Repeating the arguments from the proof of Theorem 2 we obtain the following bounds for z(n) = C(n, 0)z(0): ( z(n) D R,ε/4 z(0) exp (R(i)) + DR,ε/4 (i) exp ((i + ) ε/4) ).

872 A. Czornik et al. / Linear Algebra and its Applications 433 (200) 867 875 By (8) wehaveforλ( ) <δ< λ(z 0 ) = lim sup n The proof is completed. ε the inequality 4D R,ε/2 ln z(n, z(0)) (A) + ε + ε + ε = (A) + ε. 2 4 4 3. Formula for the exponential exponents In this paragraph we present a formula for the exponential exponents in terms of the transition matrix. We start with a result from 4] which states that we may reduce an arbitrary system to upper triangular form by an unitary transformation. Theorem 5. For each sequence (A(n)) n N there exists a sequence (U(n)) n N of orthogonal matrices such that C(n) = U T (n + )A(n)U(n) is upper triangular. Clearly the orthogonality of U(n) implies that U(A) = U(C), L(A) = L(C). In particular, the exponential exponents of (A(n)) n N and (C(n)) n N are equal. Lemma 6. If matrices (A(n)) n N are upper triangular with diagonal elements a ii (n), then U(A) = U(A d ) and L(A) = L(A d ), where A d (n) = diaga ii (n)] i=,...,n. Proof. We will proof only the part about upper sequences. The proof of the second part is similar. Denote by a ij (n) and z ij (n, k), the elements of A(n) and A(n +,k), respectively. By a straightforward induction we have that for n > m n k j p=0 l=i+ a il(n p)z lj (n p,k) n q=n p+ a ii(q) for i < j, z ij (n, k) = n q=k a ii(q) for i = j, (9) 0 for i > j. Let R U(A). By the definition of upper function, for every ε>0there is a constant D R,ε such that for all n k we have n A(n +,k) D R,ε exp R(l) + kε. (0) Because all norms in finite dimensional space are equivalent there exist positive constants D and D 2 such that D X max xij D2 X, () i,j=,...,s for every matrix X =x ij ] i,j=,...,s. Therefore from (0)wehave n z ii (n, k) D 2 D R,ε exp R(l) + kε ; max i=,...,s hence, by ()wehaver U(A d ). Let P U(A d ). By the definition of upper function, for every ε>0there is a constant D R,ε such that for all n k we have zjj (n, k) n DR,ε exp P(i) + kε (2) i=k

A. Czornik et al. / Linear Algebra and its Applications 433 (200) 867 875 873 for all j =,...,s. We will show by induction that for every ε>0there is constant D R,ε such that for all n k we have zij (n, k) n D P(l) + kε (3) R,ε exp for all j = i,...,s.fori = j it follows from (2). Suppose that (3) holds for i = j, j,..., α +. By (9) and boundednes of P we obtain zαj (n, k) n k j a αl (n p) zlj (n p,k) n a αα (q) p=0 l=α+ q=n p+ n k j n p n D R,ε a αl (n p) exp P(l) + kε exp P(i) + ε (n p + ) p=0 l=α+ i=n p+ D R,ε n n k exp j (P(l) + 2εk) a αl (n p). p=0 l=α+ From the last inequality, (3) follows by the boundedness of (A(n)) n N. Applying the equivalence of all norms in finite dimensional space we obtain from (3) that P U(A). For each real number >, define α() = lim sup lim inf n ln A ( i+ ], i ]), where x] is the integer part of x. Lemma 7. If (a(n)) n N is a bounded sequence, > and A() = lim sup M M lim sup M M In particular, lim A() exists. + = lim + A(). n ln A( i+ ], i ]) and α() = M ] M, then Proof. Suppose the sequence (a(n)) n N is bounded by D > 0. For each natural number n such that M < n M+, for certain natural number M, wehave M ] n M n M + M M n M n + M ] M i=n M n M D + n M] M D 2D ( ). M ] M

874 A. Czornik et al. / Linear Algebra and its Applications 433 (200) 867 875 Theorem 8. The limits lim α() and lim α() exist and the following equalities hold + + lim α() = (A) and lim α() = (A), + + where (A) and (A) are the upper exponential exponent and lower exponential exponent of (). Proof. By Theorem it is enough to show only the first part; moreover, according to Theorem 5, we may assume that A(n) are upper triangular with elements a ij (n). For a fixed real number >, consider sequence (R (n)) n N. defined in the following way:. If n is such that j n < j, for certain natural number j, then R (n) is equal to ln a ii (n) such that j ] k= j ] a ii(k) is maximal with respect to i =,...,s. Then we have ] ] j j exp R (k) = max a ii (k) k= j ] i=,...,s k= j ] and consequently R U(A d ), where A d (n) = diaga ii (n)] i=,...,n. According to Lemma 6 R U(A) and therefore lim sup R (i) (A). n By Lemma 7 lim lim sup + N N N ] If we define B(m, k) to be the transition matrix of (A d (n)) n N, then B ( j+ ], j ]) = max i=,...,s and consequently lim sup N N N ln R (i) (A). (4) j+ ] k= j ] By the inequality () it is also clear that lim sup N N N ln a ii (k) B ( j+ ], j ]) = lim sup N A ( j+ ], j ]) = lim sup N N N N ] N R (j). Combining (4) and (5) we obtain that the limit lim α() exists and that + lim α() (A). + Fix ε>0 and a function R ε U(A) such that ( ] ]) ln B j+, j. (5) lim sup R ε (i) < (A) + ε. (7) n By Lemma 6, R ε U(A d ) and therefore ( ] ]) B j+, j D R,ε exp j+ ] i= j ] R ε (i) + ε j] (6)

A. Czornik et al. / Linear Algebra and its Applications 433 (200) 867 875 875 for certain positive D R,ε. Hence, N N ( ] ln B j+, j ]) ln D R,ε N + N ] j+ R ε (i) + ε j] i= j ] and by (5) and (7)wehave lim α() (A). + Inequalities (6) and (8) lead to the conclusion of the theorem. (8) 4. Conclusions In this paper we have considered the influence of exponentially decreasing perturbations on the Lyapunov spectrum of a linear discrete-time system with time-varying coefficients. We have shown that the changes can be described by the exponential exponents. Finally we have presented formulas for the exponential exponents in terms of the transition matrix of the original system. Acknowledgments The research presented here by the first author was done as a part of research project No. N54 45334 and research of the second and third author was done as a part of research and development project O R00 034 09 and have been supported by Ministry of Science and Higher Education funds in the years 200 20. References ] R.P. Agarwal, Difference Equations and Inequalities. Theory, Methods, and Applications, Marcel Dekker, New York, 2000. 2] E.A. Barabanov, O.G. Vishnevskaya, Dokl. Akad. Nauk Belarusi 4 (5) (997) 29 34. 3] L. Barreira, Y.B. Pesin, Lyapunov Exponents and Smooth Ergodic Theory, University Lectures Series, vol. 23, AMS Bookstore, 200. 4] L. Barreira, C. Valls, Stability theory and Lyapunov regularity, J. Differential Equations 232 (2) (2007) 675 70. 5] B.F. Bylov, R.E. Vinograd, D.M. Grobman, V.V. Nemytskii, Theory of Lyapunov Exponents and its Applications to Stability Theory, Nauka, Moscow, 966 (Russian). 6] A. Czornik, P. Mokry, A. Nawrat, On the sigma exponent of discrete linear systems, IEEE Trans. Automat. Control, accepted for publication. 7] N.A. Izobov, The higher exponent of a system with perturbations of order higher than one, Vestn. Bel. Gos. Un-ta. Ser. I, 969, No. 3, pp. 6 9 (Russian). 8] N.A. Izobov, The higher exponent of a linear system with exponential perturbations, Differential Equations 5 (7) (969) 86 92. 9] E.K. Makarov, I.V. Marchenko, N.V. Semerikova, On an upper bound for the higher exponent of a linear differential system with integrable perturbations on the half-line, Differ. Uravn. 4 (2) (2005) 25 224. 0] E.K. Makarov, I.V. Marchenko, On an algorithm for constructing an attainable upper boundary for the higher exponent of perturbed systems, Differential Equations 4 (2) (2005) 62 634. ] V.M. Millionshchikov, A proof of the attainability of the central exponents of linear systems, Siberian Math. J. 0 () (969) 69 73. 2] F. Wirth, On the calculation of time-varying stability radii, Internat. J. Robust Nonlinear Control 8 (998) 043 058.