On the continuity of the J-spectral factorization mapping

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1 On the continuity of the J-spectral factorization mapping Orest V. Iftime University of Groningen Department of Mathematics and Computing Science PO Box 800, 9700 AV Groningen, The Netherlands Abstract The continuity of the mapping which associates a J-spectral factor to a spectral density is analyzed. For the class of essentially bounded functions on the imaginary axis that are bounded away from zero it is well nown that this mapping, even for the scalar case, is not continuous. In this paper three results concerning the continuity of the mapping which associates a J-spectral factor to a spectral density are provided for matrix-valued functions in the Wiener class. One of them is well nown, and the other two are extensions of theorems concerning the continuity of the spectral factorization mapping. KEYWORDS: Wiener class, J-spectral factorization, approximation, continuous dependence. 1 Introduction Roughly speaing, the J-spectral factorization problem is: Given a matrix-valued function Z defined on the imaginary axis, find a stable invertible matrix-valued function V with a stable inverse such that Z(s) = V T ( s)j p,q V (s) a.e. on the imaginary axis, where J p,q is a signature matrix. Such a V is nown as a J-spectral factor for Z. The J-spectral factorization problem naturally arises in control theory and it plays an important role in H -control, linear quadratic optimal control, Hanel norm approximation problem. Characterization of solution for control problems is sometimes given using the J- spectral factor(s). Consequently, it is important to provide computation algorithms for solving the J-spectral factorization problem. One would naturally thin about approximations of the J-spectral factor. This leads us to the question of whether or not the mapping which associates a J-spectral factor to the matrix-valued function to be factorized (spectral density) is continuous. An example (see [1]) shows that the spectral factorization mapping is not continuous in the -norm (see also [9]). In this paper three results concerning the continuity of the mapping which associates a J-spectral factor to a spectral density (J-spectral factorization mapping) are provided for matrix-valued functions in the Wiener class. The first result is well nown (see [3]) and states that the J-spectral factorization mapping depends continuously on the matrix-function to be factorized with respect to the 2-norm (the norm in L 2, the space of square integrable functions 1

2 on the imaginary axis). The other two results are extensions of theorems concerning the continuity of the mapping which associates a spectral factor to a spectral density (spectralfactorization mapping). One of them shows that the J-spectral factor depends continuously on the spectral density in the Wiener class topology. Moreover, if one assumes that the derivatives of the matrix-functions to be factorized exist and they are bounded in the 2-norm, the J-spectral factorization mapping is continuous in the -norm (the essential supremum norm on the imaginary axis). 2 Transfer function spaces The causal Wiener class (the class of stable transfer functions) is defined via their impulse responses. More precisely, let us consider the set A of functions f with the representation { fa (t) + f f(t) = 0 δ(t), t 0, 0, t < 0, where f 0 C (the set of complex numbers), 0 f a (t) dt <, and δ represents the delta distribution at zero. We define the causal Wiener class, denoted by Â, as the set of Laplace transform of functions in A. For a complex function f, we use the notation f to mean the following: f (s) = f( s) where by z we mean the complex conjugate of the complex number z. The Wiener class of transfer functions, denoted by Ŵ, is defined as { } Ŵ = g L (ir, C) g(i ) = g 1 (i ) + g 2 (i ), with g 1, g2 Â, where L is the space of functions bounded almost everywhere on the imaginary axis. The Wiener class is a decomposing Banach algebra continuously embedded in C(iR), the space of continuous functions on the imaginary axis with the sup-norm ( C(iR) Ŵ). Moreover, Â is a closed subalgebra of Ŵ, Â H (the set of stable functions in L ). For more information on the Wiener class we refer to [9] and the references therein (remar that we consider here only functions with no delayed impulses). We consider square matrix-functions over the Banach algebras L n n, H n n, Ŵ n n and Ân n, respectively, with an appropriate norm. 3 Continuity of the J-spectral factorization mapping Since the J-spectral factor is not unique (only up to a multiplication with a constant J-unitary matrix) we first describe more precisely what the mapping from the matrix-function to be factorized to the J-spectral factor is. For Z Ŵn n we define the operator T Z by T Z (X) := P (ZX) + Q(X), X Ŵn n, (1) where P is the continuous projection from Ŵn n onto Ân n and Q = I P. The following result is a particular case for Theorem 1.1, Clancey and Gohberg [3], page 35. Moreover the result in Corollary 1.1, page 37, gives explicit formulae for the inverse of T Z, provided that Z Ŵn n admits a J-spectral factorization. 2

3 Proposition If Z = Z Ŵn n admits a J-spectral factorization, then the operator T Z L(Ŵn n ) is invertible. The following proposition provides a formula for the J-spectral factor. For more details we refer to Clancey and Gohberg [3], Chapter II, Corollary 1.2., page 39, and Chapters III, Proposition 2.2., page 80. After a careful reading of the above mentioned proofs one can derive a theoretical algorithm for computing the J-spectral factor (an explicit K in the following proposition may be obtained). Proposition Let Z = Z Ŵn n be a matrix-valued function which admits a J-spectral factorization. Than the matrix-valued function V := KT 1 Z (I), (2) is a J-spectral factor for Z, for some invertible constant matrix K, and T Z is defined by (1). Suppose that Z = Z Ŵn n admits a J-spectral factorization. Necessary and sufficient conditions for existence of a J-spectral factorization are provided in [6]. Since the J-spectral factorization is not unique, one should specify exactly what the J-spectral factorization mapping is. Definition 3.1. We consider as the J-spectral factorization mapping the function which associate to a Z the J-spectral factor V given by (2). The following theorem shows that the J-spectral factorization mapping depends continuously on the matrix-function to be factorized with respect to the 2-norm. This theorem is an alternative formulation of a result in [3], page 205. Theorem 3.2. Assume that Z, Z Ŵn n, N, admit J-spectral factorizations and satisfy Z Z in the -norm as. Consider V, V the J-spectral factors associated to Z, Z, respectively, and assume that V V, V 1 V 1 L n n 2. Then V V and V 1 V 1 in the 2-norm as. More precisely, there exist constants c 1, c 2 > 0 such that V V 2 c 1 Z Z, and V 1 V 1 2 c 2 Z Z. A stronger continuity result for the J-spectral factorization mapping than the one stated in the above theorem is formulated below, and it is a generalization of the result obtained for (scalar) spectral factorization mapping in [7]. Theorem 3.3. Assume that Z, Z Ŵn n, N admit J-spectral factorizations and satisfy Z Z in the Ŵ-norm as. Consider V, V the J-spectral factors associated to Z, Z, respectively. Then V V and V 1 V 1 in the Ŵ-norm as. More precisely, there exist constants c 1, c 2 > 0 such that V V Ŵ c 1 Z Z Ŵ, and V 1 V 1 Ŵ c 2 Z Z Ŵ. Proof. From the definition of the J-spectral factor corresponding to the J-spectral mapping (see (2)) the first inequality is immediate. Since Z, Z Ŵn n, N admit J-spectral factorizations we now that the operators T Z and T Z are invertible over Ŵn n. Then the second inequality follows from an inversion theorem in Banach algebras [5, Chapter XXIX.4] combined with the first inequality. 3

4 As it was already said before, the scalar spectral factorization mapping is not continuous in the -norm. Let Z, Z Ŵn n, N admit J-spectral factorizations. In order to guarantee that the sequence of J-spectral factors corresponding to Z converges in the -norm to a J-spectral factor of the original matrix-function to be factorized, Z, besides the condition that Z Z in the -norm as, one needs to assume that the derivatives exist and are bounded in the 2-norm. Similar results on the continuity of the spectral factorization mapping can be found in [1, 9]. Theorem 3.4. Assume that Z, Z Ŵn n, N admit J-spectral factorizations and satisfy Z Z in the -norm as. Consider V, V the J-spectral factors associated to Z, Z, respectively, and assume that V V L2 n n, their derivative exists almost everywhere, V V L n n 2 and V V 2 < M. Then V V in the -norm as. More precisely, there is a constant c > 0 such that V V c Z Z V V 2. Proof. The J-spectral factors may be chosen to have the same constant value at infinity by suitably choice of the constant matrix which defines them. Then, for any ω, one can write [V (jω) V (jω)] [V (jω) V (jω)] = ω After derivation under the integral one can write the inequality σ([v (jω) V (jω)]) 2 ω d dα [V (jα) V (jα)] [V (jα) V (jα)] dα. σ( d dα [V (jα) V (jα)])σ([v (jα) V (jα)])dα. where σ(m) is the maximal singular value of M. Using Cauchy inequality and then taing supremum over all ω there exist a positive constant c such that V V c Z Z V V 2. Moreover, the inequality V V 2 c Z Z from Theorem 3.2 was also used. Remar that when the derivatives of the J-spectral factors exist almost everywhere then also the derivatives of the spectral densities should exist almost everywhere. 4 Conclusions The continuity of the mapping which associates a J-spectral factor to a spectral density for matrix-functions in the Wiener class was investigated. Two results, which are extensions of theorems concerning the continuity of the (scalar) spectral factorization mapping were presented. One of them shows continuous dependence in the Wiener class topology. The other one shows that the J-spectral factorization mapping is continuous in the -norm provided that the derivatives of the matrix-functions to be factorized exist and they are bounded in the 2-norm. The above results hold not only for the Wiener class but also for any decomposing Banach algebra ˆB of continuous functions on the imaginary axis. Since the set of rational functions (on ir { }) is dense in ˆB, one might use the above to provide a better understanding for problems that might occur when computing a rational approximate of a J-spectral factor. Acnowledgement: The author would lie to than Birgit Jacob from University of Dortmund for preliminary discussions. 4

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