Theorem 1 : Under assumption ( A 1 ) ( W̃ ?

This paper provides the duality structure of the optimal two-block H∞ problem. The dual description leads naturally to a numerical solution based on convex programming for LTI (including infinite dimensional) systems. Alignment conditions are obtained and show that the optimal solution is flat in general, and unique in the SISO case. It is also proved that under specific conditions a wellknown Hankel-Toepltiz operator achieves its norm on the discrete spectrum, therefore generalizing a similar result obtained formerly for finite-dimensional (rational) systems. The norm of this Hankel-Toeplitz operator corresponds to the optimal two-block H∞ performance. NOTATION R, C stand for the field of real and complex numbers respectively . < · , · > denotes either the inner or duality product depending on the context. I denotes the identity map. If B is a Banach space then B denotes its dual space. For an n-vector ζ ∈ Cn, where Cn denotes the n-dimensional complex space, |ζ| is the Euclidean norm. Cn×n is the space of n × n matrices A, where |A| is the largest singular value of A. C2n denotes the complex Banach space of 2n-vectors ζ, ζ = ( ζ1 ζ2 ) ; ζ1, ζ2 ∈ Cn with the norm |ζ| = √ |ζ1|2 + |ζ2|2 (1) Let C2n×n denote the complex Banach space of 2n × n matrices A, A = ( A1 A2 ) , A1, A2 ∈ Cn×n, with the following norm ‖A‖ := √ |A1|2 + |A2|2 (2) Let STr(A1) := Tr(A1A1) 1 2 = ∑n j=1 σj(A1), where σj(A1) is the i-th singular value of A1, and Tr(A) denotes the trace of A. STr(A1) is known as the trace-class norm of A1. The dual space of C2n×n, denoted C2n×n, is the space of matrices under the norm ‖A‖? := √ STr(A1) + STr(A2) (3) The symbol D denotes the unit disk of the complex plane, D = {z ∈ C : |z| < 1}. ∂D denotes the boundary of D, ∂D = {z ∈ C : |z| = 1}. If E is a subset The authors are with the Electrical & Computer Engineering Department, University of Tennessee, Knoxville, TN 37996-2100, [email protected], [email protected] of ∂D, then E denotes the complement of E in ∂D. m denotes the normalized Lebesgue measure on the unit circle ∂D, m(∂D) = 1. m a.e. is the label used for “Lebesgue almost everywhere”. For a matrix or vector-valued function F on the unit circle, |F | is the real-valued function defined on the unit circle by |F |(eiθ) = |F (eiθ)|, θ ∈ [0, 2π). If X denotes a finite dimensional complex Banach space, L(X), 1 ≤ p ≤ ∞, stands for the Lebesgue-Bochner space of p-th power absolutely integrable X-valued functions on ∂D under the norm