By Harm Bart, Israel Gohberg, Marinus A. Kaashoek, André C.M. Ran
The current ebook bargains with canonical factorization difficulties for di?erent periods of matrix and operator services. Such difficulties seem in a number of parts of ma- ematics and its purposes. The services we contemplate havein universal that they seem within the nation area shape or might be represented in one of these shape. the most effects are all expressed when it comes to the matrices or operators showing within the kingdom house illustration. This comprises valuable and su?cient stipulations for canonical factorizations to exist and particular formulation for the corresponding f- tors. additionally, within the functions the entries within the nation area illustration play a vital position. Thetheorydevelopedinthebookisbasedonageometricapproachwhichhas its origins in di?erent ?elds. one of many preliminary steps are available in mathematical structures idea and electric community idea, the place a cascade decomposition of an input-output procedure or a community is expounded to a factorization of the linked move functionality. Canonical factorization has a protracted and engaging background which starts off within the idea of convolution equations. fixing Wiener-Hopf imperative equations is heavily concerning canonical factorization. the matter of canonical factorization additionally looks in different branches of utilized research and in mathematical platforms conception, in H -control thought in particular.
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Additional info for A State Space Approach to Canonical Factorization with Applications
1, this anomaly disappears so that the two pole orders are the same. 1 is generally not the least possible. 3 Realization of analytic operator functions In this section we consider the realization problem for possibly non-rational operator functions. First we consider operator functions that are analytic on a bounded Cauchy domain in C. Recall from Chapter 0 that the boundary of such a Cauchy domain consists of a ﬁnite number of simple closed non-intersecting rectiﬁable curves. 2. Let Ω be a bounded Cauchy domain, and let W be an operator function with values in L(U, Y ), where U and Y are complex Banach spaces.
7) can be written as ∞ φ(t) = f (t) + γ(t, s)f (s) ds, 0 ⎧ × × ⎨ +iCe−itA ΠeisA B, where γ(t, s) = ⎩ × s < t, × −iCe−itA (In − Π)eisA B, s > t. 44 Chapter 3. Explicit solutions using realizations Proof. 5). 5). 4). 1 to prove the ﬁrst part of the theorem. Next assume that conditions (i) and (ii) hold true. 2 once again, we get the desired formulas for Im + G+ (λ), Im + G− (λ) and their inverses. The formulas for γ+ and γ− are now obtained by noticing that ∞ × eiλt e−itA Π dt = i(λ − A× )−1 Π, = −i(I − Π)(λ − A× )−1 , 0 0 −∞ × eiλt (I − Π)e−itA dt λ ∈ ρ(A× ), λ ≥ 0, λ ∈ ρ(A× ), λ ≤ 0, where I = In .
In other words C(λ − A)−1 Bu = W (λ)u − W (∞) u. As u ∈ U was taken arbitrarily, we get W (λ) = W (∞) + C(λ − A)−1 B for each λ ∈ Ω. 4 Inversion We begin with some heuristics. 4) and view W as the transfer function of the linear time invariant system ⎧ x (t) = Ax(t) + Bu(t), ⎪ ⎪ ⎨ y(t) = Cx(t) + Du(t), t ≥ 0, Σ ⎪ ⎪ ⎩ x(0) = 0. Assuming that we are in the biproper situation where D is invertible, we can solve u in terms of x and y: u(t) = −D−1 Cx(t) + D−1 y(t), t ≥ 0. Inserting this into Σ yields ⎧ x (t) = A× x(t) + BD−1 y(t), ⎪ ⎪ ⎨ u(t) = −D−1 Cx(t) + D−1 y(t), Σ× ⎪ ⎪ ⎩ x(0) = 0.