Hilbert space methods for some interpolation problems arising in - - PowerPoint PPT Presentation

Hilbert space methods for some interpolation problems arising in control engineering

Nicholas Young

Leeds and Newcastle Universities

Joint work with Jim Agler, Zina Lykova and David Brown Gargnano, June 2017

Some analytic interpolation problems

Let Ω be a domain in Cd. We seek a method to resolve the following question. Given distinct points λ1, . . . , λn in the unit disc D and target points w1, . . . , wn in Ω, does there exist an analytic map h : D → Ω such that h(λj) = wj for j = 1, . . . , n? When Ω is a matrix ball this is the Nevanlinna-Pick problem, which is solved by Pick’s Interpolation Theorem. Other domains Ω arise in the robust stabilization problem in control engineering. Of especial interest is the case that Ω is the set of N × N matrices of spectral radius < 1.

Synopsis

• The spectral Nevanlinna-Pick problem
• The symmetrized bidisc
• The rich saltire – some function spaces and mappings
• A solvability criterion for the spectral Nevanlinna-Pick

problem

• The tetrablock

These slides are accessible at http://www1.maths.leeds.ac.uk/˜nicholas/slides/2017/Gar.pdf

The spectral Nevanlinna-Pick problem

Let r(A) denote the spectral radius of a square matrix A. Given distinct points λ1, . . . , λn in the unit disc D and matri- ces W1, . . . , Wn in CN×N, determine whether there exists an analytic map F : D → CN×N such that F(λj) = Wj for j = 1, . . . , n and r(F(λ)) < 1 for all λ ∈ D. This is a long-standing problem. No satisfactory solution is currently known. We study the case that N = 2. Denote by Ω the domain {W ∈ C2×2 : r(W) < 1}.

The symmetrized bidisc

is the domain G def = {(z + w, zw) : z, w ∈ D}. It is a nonconvex, polynomially convex domain. A 2×2 matrix A belongs to Ω if and only if (tr A, det A) ∈ G. The set G ∩ R2 is the interior of an isosceles triangle:

s (2,1) (−2,1) (0,−1) p

Interpolation into Ω and G

Let λ1, . . . , λn ∈ D and W1, . . . , Wn ∈ Ω, none of them scalar

• matrices. The following statements are equivalent.

(1) There exists an analytic map F : D → Ω such that F(λj) = Wj, j = 1, . . . , n; (2) there exists an analytic map h : D → G such that h(λj) = (tr Wj, det Wj), j = 1, . . . , n. (1)⇒(2) Take h to be (tr F, det F). The converse is also easy. Thus the interpolation problems for Ω and G are equivalent.

The rich saltire∗

The following diagram summarises some spaces, mappings and correspondences related to the construction of analytic 2 × 2 matrix functions. S2×2

Left S

• Upper E
• R1

Upper W

• Right S
• Hol(D, G)

Left N

• Lower E
• S2

Lower W

• Right N
• ∗A heraldic term meaning a design formed by a bend and a bend sinister

crossing like a St. Andrew’s cross (Concise Oxford Dictionary), as in the Scottish flag

Some spaces of functions

S2×2 is the 2 × 2 matricial Schur class of the disc, that is, the set of analytic 2 × 2 matrix functions F on D such that F(λ) ≤ 1 for all λ ∈ D. S2 is the Schur class of the bidisc D2, that is, Hol(D2, D). R1 is the set of pairs (N, M) of analytic positive kernels on D2 such that the kernel defined by (z, λ, w, µ) → 1 − (1 − wz)N(z, λ, w, µ) − (1 − µλ)M(z, λ, w, µ), for all z, λ, w, µ ∈ D, is positive semidefinite on D2 and is of rank 1.

Some maps in the rich saltire

Left S : S2×2 → Hol(D, G) maps F to (tr F, det F). Left N : Hol(D, G) → S2×2 maps h to

1

2h1

f g

1 2h1

• where f, g ∈ H∞, |f| = |g| on T and fg = 1

4(h1)2 − 4h2.

Lower E : Hol(D, G) → S2 maps h to the function (z, λ) → 2zh2(λ) − h1(λ) 2 − zh1(λ) .

Right N – Agler’s realization theorem

Right N is a set-valued map from the Schur class of the bidisc to R1. It maps h ∈ S2 to the set of pairs (N, M) of Agler kernels for h, so that 1−h(w, µ)h(z, λ) = (1−wz)N(z, λ, w, µ)+(1−µλ)M(z, λ, w, µ). If h = (s, p) ∈ Hol(D, ¯ G) then Lower E ◦ Right N maps h to a nonempty set of pairs (N, M) of kernels in R1 such that, for all (z, λ), (w, µ) ∈ D2, 1−

• 2wp(µ) − s(µ)

2 − ws(µ)

• 2zp(λ) − s(λ)

2 − zs(λ) = (1 − ¯ wz)N(z, λ, w, µ) + (1 − ¯ µλ)M(z, λ, w, µ).

A necessary condition for interpolation

Suppose h ∈ Hol(D, ¯ G) satisfies h(λj) = (sj, pj), j = 1, . . . , n. Choose any three distinct points z1, z2, z3 ∈ D. Localise the last equation at the 3n points (zℓ, λi) ∈ D2 to obtain the statement there exist 3n-square positive matrices N = [Niℓ,jk]n,3

i,j=1, ℓ,k=1 ≥ 0,

M = [Miℓ,jk]n,3

i,j=1, ℓ,k=1 ≥ 0

such that rank N ≤ 1 and

 1 −

• 2zℓpi − si

2 − zℓsi

2zkpj − sj

2 − zksj

  =

• (1 − ¯

zℓzk)Niℓ,jk

• +
• (1 − ¯

λiλj)Miℓ,jk

• .

A solvability criterion for the spectral Nevanlinna-Pick problem

Let W1, . . . , Wn be nonscalar 2×2 matrices and let z1, z2, z3 ∈ D be distinct. The spectral Nevanlinna-Pick problem λj → Wj, 1 ≤ j ≤ n, is solvable if and only if there exist matrices N = [Niℓ,jk]n,3

i,j=1, ℓ,k=1 ≥ 0,

M = [Miℓ,jk]n,3

i,j=1, ℓ,k=1 ≥ 0

such that rank N ≤ 1 and

 1 −

• 2zℓpi − si

2 − zℓsi

2zkpj − sj

2 − zksj

  ≥

• (1 − ¯

zℓzk)Niℓ,jk

• +
• (1 − ¯

λiλj)Miℓ,jk

• where sj = tr Wj, pj = det Wj.

Proof of sufficiency

S2×2

Left S

• Upper E
• R1

Upper W

• Right S
• Hol(D, G)

Left N

• Lower E
• S2

Lower W

• Right N
• Given a suitable pair (N, M) ∈ R1, one constructs an inter-

polating function h ∈ Hol(D, G) using Left S ◦ Upper W. Upper W : R1 → S2×2 is a construction that uses a standard lurking isometry argument. This procedure, applied to all suitable pairs (N, M), yields all possible interpolating functions h ∈ Hol(D, ¯ G).

Interpolation into the tetrablock

The tetrablock is a domain E in C3 that plays a similar role to G for another interpolation problem arising in robust control theory. E ∩ R3 is a regular tetrahedron. There is also a rich saltire for E: simply replace Hol(D, G) by Hol(D, E) and modify the mappings in the lower triangle. There is a solvability criterion for interpolation problems in Hol(D, ¯ E). All constructions and proofs are similar to those described above. Question: for what other domains does this method work?

References

• J. Agler, Z. A. Lykova and N. J. Young, A case of mu-

synthesis as a quadratic semidefinite progam, SIAM J. Con- trol and Optimization, 51 (3) (2013) 2472-2508.

• D. C. Brown, Z. A. Lykova and N. J. Young, A rich structure

related to the construction of holomorphic matrix functions, Journal of Functional Analysis, 272(4) (2017), 1704–1754.

Thank you!