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 C d . We seek a method to resolve the following question. Given distinct points λ 1 , . . . , λ n in the unit disc D and target points w 1 , . . . , w n in Ω, does there exist an analytic map h : D → Ω such that h ( λ j ) = w j 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 λ 1 , . . . , λ n in the unit disc D and matri- ces W 1 , . . . , W n in C N × N , determine whether there exists an analytic map F : D → C N × N such that F ( λ j ) = W j 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 ∈ C 2 × 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 ∩ R 2 is the interior of an isosceles triangle: p (−2,1) (2,1) s (0,−1)

Interpolation into Ω and G Let λ 1 , . . . , λ n ∈ D and W 1 , . . . , W n ∈ Ω, none of them scalar matrices. The following statements are equivalent. (1) There exists an analytic map F : D → Ω such that F ( λ j ) = W j , j = 1 , . . . , n ; (2) there exists an analytic map h : D → G such that h ( λ j ) = (tr W j , det W j ) , 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. Upper W S 2 × 2 R 1 Upper E Right S Right N Left S Left N Lower W Hol( D , G ) S 2 Lower E ∗ 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 S 2 × 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 . S 2 is the Schur class of the bidisc D 2 , that is, Hol( D 2 , D ). R 1 is the set of pairs ( N, M ) of analytic positive kernels on D 2 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 D 2 and is of rank 1.

Some maps in the rich saltire Left S : S 2 × 2 → Hol( D , G ) maps F to (tr F, det F ). Left N : Hol( D , G ) → S 2 × 2 maps h to � 1 � 2 h 1 f 1 g 2 h 1 4 ( h 1 ) 2 − 4 h 2 . where f, g ∈ H ∞ , | f | = | g | on T and fg = 1 Lower E : Hol( D , G ) → S 2 maps h to the function ( z, λ ) �→ 2 zh 2 ( λ ) − h 1 ( λ ) . 2 − zh 1 ( λ )

Right N – Agler’s realization theorem Right N is a set-valued map from the Schur class of the bidisc to R 1 . It maps h ∈ S 2 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 R 1 such that, for all ( z, λ ) , ( w, µ ) ∈ D 2 , � � 2 wp ( µ ) − s ( µ ) 2 zp ( λ ) − s ( λ ) 1 − = 2 − ws ( µ ) 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 ( λ j ) = ( s j , p j ) , j = 1 , . . . , n. Choose any three distinct points z 1 , z 2 , z 3 ∈ D . Localise the last equation at the 3 n points ( z ℓ , λ i ) ∈ D 2 to obtain the statement there exist 3 n -square positive matrices N = [ N iℓ,jk ] n, 3 i,j =1 , ℓ,k =1 ≥ 0 , M = [ M iℓ,jk ] n, 3 i,j =1 , ℓ,k =1 ≥ 0 such that rank N ≤ 1 and   � � 2 z k p j − s j 2 z ℓ p i − s i � � � �  = (1 − ¯  1 − (1 − ¯ z ℓ z k ) N iℓ,jk + λ i λ j ) M iℓ,jk . 2 − z ℓ s i 2 − z k s j

A solvability criterion for the spectral Nevanlinna-Pick problem Let W 1 , . . . , W n be nonscalar 2 × 2 matrices and let z 1 , z 2 , z 3 ∈ D be distinct. The spectral Nevanlinna-Pick problem λ j �→ W j , 1 ≤ j ≤ n , is solvable if and only if there exist matrices N = [ N iℓ,jk ] n, 3 i,j =1 , ℓ,k =1 ≥ 0 , M = [ M iℓ,jk ] n, 3 i,j =1 , ℓ,k =1 ≥ 0 such that rank N ≤ 1 and   � � 2 z k p j − s j 2 z ℓ p i − s i � � � �  ≥ (1 − ¯  1 − (1 − ¯ z ℓ z k ) N iℓ,jk + λ i λ j ) M iℓ,jk 2 − z ℓ s i 2 − z k s j where s j = tr W j , p j = det W j .

� � � � � � � � � � Proof of sufficiency Upper W S 2 × 2 R 1 Upper E Left S Left N Right S Right N Lower W Hol( D , G ) S 2 Lower E Given a suitable pair ( N, M ) ∈ R 1 , one constructs an inter- polating function h ∈ Hol( D , G ) using Left S ◦ Upper W. Upper W : R 1 → S 2 × 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 C 3 that plays a similar role to G for another interpolation problem arising in robust control theory. E ∩ R 3 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.

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