Structural stability of (C, A)-marked and observable subspaces

Transcription

1 Universitat Politècnica de Catalunya Escola Tècnica Superior d Enginyeria Industrial de Barcelona Structural stability of (C, A)-marked and observable subspaces Albert Compta, Marta Peña Departament de Matemàtica Aplicada I Diagonal 647, Barcelona, Spain ETSEIB, UPC Regina 2005

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3 Context Deformation of f keeping the subspace invariant Deformation of invariant subspace keeping f fixed Deformation of pairs of f and invariant subspace f ~ A f ~ (C,A) 1

4 Aim F.Puerta-X.Puerta-S.Tarragona obtain the implicit form of a miniversal deformation of a (C,A)- invariant subspace with regard to the usual equivalence relation We obtain for a (C,A)-marked subspace: The explicit form of its miniversal deformation The dimension of its orbit The stable (C,A)-marked subspaces 2

5 Notation A C observable linear map defined in a subspace with Brunovsky numbers k = ( k1, k2,, k r ) Inv( kh, ) manifold of the (C,A)-invariant subspaces with Brunovsky numbers of the restriction = (,,, ) with regard to the block h h1 h2 h s equivalence 3

6 Differentiable structure of Inv(k,h) M( khset, ) of matrices of full rank with regard to the partition (k,h) formed by the matrices with constant diagonals depending on k h + 1 parameters i j X xij,,1 0 0 xij,,2 xij,,1 0 xij,,2 xij,,1 xijk,, 1,,2 M(, ) i h x = j+ ij kh 0 xijk,, i hj xijk,, i hj+ 1 Differentiable structure as an orbit space Inv( kh, ) M( kh, ) M( hh, ) (J.Ferrer-F.Puerta-X.Puerta) 4

7 Deformations Let M be a manifold and G a Lie group acting on it: G M M ( g, x) g x The orbit of the matrix x M {, } O x = g x g G is its equivalence class associated to this action Deformation Definition An l-parametrized deformation of x M is a differentiable map φ: U M where U is a neighbourhood of the origin φ (0) = x 5

8 Deformations Versal / Miniversal deformation Definition φ: U M is called versal if for any other deformation there is: ψ: V M, V a neighbourhood of the origin V' V, 0 V' a differentiable map α : V' U, α (0) = 0 k a deformation β: V ' G of I G such that ψ( v) = β( v) φ(α( v)), v V ' It is miniversal if the number of parameters is the minimum among all the versal deformations 6

9 Deformations Transversality Theorem (Arnold) A submanifold N versal deformation of x transversal to O x at x: M is a N if and only if N is N versal T ( N) + TO = T ( M) x x x x N miniversal T ( N) TO = T ( M) x x x x If N is a miniversal deformation of x M, then dimo x = dim M dim N Corollary A supplementary subspace of TxO x determines a miniversal deformation of x N 7

10 Miniversal deformation of (C,A)-invariant subspaces (C,A)-invariant subspace Definition A( W Ker C) W Implicit form deformation Theorem (F.Puerta-X.Puerta-S.Tarragona) X M( kh, ), V Sp( X) A miniversal deformation of V in Inv(k,h) with regard to the group action M( kk, ) is such that W M( k, h) and Sp( X + W ) tr( PXW ) = tr( XQW ) = 0 P M( kk, ), Q M( hh, ) 8

11 (C,A)-marked subspaces (C,A)-marked subspace Definition A (C,A)-invariant subspace is marked if there is a Brunovsky basis of the restriction extendible to a Brunovsky basis of the whole space A A 1 (i) C 1 and 2 C2 A (ii)( ) C A A 0 A C C C 0 C ( A) Brunovsky matrices similar by a permutation to a Brunovsky matrix Remark h h h > h = = h = 1 2 s s+ 1 r 0 If V Inv( k, h) is a (C,A)-marked subspace we can suppose: ki hi and ki ki+ 1 if hi = hi+ 1 9

12 (C,A)-marked subspaces Canonical matrix representation Definition (C,A) observable pair with Brunovsky numbers k V Inv( k, h) (C,A)-marked subspace, X M( kh, ) X such that Sp( X ) O ( V ) is a canonical matrix representation of O ( V ) if it is 0 Xii, = for all 1 i s I hi X = 0 in other case i, j Example k = (4,2), h = (3,1)

13 Miniversal deformation of (C,A)-marked subspaces Explicit form deformation Proposition (C,A) observable pair with Brunovsky numbers k V = Sp( X) Inv( k, h) (C,A)-marked subspace X M( kh, ) canonical matrix representation of O ( V ) W M( k, h) is a solution of tr( PXW ) = 0 P M( kk, ) tr( XQW ) = 0 Q M( h, h) if and only if w,, = 0 k h < l k h i r, 1 j s, k k w,, = 0 k h < l k h i r, 1 j s, h h i j l j j i j i j i j l i i i j i j 11

14 Miniversal deformation of (C,A)-marked subspaces Explicit form deformation Theorem (C,A) observable pair with Brunovsky numbers k V = Sp( X) Inv( k, h) (C,A)-marked subspace X M( kh, ) canonical matrix representation of O ( V ) A miniversal deformation of V is such that W M( k, h) and Sp( X + W ) wi, j, l= 0 if min( ki hi, kj hj) < l ki hj+ 1, k k or h h i j i j 12

15 Miniversal deformation of (C,A)-marked subspaces Explicit form deformation Examples k = (4,2), h = (3,1) k = (4,2), h = (2,1) w 0 0 w 1 w w w 1 3 w 0 w w2 w1 0 1 w w 0 w w4 1 13

16 Miniversal deformation of (C,A)-marked subspaces Dimension of the orbit Corollary (C,A) observable pair with Brunovsky numbers k V Inv( k, h) (C,A)-marked subspace codim O( V) = min( k h, k h ) + i r, j s k k or h h i j i j + ( k h + 1) i r, j s k > k h > h j i j i i j i i j j 14

17 Miniversal deformation of (C,A)-marked subspaces Structural stability Definition Characterization structurally stable elements: Corollary codim O ( V ) = 0 (C,A) observable pair with Brunovsky numbers k V Inv( k, h) (C,A)-marked subspace is structurally stable with regard to the considered equivalence relation if and only if k = h 1 j s j j 15

18 References [1] V.I. Arnold, On Matrices Depending on Parametres, Uspekhi Mat. Nauk., 26 (1971), p [2] A. Compta; J. Ferrer; M. Peña, Dimension of the Orbit of Marked Subspaces, Linear Algebra Appl., 379 (2004), p [3] J. Ferrer; F. Puerta; X. Puerta, Differentiable Structure of the Set of Controllable (A,B)- invariant Subspaces, Linear Algebra Appl., (1998), p [4] I. Gohberg; P. Lancaster; L. Rodman, Invariant subspaces of Matrices with Applications, Wiley, New York (1986). [5] F. Puerta; X. Puerta; S. Tarragona, Versal Deformations in Orbit Spaces, Linear Algebra Appl., 379 (2004), p

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