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On the number of control sets on projective spaces
SFSTUB h CONTIOL ELSEVIER
Systems & Control Letters 29 (1996) 21-26
On the number of control sets on projective spaces Carlos Jos6 Braga Barros a, Luiz A.B. San Martinb'*,l a Departamento de Matemtitica, Universidade Estadual de Marinod*, Cx. Postal 331, 87.020-900 Marinffd-Pr, Brazil b Instituto de Matemdttica, Universidade Estadual de Campinas, Cx. Postal 6065, 13.081-970 Campinas-SP, Brazil
Received 7 February 1995; revised 3 June 1996
Abstract The purpose of this paper is to provide an upper bound for the number of control sets for linear semigroups acting on a projective space RP d-:. These semigroups and control sets were studied by Colonius and Kliemarm (1993) who proved that there are at most d control sets. Here we apply the results of San Martin and Tonelli (1995) about control sets for semigroups in semisimple Lie groups and make a case by case analysis according to the transitive groups on RP d-: which were classified by Boothby and Wilson (1975, 1979) in order to improve that upper bound. It turns out that in some cases there are at most d/2 or d/4 control sets. Keywords: Control set; Projective space; Transitive group
1. Introduction This paper considers semigroups of invertible linear mappings and their actions on projective spaces. These semigroup actions arise naturally in the study of semilinear control systems evolving on R d as well as in the study of stability properties of non-linear systems evolving on differentiable manifolds. The issue here is about the control sets for the semigroups in the projective spaces RP d-1. These are defined as follows: let M be a differentiable manifold and S a semigroup of diffeomorphisms of M. A control set for S is a subset D C M which satisfies (1) D C cl(Sx) for all x E D, (2) intO ~ 0 and (3) D is maximal with these properties (where el and int means closure and interior, respectively). We refer the reader to [1,4, 8] for further details about the control sets and the role they play in the * Corresponding author. E-mail: [email protected]. I Research partially supported by the Brazilian Research Council-CNPq Proc. No. 301060/94-0.
study of the dynamical and geometrical properties of a control system. In the sequel we shall be interested in semigroups which satisfy the following analogue of the accessibility property of control systems: (A)
int(Sx) ~ 0
for all x E M.
For semigroups satisfying (A), it is possible to single out the class of control sets that contain points which are self-attainable and belong to the interior of their forward orbit. Explicitly, given a control set D put Do = {x E D: x E int(Sx) M int(S-lx)} . We shall refer to this subset as the set of controllability (or transitivity) of S inside D. In general, Do may be empty. Control sets for which the sets of controllability are not empty were named effective control sets in [8]. We mention that every control set is effective when the semigroup comes from a control system which satisfies the accessibility property (A). In the sequel we consider effective control sets only. Now, suppose that S is a subsemigroup of the general linear group Gl(d, R). Since the elements of S are
0167-6911/96/$12.00 Copyright (~) 1996 Elsevier Science B.V. All rights reserved SSDI 0167-6911(96)00047-3
22
C.J.B. Barros, L.A.B. San Martin~Systems & Control Letters 29 (1996) 21-26
invertible linear mappings, they map lines in ~d into lines and thus S acts in the projective space R P d - I as a semigroup of diffeomorphisms. The control sets for such a semigroup action were studied by Colonius and Kliemann [4]. One of the results of that paper states that S has at most d effective control sets in RP d-1 if it satisfies the accessibility property. This result is obtained in [4] through an analysis of the spectral decompositions of the elements of S. The purpose of this paper is to improve the estimate of the number of control sets given in [4]. We assume also accessibility of the action of S on the projective space but take an indirect approach for counting the number of control sets. This approach is based on the results of [8] about control sets for the actions of semigroups with non-empty interior in semisimpie Lie groups on the Furstenberg boundaries of the group. For these actions the number of effective control sets are computed exactly in terms of the Weyl group W of the semisimple group, and of its subgroup W(S) which depends on S and We which depends on the specific boundary under consideration (see [8, Corollary 5.2]). In particular, the number of control sets is [W/W~91if W(S) reduces to the identity and this is the maximum number of control sets in the boundary associated to We for general semigroups with non-empty interior. On the other hand, a Lie group G transitive on the projective space is reductive and hence nearly semisimple. A list of these groups was provided in [2, 3]. With the aid of this list, we make a case by case analysis according to the transitive groups. In all the cases, the projective space can be seen as a homogeneous space of G which fibers equivariantly over one of its boundaries. This fibration is in fact a principal fiber bundle with compact structure group, and this specificity implies that the quantity of control sets for a semigroup S C G in projective space coincides with the number of control sets for the same semigroup in some of the boundaries of G. This way, we get the maximum number of effective control sets for S in ~ p a - l if S is a semigroup with non-empty interior of G. It turns out that the estimate is d, as in [4] for some of the transitive groups but goes down to d/2 or d/4 for some other groups in the list.
2. Preliminaries
As mentioned above, our method consists of a case by case analysis according to the transitive groups on
the projective spaces. So we consider only the situation where G is a connected Lie group transitive on •pa-1 and S c G a semigroup with non-empty interior. This set up covers the case where the semigroup comes from a (continuous time) control system. As is well known, a right invariant control system in a Lie group generates a semigroup (control semigroup) which has non-empty interior in a Lie subgroup (see [6]). Therefore, if we consider control systems on R a which are linear in the state variables (as in [4]) we have that the system satisfies (A) on the projective space if and only if the corresponding control semigroup is a semigroup with non-empty interior in a connected Lie group transitive on •pa-1. Now, let g be the Lie algebra of G. It is a reductive Lie algebra and decomposes as
g=go•z with go semisimple and z the center o f g . Denote by Go the connected group associated to go. Then G = GoZ, where Z is the group associated to z. Moreover, Go is also transitive on the projective space, and it is either compact or non-compact. In the first ease, there is just one control set, namely, the whole RP a - l . In fact, the group Z is a direct product Z = Z1Z2 with Z1 compact and Z2 formed by multiples of the identity of R d (see [2, 3] for details). The elements of Z2 act as identity on the projective space so that the action of G factors through G/Z2 which is compact if Go is compact. Since S/Z2 is a semigroup with interior points in G/Z2 it coincides with the group hence S is transitive on RPd-1 and there is just one control set. Therefore, it is enough to consider the case where go is a non-compact semisimple Lie algebra. As mentioned before, the linear groups transitive on RP a-1 were classified by Boothby [2] (see also [3]). Table 1 has been reproduced from [3]. It contains the noncompact semisimple Lie algebras of matrices whose corresponding Lie groups are transitive on RP d-1. The column go indicates these algebras and d stands for the dimension of the underlying linear spaces. In the column z0, it is shown the centralizer of g0 in the full algebra of matrices so that the center z o f g is an abelian subalgebra of z0 if the semisimple component of g is go. Finally, in the last column appear the estimates of the number of control sets, in terms of d, the dimension of the linear spaces. The proof of these estimates is the main result of this paper. In the analysis of the control sets on the projective spaces further details about these Lie algebras will be presented.
CJ.B. Barros, L.A.B. San Martin~Systems & Control Letters 29 (1996) 21-26 Table 1 go
d
z0
No. of control sets
$1(n, R) sl(n, C)
n 2n
R C
d
sl(n, H) sp(n, R)
4n 2n
H R
sp(n, C) sl(n, H ) ~ su(2)
4n 4n
C R
d/2 d/4 d d/2 d/4
Our method for counting the control sets on the projective space is by comparing them with the control sets in some boundary of the semisimple component Go of G. In some cases (sl(n, R) and sp(n, R)), the projective space itself is a boundary of Go. In other cases, however, the projective space fibers over a boundary of Go. For these cases, the following lemma will be necessary which can be stated in a more general setting.
Lemma 2.1. Let G be a Lie group andL1 C L2 closed subgroups of G such that G/L1 is compact and Ll is normal in L2. Denote by It the equivariant fibration It: G/L1 -"} G/L2.
Let S be a semigroup with non-empty interior of G. Then It(D) is contained in an effective control set on G/L2 if D is an effective control set on G/L1. Reciprocally, assuming further that L2/LI is connected then n - l ( E ) is an effective control set on GILl if E is an effective control set on G/L2.
Proof, Let D C G/L1 be an effective control set
23
This is a semigroup with non-empty interior in L2 so that S//L 1 is a semigroup with interior points in L2/L1. The assumption that LE/L1 is connected implies then that StILl : L 2 / L 1 (see e.g. [7]). Now, the action of S' on the fiber n - l { x } factors through L1 so that it is given by the action of S'/LI. Therefore, the fact that S'/Ll =L2/L1 implies that S'y = It-l{x} for any y E It-l{x}, that is, S is controllable inside the fibers over the elements of E0. On the other hand, the fibers over the elements of E are approximately attained by elements of S and since there is controllability inside the fibers over Eo, it follows that y can be approximately attained from y' for any pair y, y' E n - l ( E ) . This implies that l t - l ( E ) is contained in a control set on G/L1 which is effective because of the controllability on a fiber It-l{x}, x E E0. []
Coronary 2.2. With the notations as in Lemma 2.1, we have that there are as many effective S-control sets in G/L1 than in G/L2 in case L2/L1 is connected. We recall now some facts about the control sets on the boundaries of the semisimple Lie groups. We refer to [8] for details. Let go be a non-compact semisimple Lie algebra and go = k @ s a Caftan decomposition with k the compactly embedded subalgebra. Let a be a maximal abelian subalgebra contained in s and denote b y / 7 the set of roots of the pair (g, a). Take a simple system of roots Z C / 7 and denote b y / / + the associated set of positive roots. A minimal parabolic subalgebra of go is given by
and pick x , y E ~(D). Then there are x ' , y ' E D with It(x') = x and I t ( J ) = y and a sequence 9, E S such that gnx' ~ y' as n ~ cx~. By the equivariance of It, we have that 9nx ~ y which shows that It(D) satisfies the first property in the definition of a control set. Hence It(D) is contained in a control set, say E, which is effective, as follows, again by the equivariance of It. Assume now that L2/L1 is connected, let E C G/L2 be an effective control set and take x E E0. Since a homogeneous space is the quotient of the group by the isotropy subgroup at any point, it can be assumed, without loss of generality, that L2 is the isotropy at x, that is
p=m@a@n,
L2 = {g E G : gx = x}.
Po = no • p.
Assuming this, we have that the group LE/L1 identifies with the fiber n-1 {x}, which is compact because G/L1 is supposed to be compact. Put S' = S M L2.
Let Go be a Lie group whose Lie algebra is go and denote by K its connected subgroup whose Lie algebra is k. The parabolic subgroups of Go are the nor-
where m is the centralizer of a in k and n is the direct sum of the root spaces corresponding to the positive roots
n= Zg~. :~6H+
The other parabolic subalgebras are conjugate to the standard ones which are built as follows: given a subset @ C Z let no be the subalgebra generated by the root spaces g_~, • E @ then the subalgebra associated to @ is
24
C.J.B. Barros, L.A.B. San Martin~Systems & Control Letters 29 (1996) 21-26
malizers of the parabolic subalgebras and the Furstenberg boundaries (also known as flag manifolds) of Go are the compact coset spaces Go/P with P a parabolic subgroup. We use the notation P e for the parabolic subgroup which is the normalizer of the standard subalgebra Pc. The empty set gives rise to the minimal parabolic subgroup P0 and the maximal flag manifold The number of control sets on the boundaries is given in terms of the Weyl group W of go which is given either as the group generated by the reflections with respect to the roots or as M'/M where M ' and M are, respectively, the normalizer and the centralizer of a inK. The Weyl group provides the number of control sets as follows: let S C Go be a semigroup with nonempty interior. In [8] it was associated to each w E W an effective control set Dw for S in the maximal flag manifold in such a way that every effective control set is Dw for some w E W. There is only one invariant control set which is Dl and the subset
W(S) = {w E W : Dw = DI } is a subgroup of W. This subgroup is a kind of kernel of the mapping w ~ Dw so that in the maximal boundary there are as many control sets as the order of W(S)\W. On the other boundaries the control sets are obtained by projections from the maximal one. "When performing a projection onto a boundary G/Pe some of the control sets collapse, so that the number of effective control sets in such a boundary is reduced to the order of the set of double cosets W(S)\W/We where We is the subgroup of W generated by the reflections with respect to the simple roots in O. This fact shows immediately that the number of control sets for S on G/Pe is at most [WI/IW~g]. From this upper bound we shall get the desired upper bound of control sets for subsemigroups of the transitive groups on Rp a-~.
3. The upper bounds
In order to count the control sets on the projective spaces it is enough to consider the case where the transitive group is semisimple. In fact, a transitive group G is of the form G = GoZ with Go semisimple and Z = exp z which in turn decomposes as Z = Z1Z2 with Zl compact and Z2 composed of multiples of the identity. Since Z2 is the identity on the projective space it can be factored out so it is possible to assume,
without loss of generality, that Z is compact. Now, let H be the isotropy of the G-action on ~pd-~ and take a semigroup S C G with int S ~ 0. Then Corollary 2.2 ensures that the number of S-control sets on RP a-1 = G/H is the same as the number of control sets on G/HZ. This homogeneous space is isomorphic to the quotient, say N, of the semisimple group Go~GoAZ G/Z by its closed subgroup H N Go~Go n Z so that the number of S-control sets on the projective space coincides with the number of control sets for the semigroup S/Z C Go~GoNZ on N. However, N is isomorphic to Go/(Go n H)(Go N Z) so that by considering the fibration 7r : Go~Go N H --+ N we get, again by Corollary 2.2, that the number of S-control sets on the projective space is bigger than the number of zt~-l(S/Z)-control sets on Go~Go n H where 7l: 1 is the canonical homomorphism ~1 : Go ---* Go~Go N Z. We have that n~-I(s/Z) is a semigroup with nonempty interior in Go and that the homogeneous space Go~Go N H is RP d-I because Go is also transitive. Therefore, the estimate given for semigroups in semisimple Lie groups is also an estimate for the general case. We consider now the different semisimple transitive groups. As mentioned before, the projective space is a boundary for the groups associated to sl(n, R) and s0(n, R), and for the other groups there exists a boundary which is related to the projective space as in Lemma 2.1. We consider first the cases where RP d- l is a boundary for the group. 1. The algebra is sl(d, ~ ) and the group Sl(d, R) with the canonical representation on R d. As the abelian subalgebra a one can take the algebra of diagonal matrices and a simple system of roots is L" = {2:. - 2j+l: 1 <<.j~d - 1}, where )-i is the linear functional on a which associates to a diagonal matrix its ith entry. The positive roots are the functionals 2j - 2k, j < k. The root space corresponding to 2i - 2j is spanned by Ejk, which is the matrix whose entries are zero except for the jkth one which is 1. Hence n is the algebra of upper triangular matrices with zeros on the main diagonal. The Weyl group W is the permutation group of { 1. . . . . d} and it acts on a by permuting the diagonal entries of its elements. Now •pd-1 = Sl(d, R)/Po where P e is the subgroup of matrices of the form
C.J.B. Barros, L.A.B. San Martin~Systems & Control Letters 29 (1996) 21-26
with R a (d - 1) x (d - 1) matrix and is the parabolic subgroup associated to the subset O c Z given by = {/~j -- ~j+l: 2<~j<~d - 1}
so that the subgroup We is the permutation group of {2 ..... d}. Hence the order of W/We is d!/(d - 1)! = d and this is the maximum number of control sets on RP a-I o f a semigroup S c Sl(d, R) with intS # 0. 2. The algebra is sp(n, R) and the group Sp(n, R) which exists in R e, d = 2n. This algebra is given by real symplectic matrices (A
B _At)
,
where A,B and C are n x n matrices with B and C symmetric. A Cartan decomposition is given by the skew-symmetric (k) and symmetric matrices (s) in sp(n, R) and a can be taken to be the subalgebra of matrices of the form A (0
0 -A)
25
which shows easily that the isotropy H of the transitive action of Sl(n, C) on the real projective space RP d-1 is composed of matrices of the form ( a bR) 0 with a E R and R a ( n - 1) x ( n - 1) complex matrix. On the other hand, the complex projective space CP n-1 is a boundary of Sl(n, C) (see [5]) and the quantity of control sets on it is given as follows: taking the Cartan decomposition of si(n, C) = k G s with k = su(n) = { X E sl(n, C): X + X* = O}
and s = {X E s I ( n , C ) : X - X * = O} one can choose the abelian subalgebra a to be the algebra of real n x n diagonal matrices in sl(n, C). With this choice a simple system of roots is Z = {2j - 2 j + l : l ~ < j ~ < n - 1}
with A diagonal n x n. The roots are 2i - 2s, i # j and 2i + 2j (where 2i is the ith coordinate of A) and a simple system of roots is z~ = {'~1 -- '~2 . . . . . "~n-I -- "~n,22n} •
The Weyl group has 2nn! elements and its action on a is given by a permutation of the entries of A followed by multiplication by (el ..... ~n), ei = + 1. Now, RP a-I is a flag manifold of Sp(n, ~) which is associated to the subset @ = {22 -- 2 3 , . . . , 2 n - 1 - - 2 n , 2 2 n }
so that the subgroup We is the subgroup which keeps fixed the first entry of A and has 2n-1 ( n - 1)! elements. Hence the upper bound on the number of control sets is 2nn! -2n=d. 2 n - l ( n - 1)! For the other transitive groups the projective space is not a boundary. For them the number of control sets is computed with the aid of Corollary 2.2. 1. The algebra is sl(n, C) and the group is SI(n,C) with the canonical representation on C" = R a, d =2n. A complex n x n matrix 9 = A + iB induces a linear map on R a given by the matrix
:)
with 2j having the same meaning as above. The Weyl group is the permutation group of {1 ..... n} and the boundary CP n-1 is associated to the parabolic subgroup P e with ~) = {~j -- 4+1" 2<~j<~n - 1}
so that We is the permutation group of {2 ..... n}. Also, Pe is composed of matrices of the form (;
W)Q
with z E C and Q a (n - 1) x (n - 1) complex matrix. Therefore, Corollary 2.2 applied to the fibration G/H--~ G/Po, with H normal in Pe (and P e / H ,~ S 1), shows that the number of control sets of a semigroup S c Sl(n, C) with int S # 0 is at most n!/(n - 1)! = n = d/2. 2. The algebra is sl(n, H ) and the group is Sl(n, H) which represents in H n = ~d, d = 4n. Here H denotes the algebra of the quaternions. Analogous to the complex case, a quaternionic n × n matrix determines a real d x d matrix through the representation
A+iB+jC+kD
(i" c A
D
-C
-D
C
A
-B
B
A
26
C.J.B. Barros, L.A.B. San Martin/Systems & Control Letters 29 (1996) 21-26
which shows that the isotropy H o f the action on ~ p d - i is composed o f matrices of the form
the group SI(n,C) the real projective space ~ p d - i is a homogeneous space S p ( n , C ) / H with H normal in P o so that there are as many control sets in ~ p d - i as in CP 2n-1 . The desired upper bound is thus 2n = d/2.
with b 6 R and R a (n - 1 ) × (n - 1 ) quaternionic matrix. On the other hand, the group P e whose matrices are o f the form
with z E H and Q a (n - 1) × (n - 1) quaternionic matrix is a parabolic subgroup of Sl(n, H). It is given by choosing k to be the subalgebra o f skew-hermitian matrices, s the subspace o f hermitian matrices, a the subalgebra o f real diagonal matrices, S = {2j --
~+1:1 <~j<,n -
!}
and O = { # - 2j+1: 2 < ~ j < ~ n - 1}. Again W is the permutation group o f { 1. . . . . n} and We the permutation group o f {2 . . . . . n}. Therefore, Corollary 2.2 applied to the fibration G / H --* G / P o with H normal in P o (and P~9/H ~ S 3) shows that there are at most n!/(n - 1 )! = n = d / 4 control sets on ~ p a - 1 = G / H . 3. The algebra is sp(n, C) and the group is Sp(n, C) which represents in C 2n = R a, d = 4n. The algebra is given by the complex symplectic matrices and as in the real case the Weyl group has 2"n! elements. The complex projective space CP 2~- ~ is a flag o f Sp(n, C) which as above is associated to the subset O such that Wo has 2 ~-1 (n - 1 )! elements so that the upper bound for the control sets in CP 2"-~ is 2n. Now, similar to
4. Finally, the algebra sl(n, H) @ su(2) represents in ~_~n= ~ d , d = 4n by left multiplication of quaternionic n × n matrices and right multiplication o f purely imaginary quaternions whose Lie algebra is isomorphic to su(2). The parabolic subalgebras o f this algebra are of the form p = q G su(2) because su(2) is compact so that it is absorbed in the centralizer m o f a. Therefore the situation here is similar to the case sl(n, H) and the upper bound o f control sets on RP d-I is n = d/4.
References [1] F. Albertini and E.D. Sontag, Some connections between chaotic dynamical systems and control systems, in: Proc. European Control Conf., Vol. 1, Grenoble (1991) 158-163. [2] W.M. Boothby, A transitivity problem from control theory, J. Differential Equations 17 (1975) 296-307. [3] W.M. Boothby and E.N. Wilson, Determination of the transitivity of bilinear systems, SlAM Z Control Optim. 17 (1979) 212-221. [4] F. Colonius and W. Kliemarm, Linear control semigroups acting on projective spaces, J. Dyn. Differential Equations 5 (1993) 495-528. [5] W. Fulton and J. Harris, Representation Theory, Graduate Texts in Mathematics, Vol. 129 (Springer, Berlin, 1991). [6] V. Jurdjevic and H.J. Sussmarm, Control systems on Lie groups, J. Differential Equations 12 (1972) 313-329, [7] L. San Martin and P.E. Crouch, Controllability on principal fibre bundles with compact structure group, Systems Control Lett. 5 (1984) 35-40. [8] L.A.B. San Martin, and P.A. Tonelli, Semigroup actions on homogeneous spaces, Semigroup Forum 50 (1995) 59-88.
Systems & Control Letters 29 (1996) 21-26
On the number of control sets on projective spaces Carlos Jos6 Braga Barros a, Luiz A.B. San Martinb'*,l a Departamento de Matemtitica, Universidade Estadual de Marinod*, Cx. Postal 331, 87.020-900 Marinffd-Pr, Brazil b Instituto de Matemdttica, Universidade Estadual de Campinas, Cx. Postal 6065, 13.081-970 Campinas-SP, Brazil
Received 7 February 1995; revised 3 June 1996
Abstract The purpose of this paper is to provide an upper bound for the number of control sets for linear semigroups acting on a projective space RP d-:. These semigroups and control sets were studied by Colonius and Kliemarm (1993) who proved that there are at most d control sets. Here we apply the results of San Martin and Tonelli (1995) about control sets for semigroups in semisimple Lie groups and make a case by case analysis according to the transitive groups on RP d-: which were classified by Boothby and Wilson (1975, 1979) in order to improve that upper bound. It turns out that in some cases there are at most d/2 or d/4 control sets. Keywords: Control set; Projective space; Transitive group
1. Introduction This paper considers semigroups of invertible linear mappings and their actions on projective spaces. These semigroup actions arise naturally in the study of semilinear control systems evolving on R d as well as in the study of stability properties of non-linear systems evolving on differentiable manifolds. The issue here is about the control sets for the semigroups in the projective spaces RP d-1. These are defined as follows: let M be a differentiable manifold and S a semigroup of diffeomorphisms of M. A control set for S is a subset D C M which satisfies (1) D C cl(Sx) for all x E D, (2) intO ~ 0 and (3) D is maximal with these properties (where el and int means closure and interior, respectively). We refer the reader to [1,4, 8] for further details about the control sets and the role they play in the * Corresponding author. E-mail: [email protected]. I Research partially supported by the Brazilian Research Council-CNPq Proc. No. 301060/94-0.
study of the dynamical and geometrical properties of a control system. In the sequel we shall be interested in semigroups which satisfy the following analogue of the accessibility property of control systems: (A)
int(Sx) ~ 0
for all x E M.
For semigroups satisfying (A), it is possible to single out the class of control sets that contain points which are self-attainable and belong to the interior of their forward orbit. Explicitly, given a control set D put Do = {x E D: x E int(Sx) M int(S-lx)} . We shall refer to this subset as the set of controllability (or transitivity) of S inside D. In general, Do may be empty. Control sets for which the sets of controllability are not empty were named effective control sets in [8]. We mention that every control set is effective when the semigroup comes from a control system which satisfies the accessibility property (A). In the sequel we consider effective control sets only. Now, suppose that S is a subsemigroup of the general linear group Gl(d, R). Since the elements of S are
0167-6911/96/$12.00 Copyright (~) 1996 Elsevier Science B.V. All rights reserved SSDI 0167-6911(96)00047-3
22
C.J.B. Barros, L.A.B. San Martin~Systems & Control Letters 29 (1996) 21-26
invertible linear mappings, they map lines in ~d into lines and thus S acts in the projective space R P d - I as a semigroup of diffeomorphisms. The control sets for such a semigroup action were studied by Colonius and Kliemann [4]. One of the results of that paper states that S has at most d effective control sets in RP d-1 if it satisfies the accessibility property. This result is obtained in [4] through an analysis of the spectral decompositions of the elements of S. The purpose of this paper is to improve the estimate of the number of control sets given in [4]. We assume also accessibility of the action of S on the projective space but take an indirect approach for counting the number of control sets. This approach is based on the results of [8] about control sets for the actions of semigroups with non-empty interior in semisimpie Lie groups on the Furstenberg boundaries of the group. For these actions the number of effective control sets are computed exactly in terms of the Weyl group W of the semisimple group, and of its subgroup W(S) which depends on S and We which depends on the specific boundary under consideration (see [8, Corollary 5.2]). In particular, the number of control sets is [W/W~91if W(S) reduces to the identity and this is the maximum number of control sets in the boundary associated to We for general semigroups with non-empty interior. On the other hand, a Lie group G transitive on the projective space is reductive and hence nearly semisimple. A list of these groups was provided in [2, 3]. With the aid of this list, we make a case by case analysis according to the transitive groups. In all the cases, the projective space can be seen as a homogeneous space of G which fibers equivariantly over one of its boundaries. This fibration is in fact a principal fiber bundle with compact structure group, and this specificity implies that the quantity of control sets for a semigroup S C G in projective space coincides with the number of control sets for the same semigroup in some of the boundaries of G. This way, we get the maximum number of effective control sets for S in ~ p a - l if S is a semigroup with non-empty interior of G. It turns out that the estimate is d, as in [4] for some of the transitive groups but goes down to d/2 or d/4 for some other groups in the list.
2. Preliminaries
As mentioned above, our method consists of a case by case analysis according to the transitive groups on
the projective spaces. So we consider only the situation where G is a connected Lie group transitive on •pa-1 and S c G a semigroup with non-empty interior. This set up covers the case where the semigroup comes from a (continuous time) control system. As is well known, a right invariant control system in a Lie group generates a semigroup (control semigroup) which has non-empty interior in a Lie subgroup (see [6]). Therefore, if we consider control systems on R a which are linear in the state variables (as in [4]) we have that the system satisfies (A) on the projective space if and only if the corresponding control semigroup is a semigroup with non-empty interior in a connected Lie group transitive on •pa-1. Now, let g be the Lie algebra of G. It is a reductive Lie algebra and decomposes as
g=go•z with go semisimple and z the center o f g . Denote by Go the connected group associated to go. Then G = GoZ, where Z is the group associated to z. Moreover, Go is also transitive on the projective space, and it is either compact or non-compact. In the first ease, there is just one control set, namely, the whole RP a - l . In fact, the group Z is a direct product Z = Z1Z2 with Z1 compact and Z2 formed by multiples of the identity of R d (see [2, 3] for details). The elements of Z2 act as identity on the projective space so that the action of G factors through G/Z2 which is compact if Go is compact. Since S/Z2 is a semigroup with interior points in G/Z2 it coincides with the group hence S is transitive on RPd-1 and there is just one control set. Therefore, it is enough to consider the case where go is a non-compact semisimple Lie algebra. As mentioned before, the linear groups transitive on RP a-1 were classified by Boothby [2] (see also [3]). Table 1 has been reproduced from [3]. It contains the noncompact semisimple Lie algebras of matrices whose corresponding Lie groups are transitive on RP d-1. The column go indicates these algebras and d stands for the dimension of the underlying linear spaces. In the column z0, it is shown the centralizer of g0 in the full algebra of matrices so that the center z o f g is an abelian subalgebra of z0 if the semisimple component of g is go. Finally, in the last column appear the estimates of the number of control sets, in terms of d, the dimension of the linear spaces. The proof of these estimates is the main result of this paper. In the analysis of the control sets on the projective spaces further details about these Lie algebras will be presented.
CJ.B. Barros, L.A.B. San Martin~Systems & Control Letters 29 (1996) 21-26 Table 1 go
d
z0
No. of control sets
$1(n, R) sl(n, C)
n 2n
R C
d
sl(n, H) sp(n, R)
4n 2n
H R
sp(n, C) sl(n, H ) ~ su(2)
4n 4n
C R
d/2 d/4 d d/2 d/4
Our method for counting the control sets on the projective space is by comparing them with the control sets in some boundary of the semisimple component Go of G. In some cases (sl(n, R) and sp(n, R)), the projective space itself is a boundary of Go. In other cases, however, the projective space fibers over a boundary of Go. For these cases, the following lemma will be necessary which can be stated in a more general setting.
Lemma 2.1. Let G be a Lie group andL1 C L2 closed subgroups of G such that G/L1 is compact and Ll is normal in L2. Denote by It the equivariant fibration It: G/L1 -"} G/L2.
Let S be a semigroup with non-empty interior of G. Then It(D) is contained in an effective control set on G/L2 if D is an effective control set on G/L1. Reciprocally, assuming further that L2/LI is connected then n - l ( E ) is an effective control set on GILl if E is an effective control set on G/L2.
Proof, Let D C G/L1 be an effective control set
23
This is a semigroup with non-empty interior in L2 so that S//L 1 is a semigroup with interior points in L2/L1. The assumption that LE/L1 is connected implies then that StILl : L 2 / L 1 (see e.g. [7]). Now, the action of S' on the fiber n - l { x } factors through L1 so that it is given by the action of S'/LI. Therefore, the fact that S'/Ll =L2/L1 implies that S'y = It-l{x} for any y E It-l{x}, that is, S is controllable inside the fibers over the elements of E0. On the other hand, the fibers over the elements of E are approximately attained by elements of S and since there is controllability inside the fibers over Eo, it follows that y can be approximately attained from y' for any pair y, y' E n - l ( E ) . This implies that l t - l ( E ) is contained in a control set on G/L1 which is effective because of the controllability on a fiber It-l{x}, x E E0. []
Coronary 2.2. With the notations as in Lemma 2.1, we have that there are as many effective S-control sets in G/L1 than in G/L2 in case L2/L1 is connected. We recall now some facts about the control sets on the boundaries of the semisimple Lie groups. We refer to [8] for details. Let go be a non-compact semisimple Lie algebra and go = k @ s a Caftan decomposition with k the compactly embedded subalgebra. Let a be a maximal abelian subalgebra contained in s and denote b y / 7 the set of roots of the pair (g, a). Take a simple system of roots Z C / 7 and denote b y / / + the associated set of positive roots. A minimal parabolic subalgebra of go is given by
and pick x , y E ~(D). Then there are x ' , y ' E D with It(x') = x and I t ( J ) = y and a sequence 9, E S such that gnx' ~ y' as n ~ cx~. By the equivariance of It, we have that 9nx ~ y which shows that It(D) satisfies the first property in the definition of a control set. Hence It(D) is contained in a control set, say E, which is effective, as follows, again by the equivariance of It. Assume now that L2/L1 is connected, let E C G/L2 be an effective control set and take x E E0. Since a homogeneous space is the quotient of the group by the isotropy subgroup at any point, it can be assumed, without loss of generality, that L2 is the isotropy at x, that is
p=m@a@n,
L2 = {g E G : gx = x}.
Po = no • p.
Assuming this, we have that the group LE/L1 identifies with the fiber n-1 {x}, which is compact because G/L1 is supposed to be compact. Put S' = S M L2.
Let Go be a Lie group whose Lie algebra is go and denote by K its connected subgroup whose Lie algebra is k. The parabolic subgroups of Go are the nor-
where m is the centralizer of a in k and n is the direct sum of the root spaces corresponding to the positive roots
n= Zg~. :~6H+
The other parabolic subalgebras are conjugate to the standard ones which are built as follows: given a subset @ C Z let no be the subalgebra generated by the root spaces g_~, • E @ then the subalgebra associated to @ is
24
C.J.B. Barros, L.A.B. San Martin~Systems & Control Letters 29 (1996) 21-26
malizers of the parabolic subalgebras and the Furstenberg boundaries (also known as flag manifolds) of Go are the compact coset spaces Go/P with P a parabolic subgroup. We use the notation P e for the parabolic subgroup which is the normalizer of the standard subalgebra Pc. The empty set gives rise to the minimal parabolic subgroup P0 and the maximal flag manifold The number of control sets on the boundaries is given in terms of the Weyl group W of go which is given either as the group generated by the reflections with respect to the roots or as M'/M where M ' and M are, respectively, the normalizer and the centralizer of a inK. The Weyl group provides the number of control sets as follows: let S C Go be a semigroup with nonempty interior. In [8] it was associated to each w E W an effective control set Dw for S in the maximal flag manifold in such a way that every effective control set is Dw for some w E W. There is only one invariant control set which is Dl and the subset
W(S) = {w E W : Dw = DI } is a subgroup of W. This subgroup is a kind of kernel of the mapping w ~ Dw so that in the maximal boundary there are as many control sets as the order of W(S)\W. On the other boundaries the control sets are obtained by projections from the maximal one. "When performing a projection onto a boundary G/Pe some of the control sets collapse, so that the number of effective control sets in such a boundary is reduced to the order of the set of double cosets W(S)\W/We where We is the subgroup of W generated by the reflections with respect to the simple roots in O. This fact shows immediately that the number of control sets for S on G/Pe is at most [WI/IW~g]. From this upper bound we shall get the desired upper bound of control sets for subsemigroups of the transitive groups on Rp a-~.
3. The upper bounds
In order to count the control sets on the projective spaces it is enough to consider the case where the transitive group is semisimple. In fact, a transitive group G is of the form G = GoZ with Go semisimple and Z = exp z which in turn decomposes as Z = Z1Z2 with Zl compact and Z2 composed of multiples of the identity. Since Z2 is the identity on the projective space it can be factored out so it is possible to assume,
without loss of generality, that Z is compact. Now, let H be the isotropy of the G-action on ~pd-~ and take a semigroup S C G with int S ~ 0. Then Corollary 2.2 ensures that the number of S-control sets on RP a-1 = G/H is the same as the number of control sets on G/HZ. This homogeneous space is isomorphic to the quotient, say N, of the semisimple group Go~GoAZ G/Z by its closed subgroup H N Go~Go n Z so that the number of S-control sets on the projective space coincides with the number of control sets for the semigroup S/Z C Go~GoNZ on N. However, N is isomorphic to Go/(Go n H)(Go N Z) so that by considering the fibration 7r : Go~Go N H --+ N we get, again by Corollary 2.2, that the number of S-control sets on the projective space is bigger than the number of zt~-l(S/Z)-control sets on Go~Go n H where 7l: 1 is the canonical homomorphism ~1 : Go ---* Go~Go N Z. We have that n~-I(s/Z) is a semigroup with nonempty interior in Go and that the homogeneous space Go~Go N H is RP d-I because Go is also transitive. Therefore, the estimate given for semigroups in semisimple Lie groups is also an estimate for the general case. We consider now the different semisimple transitive groups. As mentioned before, the projective space is a boundary for the groups associated to sl(n, R) and s0(n, R), and for the other groups there exists a boundary which is related to the projective space as in Lemma 2.1. We consider first the cases where RP d- l is a boundary for the group. 1. The algebra is sl(d, ~ ) and the group Sl(d, R) with the canonical representation on R d. As the abelian subalgebra a one can take the algebra of diagonal matrices and a simple system of roots is L" = {2:. - 2j+l: 1 <<.j~d - 1}, where )-i is the linear functional on a which associates to a diagonal matrix its ith entry. The positive roots are the functionals 2j - 2k, j < k. The root space corresponding to 2i - 2j is spanned by Ejk, which is the matrix whose entries are zero except for the jkth one which is 1. Hence n is the algebra of upper triangular matrices with zeros on the main diagonal. The Weyl group W is the permutation group of { 1. . . . . d} and it acts on a by permuting the diagonal entries of its elements. Now •pd-1 = Sl(d, R)/Po where P e is the subgroup of matrices of the form
C.J.B. Barros, L.A.B. San Martin~Systems & Control Letters 29 (1996) 21-26
with R a (d - 1) x (d - 1) matrix and is the parabolic subgroup associated to the subset O c Z given by = {/~j -- ~j+l: 2<~j<~d - 1}
so that the subgroup We is the permutation group of {2 ..... d}. Hence the order of W/We is d!/(d - 1)! = d and this is the maximum number of control sets on RP a-I o f a semigroup S c Sl(d, R) with intS # 0. 2. The algebra is sp(n, R) and the group Sp(n, R) which exists in R e, d = 2n. This algebra is given by real symplectic matrices (A
B _At)
,
where A,B and C are n x n matrices with B and C symmetric. A Cartan decomposition is given by the skew-symmetric (k) and symmetric matrices (s) in sp(n, R) and a can be taken to be the subalgebra of matrices of the form A (0
0 -A)
25
which shows easily that the isotropy H of the transitive action of Sl(n, C) on the real projective space RP d-1 is composed of matrices of the form ( a bR) 0 with a E R and R a ( n - 1) x ( n - 1) complex matrix. On the other hand, the complex projective space CP n-1 is a boundary of Sl(n, C) (see [5]) and the quantity of control sets on it is given as follows: taking the Cartan decomposition of si(n, C) = k G s with k = su(n) = { X E sl(n, C): X + X* = O}
and s = {X E s I ( n , C ) : X - X * = O} one can choose the abelian subalgebra a to be the algebra of real n x n diagonal matrices in sl(n, C). With this choice a simple system of roots is Z = {2j - 2 j + l : l ~ < j ~ < n - 1}
with A diagonal n x n. The roots are 2i - 2s, i # j and 2i + 2j (where 2i is the ith coordinate of A) and a simple system of roots is z~ = {'~1 -- '~2 . . . . . "~n-I -- "~n,22n} •
The Weyl group has 2nn! elements and its action on a is given by a permutation of the entries of A followed by multiplication by (el ..... ~n), ei = + 1. Now, RP a-I is a flag manifold of Sp(n, ~) which is associated to the subset @ = {22 -- 2 3 , . . . , 2 n - 1 - - 2 n , 2 2 n }
so that the subgroup We is the subgroup which keeps fixed the first entry of A and has 2n-1 ( n - 1)! elements. Hence the upper bound on the number of control sets is 2nn! -2n=d. 2 n - l ( n - 1)! For the other transitive groups the projective space is not a boundary. For them the number of control sets is computed with the aid of Corollary 2.2. 1. The algebra is sl(n, C) and the group is SI(n,C) with the canonical representation on C" = R a, d =2n. A complex n x n matrix 9 = A + iB induces a linear map on R a given by the matrix
:)
with 2j having the same meaning as above. The Weyl group is the permutation group of {1 ..... n} and the boundary CP n-1 is associated to the parabolic subgroup P e with ~) = {~j -- 4+1" 2<~j<~n - 1}
so that We is the permutation group of {2 ..... n}. Also, Pe is composed of matrices of the form (;
W)Q
with z E C and Q a (n - 1) x (n - 1) complex matrix. Therefore, Corollary 2.2 applied to the fibration G/H--~ G/Po, with H normal in Pe (and P e / H ,~ S 1), shows that the number of control sets of a semigroup S c Sl(n, C) with int S # 0 is at most n!/(n - 1)! = n = d/2. 2. The algebra is sl(n, H ) and the group is Sl(n, H) which represents in H n = ~d, d = 4n. Here H denotes the algebra of the quaternions. Analogous to the complex case, a quaternionic n × n matrix determines a real d x d matrix through the representation
A+iB+jC+kD
(i" c A
D
-C
-D
C
A
-B
B
A
26
C.J.B. Barros, L.A.B. San Martin/Systems & Control Letters 29 (1996) 21-26
which shows that the isotropy H o f the action on ~ p d - i is composed o f matrices of the form
the group SI(n,C) the real projective space ~ p d - i is a homogeneous space S p ( n , C ) / H with H normal in P o so that there are as many control sets in ~ p d - i as in CP 2n-1 . The desired upper bound is thus 2n = d/2.
with b 6 R and R a (n - 1 ) × (n - 1 ) quaternionic matrix. On the other hand, the group P e whose matrices are o f the form
with z E H and Q a (n - 1) × (n - 1) quaternionic matrix is a parabolic subgroup of Sl(n, H). It is given by choosing k to be the subalgebra o f skew-hermitian matrices, s the subspace o f hermitian matrices, a the subalgebra o f real diagonal matrices, S = {2j --
~+1:1 <~j<,n -
!}
and O = { # - 2j+1: 2 < ~ j < ~ n - 1}. Again W is the permutation group o f { 1. . . . . n} and We the permutation group o f {2 . . . . . n}. Therefore, Corollary 2.2 applied to the fibration G / H --* G / P o with H normal in P o (and P~9/H ~ S 3) shows that there are at most n!/(n - 1 )! = n = d / 4 control sets on ~ p a - 1 = G / H . 3. The algebra is sp(n, C) and the group is Sp(n, C) which represents in C 2n = R a, d = 4n. The algebra is given by the complex symplectic matrices and as in the real case the Weyl group has 2"n! elements. The complex projective space CP 2~- ~ is a flag o f Sp(n, C) which as above is associated to the subset O such that Wo has 2 ~-1 (n - 1 )! elements so that the upper bound for the control sets in CP 2"-~ is 2n. Now, similar to
4. Finally, the algebra sl(n, H) @ su(2) represents in ~_~n= ~ d , d = 4n by left multiplication of quaternionic n × n matrices and right multiplication o f purely imaginary quaternions whose Lie algebra is isomorphic to su(2). The parabolic subalgebras o f this algebra are of the form p = q G su(2) because su(2) is compact so that it is absorbed in the centralizer m o f a. Therefore the situation here is similar to the case sl(n, H) and the upper bound o f control sets on RP d-I is n = d/4.
References [1] F. Albertini and E.D. Sontag, Some connections between chaotic dynamical systems and control systems, in: Proc. European Control Conf., Vol. 1, Grenoble (1991) 158-163. [2] W.M. Boothby, A transitivity problem from control theory, J. Differential Equations 17 (1975) 296-307. [3] W.M. Boothby and E.N. Wilson, Determination of the transitivity of bilinear systems, SlAM Z Control Optim. 17 (1979) 212-221. [4] F. Colonius and W. Kliemarm, Linear control semigroups acting on projective spaces, J. Dyn. Differential Equations 5 (1993) 495-528. [5] W. Fulton and J. Harris, Representation Theory, Graduate Texts in Mathematics, Vol. 129 (Springer, Berlin, 1991). [6] V. Jurdjevic and H.J. Sussmarm, Control systems on Lie groups, J. Differential Equations 12 (1972) 313-329, [7] L. San Martin and P.E. Crouch, Controllability on principal fibre bundles with compact structure group, Systems Control Lett. 5 (1984) 35-40. [8] L.A.B. San Martin, and P.A. Tonelli, Semigroup actions on homogeneous spaces, Semigroup Forum 50 (1995) 59-88.