2.2 Marco teórico conceptual
2.2.2 Factores sociales
Our goal is to obtain necessary and sufficient conditions for generic accessibility and controllability of bilinear systems Σ(D,C1,...,Cm). Our first result, Theorem 5, is
in the spirit of the well-known generic controllability result for linear systems, see e.g. [21]. The proof is based on the same standard technique as in the linear case. It uses the fact that the zero set of a real analytic function is closed and nowhere dense. Our second result, Theorem 7, heavily exploits the structure theory of real semisimple Lie algebras. It extends a well-known result by Jurdjevic and Kupka [12] on generic accessibility of bilinear systems on semisimple Lie groups.
3.1 General case
Theorem 5. Let D and C1,...,Cmbe real analytic connected submanifolds of g⊂
gln(R). Then Σ(D;C1,...,Cm) is generically accessible if and only if there exists at
Festschrift in Honor of Uwe Helmke G. Dirr et al. Proof. According to Proposition 1 it suffices to show that the set
P∶= {(A0,A1,...,An) ∈ D×C1×⋯×Cm∣ ⟨A0,A1,...,Am⟩L= g} (9)
contains an open and dense subset of D×C1×⋯×Cm. To this end, let N∶= dimg and
let L′N(A0,A1,...,Am) be defined as in (6). Then Lemma 4 guarantees that
VN∶= N ∑ k=1span L ′ k(A0,A1,...,Am) (10)
coincides with⟨A0,A1,...,Am⟩L and hence ⟨A0,A1,...,Am⟩L= g is equivalent to
dimVN= N. Now, let W denote the matrix that collects all Lie words of type (7) up
to length N over the alphabet A0,A1,...,Am(as column vectors in some coordinate
representation). Since the condition dimVN= N can be expressed as a rank condition
on the matrix W , which is clearly a polynomial and thus real analytic condition, we
can conclude that the set P is open dense in D×C1×⋯×Cmif there is at least one
A0,A1,...,Amsuch that the matrix W has full rank. This, however, is guaranteed by
the assumption that there exists at least one system such that⟨A0,A1,...,Am⟩L= g.
Since the condition on W is polynomial the complement of P has also Lebesgue
measure zero in D×C1×⋯×Cm.
Proposition 2 immediately leads to the following controllability result.
Corollary 6. If Σ(D;C1,...,Cm) is a family of driftless systems, i.e. D = {0}, or if G
is compact then the condition of Theorem5 is equivalent to generic controllability of Σ(D;C1,...,Cm).
3.2 Real semisimple case
To follow and adapt the ideas by Jurdjevic and Kupka [11, 12] we first collect some basic facts on strongly regular elements of real semisimple Lie-algebra. For more details on semisimple Lie algebras we recommend in addition [14].
Let g⊂ gln(R) be a real semisimple Lie algebra and gC∶= g ⊕ ig ⊂ gln(C) be its complexification. Consider the corresponding adjoint representations
ad∶ g → End(g) and adC∶ gC→ End(gC).
For A∈ g, define
Sp(A) ∶= {λ ∈ C∖{0} ∣ ker(adCA −λIn) ≠ {0}} (11)
and EC
λ(A) ∶= ker(ad
C
A−λIn) for λ ∈ Sp(A) as the λ-eigenspace of adCA. Then, an
element A∈ gCis called strongly regular if it satisfies the following conditions: • All nonzero eigenvalues of adCA are simple, i.e. the algebraic multiplicity of
each λ∈ Sp(A) is equal to one.
• The generalized eigenspace E0C(A) ∶= ⋃n≥1ker(adCA)ndoes not contain any
It is known that strong regularity is a generic property in gCand g as well. More precisely, the set of all strongly regular elements is open and dense in gCand its intersection with g is again open and dense in g. In both cases, the complement has Lebesgue measure zero. Furthermore, the following facts about strongly regular elements are well-known and can be found e.g. in [12] or [14].
(1) With respect to a strongly regular element A∈ g, the complex Lie algebra gC
decomposes as a direct sum
gC= E0C(A)⊕ ⊕
λ∈Sp(A)
EλC(A). (12)
(2) For every λ∈ Sp(A), the set [EλC(A),E−λC(A)] is a one-dimensional vector space contained in E0C(A). The sum of all [EλC(A),E−λC(A)], λ ∈ Sp(A), equals E0C(A), i.e.
∑ λ∈Sp(A) [EC λ (A),E C −λ(A)] = E0C(A). (13)
Note that the above sum is in general not a direct sum.
(3) For λ, µ∈ Sp(A)∪{0} one has
[EC λ (A),E C µ(A)] = ⎧⎪⎪ ⎨⎪⎪ ⎩ EC
λ+µ(A) for λ + µ ∈ Sp(A)∪{0},
{0} for λ+ µ /∈ Sp(A)∪{0}. (14)
(4) It turns out that E0C(A) = ker(adCA). Moreover, E0C(A) is a Cartan subalgebra of gC, i.e. a maximal abelian subalgebra of g whose ad-action on g is simul- taneously diagonalizable. For more details on Cartan subalgebras we refer to [14].
(5) With respect to a strongly regular element A∈ g, the real Lie algebra g decom- poses as a direct sum
g= E0(A)⊕ ⊕
λ∈Sp(A), Im(λ)≥0
Eλ(A), (15)
with E0(A) ∶= E0C(A)∩g and Eλ(A) ∶= (EλC(A)+E
C
¯
λ (A))∩g, where ¯λ denotes
the complex conjugate of λ . Note that Eλ(A) = Eλ¯(A). Thus, depending on
whether λ is real or not, Eλ(A) is the real counterpart either to the eigenspace
EC
λ(A) or to the pair of eigenspaces E
C
λ(A) and E
C
¯
λ(A). Therefore, any B ∈ g
has a unique decomposition
B= B0+ ∑
λ∈Sp(A), Im(λ)≥0
Bλ, (16)
Festschrift in Honor of Uwe Helmke G. Dirr et al.
Now, for any strongly regular element A∈ gCwe define
Γ(A) ∶= {B ∈ gC∣ Bλ≠ 0 for all λ ∈ Sp(A), Im(λ) ≥ 0}. (17)
Thus, we are prepared to state our main result on generic accessibility in the semi- simple case.
Theorem 7. Let D and C be real analytic connected submanifolds of a real semisim- ple Lie algebra g⊂ gln(R). If D contains a strongly regular element (in the above
sense) and if C∩Γ(A) ≠ ∅ then Σ(D;C) is generically accessible.
Before proceeding with the proof of Theorem 7, a few comments may be helpful.
Remark8.
(a) Concerning accessibility, the role of D and C in Theorem 7 is completely interchangeable.
(b) The condition C∩ Γ(A) ≠ ∅ can be refined by the results of Gauthier and
Bornard [9], Silva-Leite and Crouch [19, 20] or Jurdjevic and Kupka [12]. However, these improvements are more of interest for analysing the accessibil- ity and controllability of an individual system. For a simple genericity test the
above condition C∩Γ(A) ≠ ∅ is usually sufficient.
For the proof of Theorem 7 we need two auxiliary results.
Lemma 9. Let A∈ g be a strongly regular element. Then one has
E0(A) ⊆ ∑
λ∈Sp(A), Im(λ)≥0
[Eλ(A),E−λ(A)]. (18)
A proof of Lemma 9 which follows straightforwardly from property (2) can be found in [15].
Lemma 10. Let A be a strongly regular element in a real semi-simple Lie algebra g and let B∈ Γ(A) Then one has ⟨A,B⟩L= g.
Proof. The inclusion⟨A,B⟩L⊂ g is trivial. Conversely, the span of adAB,...,adkABis
an invariant subspace of adAfor k sufficiently large. Therefore, we have
span{adAB,...,adkAB} = ⊕
λ∈Sp(A), Im(λ)≥0
Eλ(A) ⊂ g, (19)
since all Eλ(A) are irreducible subspaces of adA and, by assumption, Bλ ≠ 0 for
all λ∈ Sp(A). By Lemma 9, summing [Eλ(A),E−λ(A)] for all λ ∈ Sp(A) with
Im(λ) ≥ 0, we eventually generate E0(A). Hence, we obtain
⟨A,B⟩L⊃ E0(A)⊕ ⊕
λ∈Sp(A), Im(λ)≥0
Eλ(A) = g, (20)
Proof of Theorem7. Due to Theorem 5 it suffices to show that there is at least one
pair A,B∈ D×C such that ⟨A,B⟩LA= g. This, however, is guaranteed by Lemma 10
and the assumption that D contains a strongly regular A such that C∩Γ(A) ≠ ∅.
Corollary 11. (a) If G is compact then the conditions of Theorem7 are sufficient
for generic controllability of Σ(D;C).
(b) If Σ(D;C1,C2) is a family of systems with two controls then the conditions of
Theorem7 with D,C replaced by C1,C2are sufficient for generic controllability
of Σ(D;C1,C2).
The proof follows immediately from Proposition 2.
Based on Theorem 7, we can improve a result by Jurdjevic and Kupka in the sense that generic accessibility can be guaranteed for semisimple Lie algebras once one of the two sets D or C is sufficiently large.
Corollary 12. Let C≠ {0} be a real analytic connected submanifold of a real semisim- ple Lie algebra g⊂ gln(R) and let D = g. Then Σ(D;C) is generically accessible.
Proof sketch. Choose any strongly regular element A∈ g and any non-trivial B ∈ C.
Now, consider the G-orbit of B, i.e. OG(B) ∶= {XBX−1∣ X ∈ G}, where G is the
unique connected matrix Lie group with Lie algebra g. If Γ(A)∩OG(B) ≠ ∅ we are
done, because X BX−1∈ Γ(A) implies B ∈ Γ(X−1AX) and thus we can apply Theorem
7 to the strongly regular element X−1AX. Therefore, we focus on the condition
Γ(A)∩OG(B) ≠ ∅. To this end, choose any λ ∈ Sp(A). All we have to show is that
the map X↦ (XBX−1)λ does not vanish identically. The fact that a holomorphic
function vanishes identically on Ck if and only if its restriction toRk vanishes
identically allows us to pass form G to GC, the unique matrix Lie group which
corresponds to gC. Now, we can exploit familiar properties of the Cartan subalgebra E0C(A), in particular, the transitive action of the associated Weyl group of the root spaces, cf. [14].
The following final result in this section turns out to be quite useful for reductive Lie algebras, i.e. if g= g0⊕z0decomposes into a direct sum of a semisimple Lie algebra
g0and a center z0. In many cases, Proposition 13 allows to extend Theorem 7 to the
reductive case.
Proposition 13. Let g= g0⊕z0⊂ gln(R) be a real reductive Lie algebra with semi- simple component g0and center z0. Moreover, let A= A0+Z and B = B0+Z′with
A0,B0∈ g0and Z,Z′∈ z0. Then,⟨A,B⟩L= g0⊕span{Z,Z′} if and only if ⟨A0,B0⟩L=
g0.
Proof. Clearly,⟨A,B⟩Lis a subset of⟨A0,B0⟩L⊕span{Z,Z′}. Therefore, ⟨A,B⟩L=
g0⊕ span{Z,Z′} implies ⟨A0,B0⟩LA= g0. Conversely, if ⟨A0,B0⟩L= g0, then by
the semisimplicity of g0it follows[g0,g0] = g0and thus A0and B0are contained
in the commutator algebra of ⟨A0,B0⟩L. Since the two commutator algebras of
⟨A0,B0⟩L and⟨A,B⟩L obviously coincide, one has A0,B0∈ ⟨A,B⟩L and hence the
Festschrift in Honor of Uwe Helmke G. Dirr et al.