Controllability of Right-Invariant Systems on Solvable Lie Groups.
We study controllability of right-invariant control systems groups. Necessary and sufficient controllability conditions for Lie groups not coinciding with their derived subgroup are obtained in terms of the root decomposition corresponding to the adjoint operator ad B. As an application, right-invariant systems on metabelian groups and matrix groups, and bilinear systems are considered.
Introduction
Control systems with a Lie group as a state space are studied in the
mathematical control theory since the early 1970-ies.
R.W. Brockett [1] considered applied problems leading to control systems
on matrix groups and their homogeneous spaces; e.g., a model of DC to DC
conversion and the rigid body control raise control problems on the group of
rotations of the three-space SO(3) and on the group SO(3) \Theta R 3 respectively.
The natural framework for such problems are matrix control systems of the
where x(t) and A, are n \Theta n matrices. There was established
the basic rank controllability test for homogeneous systems:
such systems are controllable iff the Lie algebra generated by the matrices
has the full dimension. This test was specified for the groups of
matrices with positive determinant GL+ (n; R), the group of matrices with
1991 Mathematics Subject Classification. 93B05, 17B20.
Key words and phrases. Controllability, right-invariant systems, bilinear systems, Lie
groups.
This work was partially supported by the Russian Foundation for Fundamental Re-
search, Projects No. 96-01-00805 and No. 97-1-1a/22.
The author is a recipient of the Russian State Scientific Stipend for 1997.1079-2724/97/1000-0531$09.50/0 c
Publishing Corporation
532 YU. L. SACHKOV
determinant one SL(n; R), the group of symplectic matrices Sp(n), and the
group of orthogonal matrices with determinant one SO(n). Some controllability
conditions for nonhomogeneous matrix systems were also obtained.
The first systematic mathematical study of control systems on Lie groups
was fulfilled by V. Jurdjevic and H. J. Sussmann [2]. They noticed that the
passage from the matrix system (1) to the more general right-invariant
system
G; u(t) 2 R; (2)
are right-invariant vectorfields on a Lie group G, "in
no essential way affects the nature of the problem." The basic properties
of the attainable set (the semi-group property, path-connectedness, relation
with the associated Lie subalgebras determined by the vectorfields A,
established. The rank controllability test was proved for
system (2) in the homogeneous case and in the case of a compact group G.
Sufficient controllability conditions for other cases were also given.
V. Jurdjevic and I. Kupka [4] introduced a systematic tool for studying
controllability on Lie groups. For the control system (2) presented in the
form of the polysystem
ae L (3)
(where L is the Lie algebra of the group G) they considered its Lie saturation
LS(\Gamma) - the largest system equivalent to \Gamma. Controllability of the system \Gamma
on G is equivalent to L, and a general technique for verification of
this equality was proposed. (This technique is outlined in Subsec. 4.2 and
used in Subsecs. 4.3, 4.4 below.) In [4] sufficient controllability conditions
for the single-input systems were obtained for simple and
semi-simple groups G with the use of this technique. They were given
in terms of the root decomposition of the algebra L corresponding to the
adjoint operator ad B.
In their preceding paper V. Jurdjevic and I. Kupka [3] presented the
enlargement technique for systems on matrix groups G ae GL(n; R) and
obtained sufficient controllability conditions for
GL+ (n; R).
These results for SL(n; R) and GL+ (n; R) were generalized by J.P. Gauthier
and G. Bornard [5].
B. Bonnard, V. Jurdjevic, I. Kupka, and G. Sallet [6] obtained a characterization
of controllability on a Lie group which is a semidirect product
of a vector space and a compact group which acts linearly on the vector
CONTROLLABILITY OF RIGHT-INVARIANT SYSTEMS 533
space. The case
n\Omega s SO(n) was applied to the study of Serret-Frenet
moving frames.
The results of [4] for simple and semi-simple Lie groups were generalized
in a series of papers by J.P. Gauthier, I. Kupka, and G. Sallet [7],
R.El Assoudi and J. P. Gauthier [9], [10], F. Silva Leite and P.E. Crouch [8]:
analogous controllability conditions were obtained for classical Lie groups
with the use of the Lie saturation technique and the known structure of real
simple and semi-simple Lie algebras.
In contrast to this "simple" progress, invariant systems on solvable groups
seem not to be studied in the geometric control theory at all until 1993.
Then a complete solution of the controllability problem for simply connected
nilpotent groups G was given by V. Ayala Bravo and L. San Martin [11].
Some results on controllability of (not right-invariant) systems on Lie groups
analogous to linear systems on R n were obtained by V. Ayala Bravo and
J. Tirao [12].
Several results on controllability of right-invariant systems were obtained
within the framework of the Lie semigroups theory [13], [14]: for nilpotent
groups by J. Hilgert, K. H. Hofmann, and J. D. Lawson [15], for reductive
groups by J. Hilgert [16]. For Lie groups G with cocompact radical,
J. D. Lawson [17] proved that controllability of a system \Gamma ae L follows from
nonexistence of a half-space in L bounded by a Lie subalgebra and containing
\Gamma; if G is additionally simply connected, this condition is also necessary
for controllability. This result generalizes controllability conditions for compact
groups [2], nilpotent groups [15], and for semidirect products of vector
groups and compact groups [6].
In [18] the author characterized controllability of hypersurface right-invariant
systems, i.e., of systems \Gamma of the form (3) with the codimension one Lie
subalgebra generated by the vectorfields This gave a necessary
controllability condition for simply connected groups - the hypersurface
principle, see its formulation for single-input systems \Gamma in Proposition 2
below. In its turn, the hypersurface principle was applied and there was
obtained a controllability test for simply connected solvable Lie groups G
with Lie algebra L satisfying the additional condition: for all X 2 L the
adjoint operator ad X has real spectrum.
The aim of this paper is to give convenient controllability conditions of
single-input systems \Gamma for a wide class of Lie groups including solvable ones;
more precisely, for Lie groups not coinciding with their derived subgroups.
The structure of this paper is as follows.
We state the problem and introduce the notation in Sec. 2.
In Sec. 3 we give the necessary controllability condition for simply connected
groups G not coinciding with their derived subgroup G (1) (Theorem 1
and Corollary 1). These propositions are proved in Subsec. 3.3 after the
preparatory work in Subsec. 3.2. The main tools are the rank controllability
534 YU. L. SACHKOV
condition (Proposition 1) and the hypersurface principle (Proposition 2).
Sec. 4 is devoted to sufficient controllability conditions for the groups
G 6= G (1) . We present the main sufficient results in Subsec. 4.1. Then
we recall the Lie saturation technique in Subsec. 4.2 and prove preliminary
lemmas in Subsec. 4.3. The main results (Theorem 2 and Corollaries 2,
are proved in Subsec. 4.4.
In Sec. 5 we consider several applications of our results. Controllability
conditions for metabelian groups are obtained in Subsec. 5.1. Then controllability
conditions for some subgroup of the group of motions of the Euclidean
space are studied in detail (Subsec. 5.2) and are applied to bilinear
systems (Subsec. 5.3). Finally, the clear small-dimensional version of this
theory for the group of motions of the two-dimensional plane is presented
in Subsec 5.4.
A preliminary version of the below results was stated in [19].
2. Problem statement and definitions
Let G be a connected Lie group, L its Lie algebra (i.e., the Lie algebra
of right-invariant vector fields on G), and A, B any elements of L. The
single-input affine right-invariant control system on G is a subset of L of
the form
The attainable set A of the system \Gamma is the subsemigroup of G generated
by the set of the one-parameter semigroups
The system \Gamma is called controllable if G.
To see the relation of these notions with the standard system-theoretical
ones, let us write the right-invariant vector fields A and B as A(x) and
G. Then the system \Gamma can be written in the customary form
G:
The attainable set A is then the set of points of the state space G reachable
from the identity element of the group G for any nonnegative time. The
system \Gamma is controllable iff any point of G can be reached along trajectories
of this system from the identity element of the group G. By right-invariance
of the fields A(x), B(x), the identity element in the previous sentence can
be changed by an arbitrary one.
Our aim is to characterize controllability of the system \Gamma in terms of the
Lie group G and the right-invariant vector fields A and B.
Now we introduce the notation we will use in the sequel.
For any subset l ae L we denote by Lie (l) the Lie subalgebra of L generated
by l. Closure of a set M is denoted by cl M . The signs \Phi and
CONTROLLABILITY OF RIGHT-INVARIANT SYSTEMS 535
direct sums of vector spaces; \Phi s
and\Omega s stand for semidirect
products of Lie algebras and Lie groups correspondingly.
We denote by Id the identity operator or the identity matrix of appropriate
dimension,
sin fft cos fft
ff r
for t, ff, r 2 R. The square matrix with all zero entries except one unit in
the ith raw and the jth column is denoted by E ij .
Now we introduce the notation connected with eigenvalues and eigenspaces
of the adjoint operator ad B in L:
ffl the derived subalgebra and the second derived subalgebra:
ffl the complexifications of L and L (i) , 2:
(i)\Omega C
(the tensor products over R),
ffl the adjoint representations and operators:
ffl spectra of the operators ad Bj L (i) , 2:
\Phi a 2 C j Ker(ad c Bj L (i)
c
ffl real and complex eigenvalues of the operators ad Bj L (i) , 2:
ffl complex eigenspaces of ad c Bj L (1)
c
c
ffl real eigenspaces of ad Bj
ffl complex root subspaces of ad c Bj L (i)
c
2:
c
a
ffl real root subspaces of ad Bj L (i)
2:
c (a)
536 YU. L. SACHKOV
ffl real components of L (i) , 2:
Note that the subalgebras L (1) and L (2) are ideals of L, so they are (ad B)-
invariant, and the restrictions ad Bj L (1) and ad Bj L (2) are well defined.
In the following lemma we collect several simple statements about decomposition
of the subalgebras L (1) and L (2) into sums of root spaces and
eigenspaces of the adjoint operator ad B.
Lemma 2.1.
(2)
r ,
(3) L (2) (a) ae L (1) (a) for any a 2 Sp (2) ,
r ae L (1)
r ,
Proof. Is obtained by the standard linear-algebraic arguments. In item (5)
Jacobi's identity is additionally used.
Consider the quotient operator
defined as follows:
Analogously for a 2 Sp (1) we define the quotient operator in the quotient
root space:
and its complexification:
ad
c (a)=L (2)
c (a)=L (2)
c (a);
(ad
c (a) 8X 2 L (1)
c (a):
Definition 1. Let a 2 Sp (1) . We denote by j (a) the geometric multiplicity
of the eigenvalue a of the operator -
ad c B(a) in the vector space
c (a)=L (2)
c (a).
CONTROLLABILITY OF RIGHT-INVARIANT SYSTEMS 537
Remarks.
(a) For a 2 Sp (1) the number j (a) is equal to the number of Jordan
blocks of the operator
ad B(a) in the space L (1) (a)=L (2) (a).
(b) If an eigenvalue a 2 Sp (1) is simple, then j
Suppose that assumption will be justified by Theorem
below). Then by Lemma 2.1
that is why any element X 2 L can uniquely be decomposed as follows:
We will consider such decomposition for the uncontrolled vector field A of
the system \Gamma:
We denote by ]
A(a) the canonical projection of the vector A(a) 2 L (1) (a)
onto the quotient space L (1) (a)=L (2) (a).
Definition 2. Let We say that
a vector A has the zero a-top if
ad B(a) \Gamma a Id)(L (1) (a)=L (2) (a)):
In the opposite case we say that A has a nonzero a-top. We use the corresponding
notations: top
Remark . Geometrically, if a vector A has a nonzero a-top, then the vector
A(a) has a nonzero component corresponding to the highest adjoined vector
in the (single) Jordan chain of the operator
ad B(a). Due to nonuniqueness
of the Jordan base, this component is nonuniquely determined, but its property
to be zero is basis-independent.
Definition 3. A pair of complex numbers (ff; fi), Re ff - Re fi, is called
an N-pair of eigenvalues of the operator ad B if the following conditions
hold:
(2) L (2) (ff) 6ae
Re a; Re
(3) L (2) (fi) 6ae
Re a; Re
\Psi .
538 YU. L. SACHKOV
Remarks.
(a) In other words, to generate the both root spaces L (2) (ff) and L (2) (fi)
for an N-pair (ff; fi), we need at least one root space L (1) (fl) with
Re fi]. The name is explained by the fact that N-pairs can
NOT be overcome by the extension process described in Lemma 4.2:
they are the strongest obstacle to controllability under the necessary
conditions of Theorem 1.
(b) The property of absence of the real N-pairs will be used to formulate
sufficient controllability conditions in Theorem 2. In some generic
cases this property can be verified by Lemma 4.3.
3. Necessary controllability conditions
x 3.1. Main theorem and known results. It turns out that controllability
on simply connected Lie groups G with G 6= G (1) is a very strong
property: it imposes many restrictions both on the group G and on the
system \Gamma.
Theorem 1. Let a Lie group G be simply connected and its Lie algebra
L satisfy the condition L 6= L (1) . If a system \Gamma is controllable, then:
(1) dimL
(3) L (2)
r ,
r ,
r ae
The notations j (a) and top (A; a) used in Theorem 1 are explained in
Definitions 1 and 2 in Sec. 2.
Remarks.
(a) The first condition is a characterization of the state space G but
not of the system \Gamma. It means that no single-input system
can be controllable on a simply connected Lie group G
with dimG (1) ! dimG \Gamma 1. That is, to control on such a group,
one has to increase the number of inputs. There is a general lower
for the number of the controlled
vectorfields necessary for controllability of the multi-input
system (3) on a simply connected group G [18].
(b) Conditions (3)-(7) are nontrivial only for Lie algebras L with L (2) 6=
L (1) (in particular, for solvable noncommutative L). If L
then these conditions are obviously satisfied.
CONTROLLABILITY OF RIGHT-INVARIANT SYSTEMS 539
(c) The third condition means that j
r , that is
why condition (6) is nontrivial only for a 2
c .
(d) By the same reason, in condition 7 the inclusion a 2 Sp (1) can be
changed by a 2
c . Note that if j (a) = 0, then by the formal
Definition 2 the vector A has the zero a-top.
(e) The fourth and fifth conditions are implied by the third one but are
easier to verify. The simple (and strong) "arithmetic" necessary
controllability condition (5) can be verified by a single glance at
spectrum of the operator ad Bj L (1) .
(f) For solvable L under conditions (1), (2) the spectrum
is the same for all
homotheties. Then conditions
(3)-(5) depend on L but not on B.
(g) For the case of simple spectrum of the operator ad Bj L (1)
the necessary
controllability conditions take respectively the more simple
Corollary 1. Let a Lie group G be simply connected and its Lie algebra
L satisfy the condition L 6= L (1) . Suppose that the spectrum Sp (1) is simple.
If a system \Gamma is controllable, then:
(1) dimL
r ,
r ae
Theorem 1 and Corollary 1 will be proved in Subsec. 3.3.
Remark . Now we discuss the condition L (1) 6= L essential for this work
and motivated by its initial focus - solvable Lie algebras L. Consider a
Levi decomposition
It is well known (see, e.g., [20], Theorem 3.14.1) that the Levi decomposition
of the derived subalgebra is then
This means that
If a Lie algebra L is semisimple (i.e., rad obviously L
The converse is generally not true (although this is asserted by [21], Sec. 87,
Corollary 3). For example, for the Lie algebra R 3 \Phi s so(3) (which is the Lie
540 YU. L. SACHKOV
algebra of the Lie group of motions of the three-space) its derived subalgebra
coincides with the algebra itself. (This example was kindly indicated to the
author by A. A. Agrachev).
The main tools to obtain the necessary controllability conditions given
in Theorem 1 is the rank controllability condition and the hypersurface
principle.
The system \Gamma is said to satisfy the rank controllability condition if the
Lie algebra generated by \Gamma coincides with L:
Proposition 1. (Theorem 7.1, [2]). The rank controllability condition
is necessary for controllability of a system \Gamma on a group G.
Generally, the attainable set A lies (and has a nonempty interior, which
is dense in A) in the connected subgroup of G corresponding to the Lie
algebra Lie (A; B).
The hypersurface principle is formulated for the system \Gamma as follows:
Proposition 2. (Corollary 3.2, [18]). Let a Lie group G be simply con-
nected, L, and let the Lie algebra L have a codimension one subalgebra
containing B. Then the system RB is not controllable on
G.
The sense of this proposition is that under the hypotheses stated there
exists a codimension one subgroup of the group G which separates G into
two disjoint parts, is tangent to the field B, and is intersected by the field
A in one direction only. Then the attainable set A lies "to one side" of this
subgroup.
Notice that the property of absence of a codimension one subalgebra of
containing B is sufficient for controllability of \Gamma on a Lie group G with
cocompact radical; if G is additionally simply connected, this condition is
also sufficient (Corollary 12.6, [17]).
x 3.2. Preliminary lemmas. First we obtain several conditions sufficient
for existence of codimension one subalgebras of a Lie algebra L containing
a vector B 2 L.
Lemma 3.1. Suppose that L (1) +RB 6= L. Then there exists a codimension
one subalgebra of L containing B.
Proof. Denote by l the vector space L (1) +RB. We have [l; l] ae L (1) ae l, that
is why l is a subalgebra; any vector space containing l is a subalgebra of L
too. Since l 6= L, there exists a codimension one subspace l 1 of L containing
l. Then l 1 is the required codimension one subalgebra of L containing B.
CONTROLLABILITY OF RIGHT-INVARIANT SYSTEMS 541
Lemma 3.2. Let L (1) \Phi
r 6= L (1)
r , then there exists a
codimension one subalgebra of L containing B.
Proof. If L (1)
r 6= L (2)
r , then there exists a real eigenvalue a 0 2
r such
that L (1) (a 0
be a Jordan base of the operator
ad B(a 0
(We suppose, for simplicity, that the eigenvalue a 0 of the operator -
ad
is geometrically simple, i.e., matrix of this operator is a single Jordan block;
for the general case of several Jordan blocks the changes of the proof are
obvious.)
Consider the vector space
It follows from (4), (5) that the space l 1 is (ad B)-invariant. Additionally,
we have dim l
Then we define the vector spaces
f L (1) (a) j a 2
First, dim l that is why
Third, the space l 2 is (ad B)-invariant. That is why, by virtue of (6) and
(8), we obtain the chain
Hence, l 3 is the required subalgebra of L: it has codimension one (see (7))
and contains the vector B (see (6)).
In the following three lemmas we obtain conditions sufficient for violation
of the rank controllability condition, i.e., necessary for controllability.
542 YU. L. SACHKOV
Lemma 3.3. Suppose that
2 L (1) and let there exist a vector subspace
l 1 ae L such that the following relations hold
(1) L (2) ae l 1 ae L (1) ,
Then Lie
Proof. By condition (1),
i.e., l 1 is a Lie subalgebra.
Consider the vector space l = RB \Phi l 1 . We have (in view of condition
so l is a Lie subalgebra too.
By condition (3) we have Lie (A; B) ae l, and condition (2) implies l 6= L.
Hence, Lie (A; B) 6= L.
Lemma 3.4. Let
Proof. Consider the case of the complex a 0 2
c first. By the condition
the quotient operator
ad c B(a 0 ) has at least two cyclic
spaces V; W ae L (1)
c (a 0 )=L (2)
c (a 0 ). That is, there are two Jordan chains
and in these bases matrices of the operators -
ad c B(a 0 )j V and -
ad c B(a 0 )j W
are the Jordan blocksB
a
. a 0 0
a
. a 0 0
(Obviously, we can assume that the complex conjugate bases fv
and Jordan chains of the operator -
ad c B(a 0 ) in the complex
conjugate spaces V ; W ae L (1)
c (a 0 )=L (2)
c (a 0 ).)
CONTROLLABILITY OF RIGHT-INVARIANT SYSTEMS 543
Notice that
ad c B(a 0
ad c B(a 0
For the direct sum
c (a 0 )=L (2)
(other cyclic spaces of -
ad c B(a 0
consider the decomposition
(components in other cyclic spaces of -
ad c B(a 0 )):
We can assume that
This will be proved at the end of this proof. Now suppose that condition (14)
holds and, for definiteness, That is why
(other cyclic spaces of -
ad c B(a 0 ));
c (a 0 )=L (2)
and, in view of (10), (11),
ad c B(a 0
Now let l ae L (1)
c (a 0 ) be the canonical preimage of the space ~ l. Obviously,
(ad c B) l ae l;
c (a 0
Then we pass to realification:
544 YU. L. SACHKOV
Finally, for the space l 1 := l r \Phi
we obtain
(ad
all conditions of Lemma 3.3 are satisfied, and Lie (A; B) 6= L modulo the
unproved condition (14).
To prove this condition, suppose that A v1 6= 0 and Aw1 6= 0 in decomposition
(13). In view of symmetry between V and W , we can assume that
. Define the new basis in
A v1
It is easy to see that f~v is a basis of V , and
for the new basis f~v and the old basis fw g.
Now we show that the new basis is a Jordan one.
ad c B(a 0 )
ad c B(a 0 )
A v1
ad c B(a 0 )
A v1
(a
A v1
A v1
ad c B(a 0 )
ad c B(a 0 )
A v1
ad c B(a 0 )
A v1
a
A v1
ad c B(a 0 )
ad c B(a 0 )
A v1
ad c B(a 0 )
A v1
a
A v1
CONTROLLABILITY OF RIGHT-INVARIANT SYSTEMS 545
And if q
ad c B(a 0 )
ad c B(a 0 )
Finally, if
ad c B(a 0 )
ad c B(a 0 )
is a Jordan basis for -
ad c B(a 0 )j V , and
(13), as was claimed.
So the lemma is proved for the case of the complex eigenvalue a 0 .
And if a 0 is real, then the proof is analogous and easier: there is no need
in complexification and further realification.
Lemma 3.5. Let
Proof. The Jordan base of the operator -
ad c B(a 0 ) consists of one Jordan
chain
c (a 0 )=L (2)
and moreover, in the decomposition
we have A
ad c B(a) \Gamma a 0 Id)(L (1)
c (a 0 )=L (2)
Then in the same way as in Lemma 3.4 we denote by l the preimage of the
space under the projection L (1)
c (a 0 )=L (2)
and by l the complex conjugate to l in L c . Then the space
satisfies all hypotheses of Lemma 3.3, that is why Lie (A; B) 6= L.
x 3.3. Proofs of the necessary controllability conditions.
Proof of Theorem 1. Suppose that the system \Gamma is controllable on the group
G.
Items (1) and (2). If dimL (1)
L. It follows from Lemma 3.1 and the hypersurface principle (Proposition 2)
that \Gamma is not controllable. This contradiction proves items (1) and (2), and
allows to assume below in the proof that L (1) \Phi
(3). If L (2)
r 6= L (1)
r , then it follows from Lemma 3.2 and the hyper-surface
principle that \Gamma is not controllable.
immediately from item (3).
546 YU. L. SACHKOV
(5). From the previous item we have
r . But
consequently,
r ae
Items (6), (7) follow from Lemmas 3.4, 3.5 and from the rank controllability
condition (Proposition 1).
Proof of Corollary 1. If the spectrum Sp (1) is simple, then L (1)
for all a 2
r or Sp (1)
c respectively.
Further, L (2)
r is equivalent to Sp (2)
r , and top (A; a) 6= 0 iff
A(a) 6= 0, a 2 Sp (1) . Now Corollary 1 follows immediately from Theorem
1.
4. Sufficient controllability conditions
x 4.1. Main results. Under the necessary assumptions of Theorem 1, we
can give wide sufficient controllability conditions. Notice that the assumption
of simple connectedness can now be removed. So the below sufficient
conditions are completely algebraic; this is in contrast with the geometric
assumption (the finiteness of center of G) essential for the sufficient controllability
conditions for simple and semi-simple Lie groups G [4].
Theorem 2. Suppose that the following conditions are satisfied for a Lie
algebra L and a system \Gamma:
(1) dimL
(3) L (2)
r ,
c ,
c ,
(6) the operator adBj L (1) has no N-pairs of real eigenvalues.
Then the system \Gamma is controllable on any Lie group G with the Lie algebra
L.
The notation top (A; a) and the notion of N-pair used in Theorem 2 are
explained in Definitions 2 and 3 in Sec. 2.
Remarks. (a) Conditions (1)-(3) are necessary for controllability in the
case of a simply connected G 6= G (1) (by Theorem 1).
(b) Conditions (4) and (5) are close to the necessary conditions (6) and
(7) of Theorem 1 respectively. Notice that the fourth condition means that
all complex eigenvalues of ad Bj L (1)
are geometrically simple.
(c) Conditions (2) and (5) are open, i.e., they are preserved under small
perturbations of A and B.
CONTROLLABILITY OF RIGHT-INVARIANT SYSTEMS 547
(d) The most restrictive of conditions (1)-(6) is the last one. It can be
shown that the smallest dimension of L (1) in which this condition is satisfied
and preserved under small perturbations of spectrum of ad Bj L (1)
for
solvable L is (6). This can be used to obtain a classification of controllable
systems \Gamma on solvable Lie groups G with small-dimensional derived
subgroups G (1) .
(e) The technically complicated condition (6) can be changed by more
simple and more restrictive one, and sufficient conditions can be given as in
Corollary 2 below.
(f) Under the additional assumption of simplicity of the spectrum
the sufficient controllability conditions take the even more simple form presented
in Corollary 3 below.
Corollary 2. Suppose that the following conditions are satisfied for a
Lie algebra L and a system \Gamma:
(1) dimL
(3) L (2)
r ,
c ,
c ,
0g.
Then the system \Gamma is controllable on any Lie group G with the Lie algebra
L.
Corollary 3. Suppose that the following conditions are satisfied for a
Lie algebra L and a system \Gamma:
(1) dimL
(3) the spectrum Sp (1) is simple,
r ,
c ,
0g.
Then the system \Gamma is controllable on any Lie group G with the Lie algebra
L.
Theorem 2 and Corollaries 2, 3 will be proved in Subsec. 4.4.
x 4.2. Lie saturation. To prove the above sufficient conditions we use
the notion of the Lie saturation of a right-invariant system introduced by
V. Jurdjevic and I. Kupka. Now we recall the basic definition and properties
necessary for us (see details in [4], pp. 163-165).
548 YU. L. SACHKOV
Given a right-invariant system \Gamma ae L on a Lie group G, its Lie saturation
ae L is defined as follows:
cl A 8t 2 R+ \Psi
LS(\Gamma) is the largest (with respect to inclusion) system having the same
closure of the attainable set as \Gamma.
Properties of Lie saturation:
(2) LS(\Gamma) is a convex closed cone in L,
controllability condition:
x 4.3. Preliminary lemmas. In this section we assume that L 6= L (1)
(this condition holds, e.g., for solvable L). In view of Theorem 1, we suppose
additionally that dimL
First we present a necessary technical lemma.
Lemma 4.1. Let
lim
Z
(j=2)
lim
Z
Proof. Is obtained by the direct computation.
Now we prove the proposition that plays the central role in obtaining our
sufficient controllability conditions (Theorem 2). It is analogous to item (a)
of Proposition 11, [4].
Lemma 4.2. Let C 2 LS(\Gamma) " L (1) . Suppose that for any a 2
c the
following conditions hold :
(2) top (C; a) 6= 0 or L (1) (a) ae LS(\Gamma).
Suppose additionally that for the number
(or
CONTROLLABILITY OF RIGHT-INVARIANT SYSTEMS 549
we have
LS(\Gamma) oe
Proof. For simplicity suppose that
where
there are more
than two pairs of complex conjugate eigenvalues at the line f Re
the proof is analogous; if there are less than two pairs, then the proof is
obviously simplified.
So we have
(notice that if
respectively,
For any element D 2 L, nonnegative function g(t), and natural number
consider the limit
Z
It follows from the properties of the cone LS(\Gamma) (see Subsec. 4.2) that if D 2
LS(\Gamma) and the limit I(D;
Z
if the limit exists.
Introduce the notation
C a
Notice that
Z
For any bounded nonnegative function g(t) and any p 2 N we have
is equal to the
size of the maximal Jordan block of the operator ad c Bj L (1)
c
corresponding
to the eigenvalues c 2 Sp (1) with Re c ! r. That is why
lim
Z
Consequently,
Z
if the limit exists.
Now we choose the bases f x in
the spaces L (1) (a) and L (1) (b) in which matrices of the operators adBj L (1) (a)
and adBj L (1) (b) are the Jordan blocksB
. M r;fi 0
CONTROLLABILITY OF RIGHT-INVARIANT SYSTEMS 551
where
fi. In the bases f x and
we have
(the prime denotes transposition
of vectors and matrices). Then in the base
of the space L (1) (a) \Phi L (1) (b) we have
C a (t)
e
oe ff (t)C X1
oe ff (t)(tC X1
oe ff (t)
oe fi (t)C Z1
oe fi (t)(tC Z1 +CZ2 )
oe fi (t)
l, then the below argument can easily be modified).
(A) Now we show that span(x k ; y k
According to the hypotheses of this lemma, we have L (1) (a) ae LS(\Gamma)
or top (C; a) 6= 0. If L (1) (a) ae LS(\Gamma), then span(x That
is why we suppose below that top (C; a) 6= 0, which means in the base
that CX1 6= 0.
Taking
into account (21), (22), and Lemma 4.1, we obtain
Z
where
552 YU. L. SACHKOV
By virtue of the fact that the convex conic hull of vectors (23) for the
matrices M of the form (24) and (25), jjj - 1, is the plane span(x k ; y k ), we
We take v(t) 2 LS(\Gamma) equal to
i.e., to the component of vector (22) in the plane span(x k ; y k ), and repeat
the limit passage described in (A) replacing I(C; g;
and obtain span(x
We repeat process (B) with I(C; g; p; v), where v(t) is the component
of vector (22) in the plane span(x is decreasing from
obtain the inclusion span(x l+1 ; y
(D) We apply process (C) with l and using the functions g(t) of
the
We decrease p and repeat procedure (D) until
In view of (16), the proof of the lemma is completed.
We can give several sufficient conditions for an element B not to have
real N-pairs of eigenvalues. These conditions can be verified simply by the
picture of spectrum of the operator ad Bj L (1)
in the complex plane. We need
them to obtain Corollary 2.
Lemma 4.3. Suppose that
r . Then any one of
the following conditions is sufficient for the operator ad Bj L (1) not to have
real N-pairs of eigenvalues:
(1)
(2)
(3) 0g.
Proof. The first case (Sp (1)
obvious as there are no real eigenvalues
at all.
Case 2. Let (ff; fi) be a real N -pair, 0 ! ff - fi. We have
Jacobi's identity implies that [L (1) (a); L (1) (b)] ae L (1) (a + b), and the spaces
direct sum, that is why
L (2) (ff) ae
CONTROLLABILITY OF RIGHT-INVARIANT SYSTEMS 553
From the conditions a
ff. That is why (26) gives
L (2) (ff) ae
This contradicts item (2) of Definition 3.
Case 3 is considered analogously.
x 4.4. Proofs of the sufficient controllability conditions.
Proof of Theorem 2. We show that LS(\Gamma) oe L (1) .
Introduce the following numbers and sets:
Suppose that LS(\Gamma) 6oe L (1) , then
Recall that we have the following decomposition of the vector A corresponding
to root subspaces of the operator adBj L (1) :
Define the element
Notice that A 1 2 LS(\Gamma) since all terms in the right-hand side belong to
LS(\Gamma). In addition, we have A 1 2 L (1) . Consider the decomposition
For any a 2 Sp (1) we have
Re a 2 [n; m]
Re
According to condition 6 of this theorem, the pair of real numbers (n; m)
is not an N -pair. That is why at least one of conditions (1)-(3) of Definition
3 is violated. Now we consider these cases separately and come to a
contradiction.
(1) Let condition (1) of Definition 3 be violated, i.e., n 62 Sp (1) or m 62
. Suppose, for definiteness, that m 62
Apply Lemma 4.2 with m. Then we have
LS(\Gamma) oe
f L (1) (a) j a 2
f L (1) (a) j a 2
which is a contradiction to (28).
If n 62 we come to a contradiction with (27) analogously. That is
why case (1) is impossible.
(2) Let now n; m 2 Sp (1) and let condition (2) of Definition 3 be violated,
i.e.,
But for Re -;
we have L (1) (-); L (1) (-) ae LS(\Gamma) (by definitions
(27) and (28)), consequently, L (2) (n) ae LS(\Gamma). According to hypotheses of
this theorem, L (1) that is why
Consider the vector A
Now we apply Lemma 4.2 with
LS(\Gamma) oe
f L (1) (a) j a 2
Then, by virtue of (29), we have
LS(\Gamma) oe
f L (1) (a) j a 2
which is a contradiction with (27). That is why case (2) is impossible, and
condition (2) of Definition 3 cannot be violated.
(3) We prove analogously that condition (3) of Definition 3 cannot be
violated as well.
Hence, all three conditions of Definition 3 hold, and (n; m) is a real N-pair
of eigenvalues. This is a contradiction with condition (6) of this theorem.
That is why LS(\Gamma) oe L (1) . But
controllable by the controllability condition (15).
Proof of Corollary 2 follows immediately from Theorem 2 and Lemma 4.3.
Proof of Corollary 3 is obvious in view of Corollary 2.
CONTROLLABILITY OF RIGHT-INVARIANT SYSTEMS 555
5. Examples and applications
x 5.1. Metabelian groups. Solvable Lie algebras L having the derived
series of length 2:
are called metabelian. A Lie group with a metabelian Lie algebra is also
called metabelian.
Our previous results make it possible to obtain controllability conditions
for metabelian Lie groups.
Theorem 3. Let G be a metabelian Lie group. Then the following conditions
are sufficient for controllability of a system \Gamma on G:
(1) dimL
(3)
c ,
c .
If the group G is simply connected, then conditions (1)-(5) are also necessary
for controllability of the system \Gamma on G.
The notation top (A; a) used in Theorem 3 is explained in Definition 2 of
Sec. 2.
Proof. The sufficiency follows from Corollary 2.
In order to prove the necessity for the simply connected G suppose that
\Gamma is controllable.
Conditions (1) and (2) follow then from items (1) and (2) of Theorem 1.
Condition (3) follows from item (3) of Theorem 1 and from the metabelian
property of G:
Condition (4). For any a 2
c we have L (2) f0g, that is why
j (a) is equal to geometric multiplicity of the eigenvalue a of the operator
ad Bj L (1) (a) , i.e., to dimL c (a). By item (6) of Theorem 1, we have
that is why dimL c (a) = 1.
Condition (5). For any a 2
c we have j (a) = 1, then, by item (7) of
Theorem 1, we obtain top (A; a) 6= 0.
Example. Let l be a finite-dimensional real Lie algebra acting linearly in
a finite-dimensional real vector space V . Consider their semidirect product
l. It is a subalgebra of the Lie algebra of affine transformations
of the space V since L ae V \Phi s gl (V ). If l is Abelian, then L is metabelian:
In the following subsection we study in detail a particular case when l is
one-dimensional.
x 5.2. Matrix group. Now we apply the controllability conditions from
the previous subsection to some particular metabelian matrix group. To
begin with we describe this group.
Let V be a real finite-dimensional vector space,
linear operator in V . The required metabelian Lie algebra is the semi-direct
product
RM
(compare with the example at the end of the previous subsection).
Now we choose and fix a base in V , and denote the matrix of the operator
M in this base by the same letter M . Then L(M ) can be represented as
the subalgebra of gl (n generated by the following matrices:
(Recall that E ij is the n \Theta n matrix with the only unit entry in the ith line
and the jth raw.) Obviously, we have
Notice also that [y and M is the matrix of
the adjoint operator ad xj L (1) in the base f y g. In the sequel we
consider the Lie algebra L(M in this matrix representation.
Let G(M ) be the connected Lie subgroup of GL(n corresponding
to L(M ). The group G(M ) can be parametrized by the matrices
It is a semidirect product:
n\Omega
The group G(M ) is not simply connected iff the one-parameter subgroup G 1
is periodic, which occurs iff the matrix M has purely imaginary commensurable
spectrum. More precisely, we say that a set of numbers (b
R n is commensurable if
(b
CONTROLLABILITY OF RIGHT-INVARIANT SYSTEMS 557
And the group G(M ) is not simply connected iff
the set Im(Sp(M )) is commensurable:
oe
Before studying controllability conditions for the group G(M ) we present
an auxiliary proposition, which translates the Kalman condition (equivalent
both to controllability and to rank controllability condition for linear systems
into the language of eigenvalues of
the matrix A and of components of the vector b in the corresponding root
spaces). We will apply this proposition below to reformulate our controllability
conditions for right-invariant and bilinear systems.
Lemma 5.1. Let A be a real n \Theta n matrix, b 2 R n . Then the Kalman
condition
rank (b;
is equivalent to the following conditions:
(1) the matrix A has a geometrically simple spectrum,
(2) top (b; -) 6= 0 for any eigenvalue - 2 Sp(A).
By analogy with Definition 2 in Sec. 2, we say that top (b; -) 6= 0 if the
component b(-) of the vector b in the root space R n (-) corresponding to
the eigengalue - satisfies the condition
i.e., the vector b(-) has a nonzero component corresponding to the highest
adjoined vector in the (single) Jordan chain of the operator A corresponding
to -.
To prove Lemma 5.1, we cite the following
Proposition 3. (Hautus Lemma, [22], Lemma 3.3.7.) Let A be a complex
n \Theta n matrix, b 2 C n . Then the Kalman condition (31) is equivalent
to the condition
Proof of Lemma 5.1. In view of Proposition 3, we prove that condition (32)
is equivalent to conditions (1), (2) of Lemma 5.1.
First, we suppose that all eigenvalues of A are real; otherwise we pass
to complexification. Second, the Kalman condition (31) preserves under
558 YU. L. SACHKOV
changes of base in R n . That is why we assume that the matrix A is in the
Jordan normal form:
A =B @
. - l 0
Then the n \Theta (n + 1) matrix in condition (32) is represented as
denotes projection of the vector b onto the root
space of the matrix A corresponding to the eigenvalue - l .
Necessity. We assume that rank
conditions (1), (2) of Lemma 5.1.
1. If spectrum of A is not geometrically simple, then -
j. Then the matrix OE(- i ) has two zero columns, and rank OE(-
2. Suppose that the vector b has the zero -top for some - 2 Sp(A); for
definiteness, let top (b; the first component of b in the chosen
Jordan base equals to zero, and the first raw of the matrix OE(- 1 ) is zero.
Hence rank OE(- 1
Sufficiency. If conditions (1), (2) of Lemma 5.1 hold, then it is easy to
see from representation (33) that all matrices OE(- l ),
linearly independent columns and condition (32) is satisfied.
Now we obtain controllability conditions for the universal covering
and for the group G(M ) itself.
Theorem 4. Let M be an n \Theta n matrix,
is controllable on G if and only if the following
conditions hold :
(1) the matrix M has a purely complex geometrically simple spectrum,
CONTROLLABILITY OF RIGHT-INVARIANT SYSTEMS 559
For the group G(M ) conditions (1)-(3) are sufficient for controllability; if
conditions (30) are violated, then (1)-(3) are equivalent to controllability on
The notation top (A; -) used in Theorem 4 is explained in Definition 2 in
Sec. 2.
Remark . By Lemma 5.1, conditions (1)-(3) of the above theorem are
equivalent to the following ones:
(1) the matrix M has a purely complex spectrum,
Proof of Theorem 4. Theorem 3 (see Subsec. 5.1) is applicable to the group
G(M ), and condition (1) of Theorem 3 is satisfied.
Decompose the vector B 2 L using the base of L:
2 L (1) is equivalent to B x 6= 0. Moreover, in view of the metabelian
property of L,
By virtue of Theorem 3, the system \Gamma is controllable on G if and only if
the following conditions hold:
(3) the matrix M has a geometrically simple spectrum,
Now the proposition of the current theorem for -
For G(M ), controllability is implied by controllability on its universal
covering
conditions (30) are violated, then G(M
Let now conditions (30) be satisfied. Then the group G(M ) is a semi-direct
product of the vector group R n and the one-dimensional compact
group G 1 . But controllability conditions on such semi-direct products were
obtained by B. Bonnard, V. Jurdjevic, I. Kupka, and G. Sallet [6]: if the
compact group has no fixed nonzero points in the vector group (which is just
the case), then the controllability is equivalent to the rank controllability
condition (Theorem 1, [6]).
560 YU. L. SACHKOV
So we have complete controllability conditions of systems of the form
on the group G(M ) and its simply connected covering
In the simply connected case (i.e., when conditions (30) are violated) we
have Theorem 4, and otherwise the theorem of B. Bonnard, V. Jurdjevic,
I. Kupka, and G. Sallet [6] works.
x 5.3. Bilinear system. Now we apply the controllability conditions for
the group G(M ) and study global controllability of the bilinear system
where A is a constant real n \Theta n matrix and b 2 R n .
Theorem 5. The system \Sigma is globally controllable on R n if and only if
the following conditions hold :
(1) the matrix A has a purely complex spectrum,
Remark . By Lemma 5.1, conditions (1)-(2) of this theorem can equivalently
be formulated as follows:
(1) the matrix A has a purely complex geometrically simple spectrum,
Proof of Theorem 5. We use the hypotheses of this theorem in the equivalent
form given in the above remark.
Sufficiency. Consider the bilinear system
where
are (n matrices. It is easy to see that the system \Sigma is
globally controllable on R n iff the system \Sigma is globally controllable in the
n-dimensional affine plane
(R
Consider the matrix Lie algebra L(A) and the corresponding Lie group
described in the previous subsection. We have
ae L(A) is a right-invariant system on the group G(A). Theorem 4
ensures that under hypotheses (1), (2) of the current theorem the system
is controllable on the group G(A). But the group G(A) acts transitively
in the plane (R , and the bilinear system \Sigma is the projection of the
right-invariant system \Gamma from the group G(A) onto the plane (R That
is why controllability of \Gamma on G(A) implies controllability of \Sigma on (R
Thus \Sigma is globally controllable on R n .
CONTROLLABILITY OF RIGHT-INVARIANT SYSTEMS 561
Necessity. Assume that \Sigma is globally controllable on R n .
(1a) First we show that the matrix A has no real eigenvalues. Suppose
there is at least one eigenvalue a 2 We choose a Jordan base
of the matrix A and denote by f x the corresponding
coordinates in R n . Let e k denote the maximum order root vector
coresponding to the eigenvalue a:
and k is the maximal possible integer. Then the system \Sigma implies
where b k is the kth coordinate of the vector b in the base f e g.
Now it is obvious that at least one of the half-spaces f x
is positive invariant for the system \Sigma, i.e., this system is not controllable.
(1b) Now we show that the spectrum Sp(A) is geometrically simple. Suppose
that for some (complex) eigenvalue - 2 Sp(A) there are at least two
linearly independent eigenvectors. Then we apply the same transformation
of Jordan chains as in Lemma 3.4 to obtain the zero component of the vector
b in the two-dimensional subspace of R n spanned by the pair of the highest
order root vectors of the matrix A (see conditions (13), (14)). Now if x k , y k
are the coordinates in R n in the transformed Jordan base corresponding to
the above-mentioned two-dimensional subspace, then the system \Sigma yields
Hence it follows that the codimension two
subspace f x is (both positive and negative) invariant for the
system \Sigma, and so it is not controllable.
(2) Finally, we show that the vector b has a nonzero -top for any eigenvalue
this is not the case, we choose any Jordan chain in the
root space corresponding to -, apply the argument from item 1.b) above,
and show that \Sigma is not controllable.
The necessity and sufficiency are now completely proved.
x 5.4. The Euclidean group in two dimensions. It is interesting to
consider the work of the above general theory for the visual three-dimensional
case.
E(2) be the Euclidean group of motions of the plane
R 2 . E(2) is connected but not simply connected. It can be represented as
the group of 3 \Theta 3 matrices of the form
sin t cos t s 2
562 YU. L. SACHKOV
where
sin t cos t
The corresponding matrix Lie algebra L is spanned by the matrices
Consider the system
E(2) - the universal covering of
E(2). A complete characterization of controllability of \Gamma on g
E(2) is derived
Theorem 4.
Theorem 6. The system \Gamma is controllable on g
E(2) if and only if the
vectors A, B are linearly independent and
Let us compare the controllability conditions for g
E(2) with the following
conditions for E(2) derived from Theorem 1, [6]:
Theorem 7. The system \Gamma is controllable on E(2) if and only if the
vectors A, B are linearly independent and span(A; B) 6ae span(y; z).
Finally, Theorem 5 gives the following geometrically clear proposition.
Theorem 8. The system
is controllable on the plane R 2 if and only if :
(1) the matrix A has a purely complex spectrum,

Acknowledgment

. The author thanks Professor G'erard Jacob and Laboratoire
d'Informatique Fondamentale de Lille, Universit'e Lille I, where this
paper was started, for hospitality and excellent conditions for work. The
author is also grateful to Professor A. A. Agrachev for valuable discussions
of the results presented in this work.
CONTROLLABILITY OF RIGHT-INVARIANT SYSTEMS 563



--R

System theory on group manifolds and coset spaces.
Control systems on
Control systems subordinated to a group action: Acces- sibility


Transitivity of families of invariant vector fields on the semidirect products of
Controllability of right invariant systems on real simple
Controllability on classical
Controllability of right invariant systems on real simple

Controllability of nilpotent systems.
Controllability of linear vector fields on
Foundations of
Notes Math.
Controllability of systems on a nilpotent Lie group.
Controllability on real reductive
Maximal subsemigroups of
Controllability of hypersurface and solvable invariant systems.


Mathematical control theory: Deterministic finite dimensional systems.
--TR
Mathematical control theory: deterministic systems

--CTR
Dirk Mittenhuber, Controllability of Solvable Lie Algebras, Journal of Dynamical and Control Systems, v.6 n.3, p.453-459, July 2000
Yu. L. Sachkov, Classification of Controllable Systems on Low-Dimensional Solvable Lie Groups, Journal of Dynamical and Control Systems, v.6 n.2, p.159-217, April 2000
Dirk Mittenhuber, Controllability of Systems on Solvable Lie Groups: The Generic Case, Journal of Dynamical and Control Systems, v.7 n.1, p.61-75, January 2001
right-invariant systems;controllability;lie groups;bilinear systems