Control sets of linear systems on semi-simple Lie groups
Victor Ayala, Adriano Da Silva, Philippe Jouan, Guilherme Zsigmond

TL;DR
This paper investigates the properties of control sets with nonempty interior for linear control systems on semisimple Lie groups, revealing differences from solvable cases and describing their structure.
Contribution
It characterizes the structure and number of control sets with nonempty interior for linear systems on semisimple Lie groups, highlighting key differences from solvable groups.
Findings
Multiple control sets with nonempty interior can exist on semisimple Lie groups.
Control sets are contained in right translations of a fundamental control set.
Distinct from solvable cases, the number of control sets can be greater than one.
Abstract
In this paper we study the main properties of control sets with nonempty interior of linear control systems on semisimple Lie groups. We show that, unlike the solvable case, linear control systems on semisimple Lie groups may have more than one control set with nonempty interior and that they are contained in right translations of the one around the identity.
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Control sets of linear systems on semi-simple Lie groups
Víctor Ayala
Instituto de Alta Investigación
Universidad de Tarapacá, Arica, Chile and
Departamento de Matemáticas
Universidad Católica del Norte, Antofagasta, Chile
Supported by Proyecto Fondecyt n∘ 1150292. Conicyt, Chile.
Adriano Da Silva
Instituto de Matemática
Universidade Estadual de Campinas, Brazil Supported by Fapesp grant 2016/11135-2 and 2018/10696-6
Philippe Jouan
Laboratoire de Mathématiques Raphaël Salem
CNRS UMR 6085
Université de Rouen, France
Guilherme Zsigmond
Departamento de Matemáticas
Universidad Católica del Norte, Antofagasta, Chile and
Laboratoire de Mathématiques Raphaël Salem
CNRS UMR 6085
Université de Rouen, France
Supported by Capes grant BEX 1041-14-2
Abstract
In this paper we study the main properties of control sets with nonempty interior of linear control systems on semisimple Lie groups. We show that, unlike the solvable case, linear control systems on semisimple Lie groups may have more than one control set with nonempty interior and that they are contained in right translations of the one around the identity.
1 Introduction
Linear control systems on Lie groups appear as a natural extension of linear control systems on Euclidean spaces. Several works addressing the main issues in control theory for such systems, such as controllability, observability and optimization appeared over the years. In [7] P. Jouan showed that such generalization is also important for the classification of general control systems on abstract connected manifolds.
On the other hand it is meaningful to deal with restricted inputs because that corresponds to the practical case. However many systems are not controllable for restricted inputs. It is therefore natural to study the maximal regions of controllability, i.e. the control sets (see Section 2.1 for the definition). In the paper at hand we analyze the control sets of linear control systems on semisimple Lie groups. In [2], several topological properties of such sets were proven in the solvable case. By using the close relationship between the dynamics of the drift and the behaviour of the control system (see [1] and [6]) the authors were able to prove boundedness and uniqueness of the control sets. The richness of the geometry of semisimple Lie groups modifies completely the picture. Indeed more than one control set with nonempty interior may exist. They are contained in right translations of the existent control set around the identity. Moreover, the existence of an invariant control set implies global controllability, showing how semisimplicity strongly influences the behaviour of the control system.
The paper is structured as follows: Section 2 introduces the main definitions and principal properties concerning control systems, control sets, linear vector fields and linear control systems on Lie groups. Since our work is devoted to semisimple Lie groups, we also provide in Section 2 a small subsection about semisimple theory in order to make the paper self-contained. Section 3 contains the main results concerning control sets with nonempty interior of a linear control system on a connected semisimple Lie group. We show that all the possible control sets with nonempty interior of the system are contained in the right translations of the control set around the identity. The particular case where the drift of the system has trivial nilpotent part, these right translations are precisely the control sets of the system. In this section we also show that for linear control systems on semisimple Lie groups, the only possible invariant control set is the whole group. Section 4 is devoted to illustrating the paper with an example in .
Notations: Let be a connected Lie group. We denote by the identity element of . For any element , the maps and stand for the left and right translations in , respectively. By we denote the conjugation of . By we denote the group of automorphisms of . If is a 1-parameter subgroup, its orbit from is the subset . We say that a subset is -invariant if for any .
2 Preliminaries
2.1 Control systems
Let be a -dimensional smooth manifold. A control system on is a family of ordinary differential equations
[TABLE]
where is a smooth map and
[TABLE]
is the set of the admissible control functions, with a bounded subset of such that . For any and we denote by the unique solution of (2.1) with initial value . We use to denote the diffeomorphism . Given and we have that
[TABLE]
where is the *concatenation * of and define by
[TABLE]
The set of points reachable from at time exactly , the set of points reachable from up to time and the reachable set from are respectively denoted by
[TABLE]
By , and we denote the corresponding sets for the time-reversed system. We say that the system (2.1) is locally accessible from if and are nonempty for all . The system is said to be locally accessible if it is locally accessible from any . A sufficient condition for local accessibility is the Lie algebra rank condition (LARC). It is satisfied if the Lie algebra generated by the vector fields , for , satisfies for all .
A subset is a control set of 2.1 if it is maximal w.r.t. set inclusion with the following properties:
- (i)
is controlled invariant, i.e., for each there is with . 2. (ii)
Approximate controllability holds on , i.e., for all .
Following [9], Proposition 3.2.4., any subset of with nonempty interior that is maximal with property (ii) in the above definition is a control set.
The next result (see Lemma 3.2.13 of [9]) states the main properties of control sets with nonempty interior.
2.1 Lemma:
Let be a control set of (2.1) with nonempty interior. It holds:
- (i)
If the system is locally accessible from all , then is connected and ;
- (ii)
If is locally accessible, then for all ;
- (iii)
If the system is locally accessible from all , then for all and for every one has
[TABLE]
In particular, exact controllability holds on .
We say that a control set is positively-invariant (resp. negatively-invariant) it for any and (resp. ).
2.2 Semisimple theory
Standard references for the theory of semisimple Lie groups are Duistermat-Kolk-Varadarajan [10], Helgason [11], Knapp [12] and Warner [14]. In the sequel, we only provide a brief review of the concepts used in this paper.
Let be a connected semisimple non-compact Lie group with finite center and Lie algebra . We choose a Cartan involution and denote by the associated inner product, where is the Cartan-Killing form. If and stand, respectively, for the eigenspaces of associated with and , the Cartan decompositions of and are given, respectively, by
[TABLE]
Fix a maximal abelian subspace and denote by the set of roots for this choice. If , where is the set of positive roots and
[TABLE]
is the root space associated with , the Iwasawa decompositions of and are given, respectively, by
[TABLE]
Let be the positive Weyl chamber associated with the above choices and consider . The eigenspaces of in are given by , and . The centralizer of in is given by
[TABLE]
and the centralizer in by . They are, respectively, the Lie algebra of the centralizer of in , , and in , . Since has finite center, the centralizer of in is a compact subgroup of . This fact, together with the equality implies that has a finite number of connected components. The finite subgroup parametrizes the connected components of and hence, will stand for the connected component of related with .
The negative and positive nilpotent subalgebras of type are given by
[TABLE]
The parabolic subalgebra and the negative parabolic subalgebra of type are given, respectively, by
[TABLE]
At the group level, and stand for the connected nilpotent Lie subgroup associated with and , respectively. The parabolic subgroups and are, respectively, the normalizer of and in . It holds that is a normal subgroup of and the same is true for and . In particular, it holds that
[TABLE]
Moreover, the set
[TABLE]
is an open and dense subset of .
2.3 Linear vector fields on semisimple Lie groups
A vector field on a connected Lie group is said to be linear if its flow is a -parameter subgroup of . Associated to any linear vector field there is a derivation of defined by the formula
[TABLE]
The relation between and is given by the formula
[TABLE]
In particular, it holds that
[TABLE]
Let be semisimple and consider to be a linear vector field on . If stands for the derivation associated with , the fact that is a semisimple Lie algebra implies that is inner, that is, there exists such that .
By equation (2.3) we get that
[TABLE]
and since is connected, we conclude that , where the minus sign on the above formula is connected with the choice of right-invariant vector fields.
Following [11] (Chapter 9, Lemma 3.1) the Jordan decomposition of an element is the commuting decomposition where , and is nilpotent. In particular, the Lie subalgebras , and coincide, respectively, with the sum of the real generalized eigenspaces of associated with the eigenvalues with positive, zero and negative real parts.
We call the elements , and obtained from the Jordan decomposition of the elliptic, hyperbolic and nilpotent parts of , respectively. Moreover, the Jordan decomposition of implies that the flow of is given by the commutative product
[TABLE]
A simple calculation shows that , and are -invariant.
2.4 Linear control systems on semisimple Lie groups
A linear control system on a connected Lie group is a family of ordinary differential equations of the form
[TABLE]
where is a linear vector field, are right-invariant vector fields and . The solutions of linear control systems are related to the flow of by the formula
[TABLE]
Let us denote by and the sets and , respectively.
The next proposition states the main properties of the reachable sets of linear control systems (see for instance [8], Proposition 2).
2.2 Proposition:
With the previous notations it holds:
;
- 2.
- 3.
The next result shows that the set is invariant by right translations of elements whose -orbits are contained in ([6], Lemma 3.1).
2.3 Lemma:
Let and assume that . Then .
Let to be a connected semisimple Lie group with finite center. By using the notations introduced in Section 2.2 for the semisimple case, Theorem 3.9 of [1] gives us a strict relation between the subgroups and and the linear control system as follows.
2.4 Theorem:
Let be a connected semisimple Lie group with finite center. If is open, then and .
Next we extend the above results relating with the other connected components of .
2.5 Proposition:
Let be a connected semisimple Lie group with finite center and assume that is open. For any we have that
[TABLE]
Proof.
In fact, let . Since we get that any can be written as with and hence
[TABLE]
Let us notice that and then, if we have by the -invariance of and Lemma 2.3 that . Since is finite, the above is true for any showing the result. ∎
The next technical lemma will be useful ahead.
2.6 Lemma:
Let and .
- (i)
It holds that ;
- (ii)
If then .
Proof.
(i) For any consider and with . By the -invariance of , there exists such that and hence
[TABLE]
where for the inclusion we used that .
(ii) Let and write it as for some and . Then
[TABLE]
and by the arbitrariness of we get concluding the proof. ∎
2.7 Remark:
It is important to notice that the assumption that is open is equivalent to the existence of some such that (see [9], Lemma 4.5.2). In particular, if is open the system is locally accessible (see Theorem 3.3 of [5]) and is also open. There is an easily checkable algebraic condition that ensures the openness of called the ad-rank condition (see for instance [8], Proposition 6).
3 Control sets of linear systems on semisimple Lie groups
In this section will be assumed that is a connected semisimple Lie group with finite center and that is a linear control system such that is open. Let us denote by the hyperbolic part of the linear vector field , drift of the system .
By considering the homogeneous space we have, by the -invariance of , a well defined system on induced by the linear control system (see [7], Proposition 4). We aim to show that there is a strict relation between the control sets of and the ones of .
If is the canonical projection and the flow induced by on we have that
[TABLE]
Moreover, if stands for the left translation in given by we have that the solutions of satisfy
[TABLE]
In particular, for any
[TABLE]
are the reachable sets for from for the induced system. By our assumption on the openness of , there exists a control set of with nonempty interior containing the identity in its interior (see Corollary 4.5.11 of [9]). By Lemma 2.1 it is equal to and is denoted by in the sequel.
3.1 Theorem:
The projection is a control set for and satisfies .
Proof.
Since is an open map, it holds that has nonempty interior. Moreover, by (3.4) we have that
[TABLE]
and therefore, is contained in a control set for the control system .
The result is proved if we show that . However, since it is enough to show that
[TABLE]
Let then . Denote by . Since exact controllability holds on and , there exist and such that
[TABLE]
for some . On the other hand, the openness of implies and hence there exist and such that
[TABLE]
Therefore, for any . Using the fact that is dense in and by maximility of we get the desired result. ∎
We define now a group of homeomorphisms in . For any let us define the map by , where satisfies . Since is a normal subgroup of the map is a well defined homeomorphism of satisfying:
- (i)
for all ;
- (ii)
for ;
3.2 Lemma:
For any it holds that
[TABLE]
In particular, for any and .
Proof.
Since for any it holds that
[TABLE]
[TABLE]
and the result follows. ∎
A direct consequence of Lemma 3.2 is that is a control set with nonempty interior of , where is the projection of the control set of as proved in Theorem 3.1. The set
[TABLE]
is a subgroup of and the map given by is a well-defined injective map, since
[TABLE]
The next result shows that the control sets of a linear control system are related with the control sets for .
3.3 Lemma:** **(Fundamental Lemma)
If is a control set of with nonempty interior then
[TABLE]
Proof.
Let be a control set with nonempty interior of . By equation (2.2) the set
[TABLE]
is an open and dense subset of and hence for some . There exist , with . Since in we have exact controllabillity, there exists and such that
[TABLE]
If stands for a left invariant Riemannian metric on we get
[TABLE]
Since and has only eingevalues with negative real part, we have that as On the other hand, the fact that implies that for each , where . Hence, implying that . Let then and . By exact controllability there exists and with . If we write with we get
[TABLE]
where for the last inclusion we used Lemma 2.3. By the arbitrariness of we conclude that
[TABLE]
By arguing analogously for we assure the existence of such that where . Therefore,
[TABLE]
In particular, . Let then and consider and such that
[TABLE]
where and . Therefore, implying that . By Proposition 2.5 we get
[TABLE]
Since, and , equation (3.7) implies
[TABLE]
and using equation (3.6) we get
[TABLE]
concluding the proof. ∎
Now we can prove our main result.
3.4 Theorem:
If is a control set with nonempty interior of then
[TABLE]
Proof.
In fact, for any , it holds that and consequently
[TABLE]
In fact,
[TABLE]
[TABLE]
and the result follows from Lemma 3.3. ∎
The next result shows that for linear control system whose drift has trivial nilpotent part the right translations of coincides with the control sets of . This case is particularly important because generically linear vector fields have trivial nilpotent part since it is true as soon for .
3.5 Theorem:
If is open and has trivial nilpotent part, then
[TABLE]
are all the control sets with nonempty interior of . In this case admits exactly control sets with nonempty interior.
Proof.
Since we already have that any control set with nonempty interior is contained in for some it is enough to show that, if the nilpotent part of is trivial, we actually have that is a control set of for any .
By the assumption on the linear vector field we have that the flow of the linear vector field restricted to is given by . Since and is compact, we get that is bounded and hence, if we have that is a compact subset. Therefore, for any and any we obtain by Corollary 4.5.11 of [9] that there exists a control set with nonempty interior such that
[TABLE]
In particular but implies by item (ii) of Lemma 2.6 that for any . On the other hand, for any there is with which gives us
[TABLE]
and hence . Analogously, for any we get implying that . Moreover,
[TABLE]
Using that and that is dense in we have as stated. ∎
3.6 Remark:
By following the idea of the proof of the above theorem, it is not hard to show that is a control set as soon as possesses a fixed or a periodic point for the flow of , even when the nilpotent part of is not trivial.
We finish this section by showing that the existence of an invariant control set is equivalent to the controllability of .
3.7 Theorem:
The only possible positively-invariant (resp. negatively-invariant) control set of with nonempty interior is .
Proof.
Our proof is divided in two steps:
Step 1: if and only if
Since both cases are analogous we will only show that . Recall that being semisimple the derivation is inner and equal to for some right-invariant vector field . We can consequently define an invariant system by:
[TABLE]
The solutions of and of are related by
[TABLE]
An easy proof of equation 3.8 can be found in [8], Proposition 8. As consequence the reachable set at time from the identity for is .
The reachable set from the identity is a semigroup with nonempty interior. It is said to be left reversible (resp. right reversible) if (resp. ). Following Theorem 6.7 of [13], if is a connected semisimple Lie group with finite center then itself is the only subsemigroup with nonempty interior which is left or right reversible.
Assume then that . The first thing to show is that . Let . Since there exists such that . This implies . However, since we obtain and hence .
We can now prove that . Let . There exists such that or equivalently . Consequently:
[TABLE]
But and there exists and such that , so that:
[TABLE]
and hence concluding the proof.
Step 2: If admits a positively-invariant (resp. negatively-invariant) control set with nonempty interior then .
In fact, let us assume that is a positively-invariant control set of with nonempty interior. By Theorem 3.3 we get that for some . Since and exact controllability holds on , we can always build an periodic orbit passing for a given point in and intersecting which by the positively-invariance of implies that and hence that is positively-invariant. Since we get that is also positively-invariant and by Theorem 3.1, the same holds for . In particular, is closed which by Theorem 3.6 of [2] implies that and by the previous step that which implies the result. ∎
3.8 Remark:
It is important to remark that Step 1 on the previous proof was first stated in [8] but for unrestricted inputs. Moreover, the idea of the “reversible semigroups” trick comes from [4].
4 Example
Let be the three-dimensional semisimple Lie group of the matrices with determinant equal to one. Its Lie algebra is given by , the set of the matrices with zero trace.
Denote any element by , where satisfies . Here is the orthogonal vector of obtained by a counter-clockwise rotation of . If we have the diffeormorphism
[TABLE]
Note that
[TABLE]
and hence, the flow of any right-invariant vector field , , passes to as
[TABLE]
where we used that .
A simple calculation shows that the flow (4.9) is associated with the vector field
[TABLE]
We are interested in the dynamical behaviour of the flow of where has a pair of distinct real eigenvalues. For such case, there is a basis of eigenvalues of that we always assume ordered such that is associated with the positive eigenvalue.
For such case, the first component of the solution is given by
[TABLE]
and its dynamical behaviour in the circle is given as in the picture ahead (see Figure 1).
The second component of the solution of is then
[TABLE]
where is the positive eigenvalue of and is defined by where is the inner product that makes orthogonal. The dynamical behaviour of is given by the right Figure 1.
Let us consider now a linear control system given by with and the associated derivation. Assume that
is a nonzero diagonal matrix;
- 2.
has a pair of distinct real eigenvalues for any ;
- 3.
is a basis for 111A pair of matrices satisfying the conditions are, for instance, H=\left(\begin{array}[]{cc}1&0\\ 0&-1\end{array}\right)\;\;\mbox{ and }\;\;X=\left(\begin{array}[]{cc}1&1\\ 1/2&-1\end{array}\right)..
The above conditions imply that the system is not controllable (see [3]), that is open and that . Moreover, the fact that implies that the induced control on coincides with the associated invariant system. Moreover, the fact that the piecewise constant control functions are dense in , implies that we only need to analyze how the concatenations of for acts on .
Following [9], Chapter 6, the system on given by has, for small , four control sets where the one containing is the closed interval on given by , where , is the attractor of . The control set that contains is then given in Figure 2. Moreover, since implies that and hence (see Figure 2).
By Theorem 3.5 we have that the control sets with nonempty interior of are given by and .
4.1 Remark:
It is not hard to see that there are control sets around the points and who are still related by the map . That shows that the induced control system on can have more control sets than the ones given by , .
The reference list from the paper itself. Each links out to its DOI / PubMed record.
- 1[1] V. Ayala and A. Da Silva , Controllability of Linear Control Systems on Lie Groups with Semisimple Finite Center , SIAM Journal on Control and Optimization 55 No 2 (2017), 1332-1343.
- 2[2] V. Ayala, A. Da Silva and G. Zsigmond. Control sets of linear systems on Lie groups. Nonlinear Differential Equations and Applications, Vol. 24:8 (2017)
- 3[3] V. Ayala and L.A.B. San Martin. Controllability of two-dimensional bilinear systems: restricted controls and discrete-time . Proyecciones, vol. 18 n.2 (1999), pp. 207-233.
- 4[4] V. Ayala and L.A.B. San Martin. Controllability properties of a class of control systems on Lie Groups Nonlinear control in the year 2000, Vol. 1 (Paris), 83–92, Lecture Notes in Control and Inform. Sci., 258, Springer, London, 2001.
- 5[5] V. Ayala and J. Tirao. Linear Control Systems on Lie Groups and Controllability. American Mathematical Society, Series: Symposia in Pure Mathematics, Vol. 64, pp. 47-64, 1999.
- 6[6] A. Da Silva. Controllability of linear systems on solvable Lie groups , SIAM Journal on Control and Optimization 54 No 1 (2016), 372-390.
- 7[7] Ph. Jouan. Equivalence of Control Systems with Linear Systems on Lie Groups and Homogeneous Spaces , ESAIM: Control Optimization and Calculus of Variations, 16 (2010) 956-973.
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