Combinatorics-Based Approaches to Controllability Characterization for Bilinear Systems
aa r X i v : . [ m a t h . O C ] S e p COMBINATORICS-BASED APPROACHES TO CONTROLLABILITYCHARACTERIZATION FOR BILINEAR SYSTEMS
GONG CHENG ∗ , WEI ZHANG ∗ , AND
JR-SHIN LI ∗ Abstract.
The control of bilinear systems has attracted considerable attention in the field ofsystems and control for decades, owing to their prevalence in diverse applications across scienceand engineering disciplines. Although much work has been conducted on analyzing controllabilityproperties, the mostly used tool remains the Lie algebra rank condition. In this paper, we developalternative approaches based on theory and techniques in combinatorics to study controllability ofbilinear systems. The core idea of our methodology is to represent vector fields of a bilinear sys-tem by permutations or graphs, so that Lie brackets are represented by permutation multiplicationsor graph operations, respectively. Following these representations, we derive combinatorial char-acterization of controllability for bilinear systems, which consequently provides novel applicationsof symmetric group and graph theory to control theory. Moreover, the developed combinatorial ap-proaches are compatible with Lie algebra decompositions, including the Cartan and non-intertwiningdecomposition. This compatibility enables the exploitation of representation theory for analyzingcontrollability, which allows us to characterize controllability properties of bilinear systems governedby semisimple and reductive Lie algebras.
Key words.
Bilinear systems, Lie groups, graph theory, symmetric groups, representationtheory, Cartan decomposition
1. Introduction.
Bilinear systems, a class of nonlinear systems, emerge nat-urally as mathematical models to describe the dynamics of numerous processes inscience and engineering. Prominent examples include the Bloch system governing thedynamics of spin- nuclei immersed in a magnetic field in quantum physics [11, 19, 20],the compartmental model describing the movement of cells and molecules in biology[22, 8, 21], and the integrate-and-fire model characterizing the membrane potential ofa neuron under synaptic inputs and injected current in neuroscience [7, 10]. The preva-lence of bilinear systems has been actively promoting the research in control theoryand engineering concerning the analysis and manipulation of such systems for decades.The initial investigation into control problems involving bilinear systems traces backto the year of 1935, when the Greek mathematician Constantin Carathéodory studiedoptimal control of bilinear systems presented in terms of Pfaffian forms by using calcu-lus of variations and partial differential equations [5]. However, research in systematicanalysis of fundamental properties of bilinear control systems was not prosperous untilthe early 1970s, when leading control theorists, such as Brockett, Jurdjevic, and Suss-mann, developed geometric control theory for introducing techniques in Lie theoryand differential geometry to classical control theory [4, 2, 16, 13, 3, 12].One of the most remarkable results in geometric control theory is the Lie algebrarank condition (LARC), which establishes an equivalence between controllability ofcontrol-affine systems defined on smooth manifolds and Lie algebras generated by thevector fields governing the system dynamics [2, 14, 15]. In our recent work, basedon the LARC, we developed a necessary and sufficient controllability condition forbilinear systems by using techniques in symmetric group theory [27]. In particular,we introduced a monoid structure on symmetric groups so that Lie bracket operationsare compatible with monoid operations. This then resulted in a characterizationof controllability in terms of elements in “symmetric monoids” for bilinear systems, ∗ Department of Electrical and Systems Engineering, Washington University, St. Louis, MO 63130([email protected], [email protected], [email protected]). Questions, comments, or correctionsregarding this document may be directed to Li. 1 hich also offered an alternative to the LARC and further shed light on interpretinggeometric control theory from an algebraic perspective.In this paper, we propose a combinatorics-based framework to analyze controlla-bility of bilinear systems defined on Lie groups by adopting techniques in symmetricgroup theory and graph theory. Specifically, the main idea is to associate such sys-tems with permutations or graphs, so that Lie bracket operations of the vector fieldsgoverning the system dynamics can be represented by permutation multiplicationsand edge operations on the graphs. This combinatorics approach immediately leadsto the characterizations of controllability in terms of permutation cycles and graphconnectivity. In particular, we identify the classes of bilinear systems, for which con-trollability has equivalent symmetric group and graph representations. A prominentexample is the system defined on
SO( n ) , the special orthogonal group, for which wereveal a correspondence between permutation cycles in the symmetric group and treesin the graph associated with these systems. It is worth noting that, different fromour previous work on the symmetric group method [27], the correspondence betweenLie bracket operations and permutation multiplications established in this paper donot require any monoid structure on symmetric groups. On the other hand, the ap-plication of graph theory in the developed combinatorics-based framework offers adistinct viewpoint to the field of control theory. Specifically, in the existing literature,graphs are naturally used in the context of networked and multi-agent systems, e.g.,for describing the coupling topology and deriving structural controllability conditions[23, 24, 25], while, in this work, we establish a non-trivial relationship between graphconnectivity and controllability for a single bilinear system.Moreover, a great advantage of the developed framework is its compatibility withvarious Lie algebra decomposition techniques in representation theory. In particular,we illustrate the application of these methods to systems of which the underlying Liealgebras are semisimple or reductive, while in these cases, the correspondence betweenLie bracket operations and permutation multiplications as well as graph operationsis elusive due to their complicated algebraic structures. In this work, we exploit theCartan and non-intertwining decompositions to decompose the system Lie algebrasinto simple components, so that the combinatorics-based controllability analysis isequivalently carried over to these components.This paper is organized as follows. In Section 2, we provide the preliminariesrelevant to our developments, including the LARC for systems on Lie groups and abrief review of the Lie algebra so ( n ) . In Section 3, we establish the symmetric groupand graph-theoretic methods based upon the study of bilinear systems on SO( n ) .In Section 4, we introduce the notions and tools of Cartan and non-intertwiningdecompositions for decomposing the system Lie algebras into simpler components,which enables and facilitates the generalization of the combinatorics-based frameworkto broader classes of bilinear systems. A brief review of the basics of symmetric groupsand Lie algebra decompositions can be found in the appendices.
2. Preliminaries.
To prepare for our development of the combinatorial con-trollability conditions, in this section, we briefly review the Lie algebra so ( n ) and theLARC for right-invariant bilinear systems. Meanwhile, we introduce the notations weuse throughout this paper. The LARC has been the most rec-ognizable tool, if not unique, for analyzing controllability of bilinear systems since the1970s. It establishes a connection between controllability and the Lie algebra gener-ated by the vector fields governing the system dynamics. In this paper, we primarily ocus on the bilinear system evolving on a compact and connected Lie group of theform,(2.1) ˙ X ( t ) = B X ( t ) + (cid:16) m X i =1 u i ( t ) B i (cid:17) X ( t ) , X (0) = I, where X ( t ) ∈ G is the state on a compact and connected Lie group G , I is the identityelement of G , B i are elements in the Lie algebra g of G , and u i ( t ) ∈ R are piecewiseconstant control inputs. For any subset Γ ⊆ g , we use Lie (Γ) to denote the Liesubalgebra generated by Γ , i.e., the smallest vector subspace of g containing Γ that isclosed under the Lie bracket defined by [ C, D ] := CD − DC for C, D ∈ g . With thesenotations, the LARC for the system in (2.1) can be stated as follows. Theorem
The system in (2.1) is controllable on G if and only if Lie (Γ) = g , where Γ = { B , B , . . . , B m } .Proof. See [2]. so ( n ) . The Lie algebra so ( n ) is a vector spaceof dimension n ( n − / , which consists of all n -by- n real skew-symmetric matrices.In particular, if we use Ω ij to denote the skew-symmetric matrix with in the ( i, j ) -thentry, − in the ( j, i ) -th entry, and elsewhere, then the set B = { Ω ij ∈ R n × n : 1 i < j n } forms a basis of so ( n ) , which we refer to as the standard basis of so ( n ) .The following lemma then reveals the Lie bracket relations among elements in B . Lemma
The Lie bracket of Ω ij and Ω kl satisfies the relation [Ω ij , Ω kl ] = δ jk Ω il + δ il Ω jk + δ jl Ω ki + δ ik Ω lj , where δ is the Kronecker delta function defined by δ mn = ( , if m = n ;0 , otherwise . Proof.
The proof follows directly from computations.The relations in Lemma 2.2 can also be equivalently expressed as [Ω ij , Ω kl ] = 0 if and only if i = k , i = l , j = k , or j = l . This algebraic structure facilitatescontrollability characterization of the bilinear system governed by the vector fieldsrepresented in the standard basis B , which is the main focus of the next section.
3. Combinatorics-Based Controllability Analysis for Bilinear Systems.
In this section, we introduce a combinatorics-based framework to characterize control-lability of bilinear systems. Within this framework, we adopt tools in two subfields ofcombinatorics - the symmetric group theory and graph theory, and build connectionsof Lie brackets of vector fields to permutation multiplications in symmetric groupsand operations on graph edges, respectively. Such connections enable us to charac-terize controllability in terms of permutation cycles and graph connectivity. Here, wewill investigate bilinear systems defined on
SO( n ) , given by(3.1) ˙ X ( t ) = Ω i j X + (cid:16) m X k =1 u i ( t )Ω i k j k (cid:17) X, Ω i k j k ∈ B , X (0) = I. as building blocks to establish this framework. Furthermore, we will show that owingto the special algebraic structure of so ( n ) presented in Lemma 2.2, the symmet-ric group and the graph-theoretic approach, when applied to (3.1), give an equiva-lent characterization of controllability through an interconnection between symmetricgroups and graphs. .1. The Symmetric Group Method for Controllability Analysis. In thissection, we introduce the symmetric group method for analyzing controllability of thesystem in (3.1). In this approach, a subset of vector fields in B is represented usinga permutation in S n , the symmetric group of n letters. Through this representation,we connect the Lie brackets of vector fields to permutation multiplications, so thatcontrollability is determined by the length of permutation cycles. For a brief summaryof symmetric groups and permutations, see Appendix A. To establish a relation fromLie brackets to permutation multiplications, we first define a relation between subsetsof B and permutations in S n by(3.2) ι : P ( B ) → S n , ι ( { Ω i j , Ω i j , . . . , Ω i m j m } ) = ( i j )( i j ) · · · ( i m j m ) . Because every permutation can be decomposed into a product of transpositions ( -cycles), the relation ι is surjective so that every subset of B admits a permutationrepresentation.To see how Lie bracket operations are related to permutation multiplications by ι , we illustrate the idea using two elements Ω ij , Ω kl ∈ B . On the Lie algebra level, if [Ω ij , Ω kl ] = 0 , then Lemma 2.2 implies that { i, j } and { k, l } have a common index.Without loss of generality, we may assume j = k and i = l , so that [Ω ij , Ω jl ] = Ω il .Meanwhile, on the symmetric group level, we have ι ( { Ω ij , Ω jl } ) = ( ij )( jl ) = ( ijl ) , sothe permutation multiplication increases the cycle length by , from the -cycle factors ( ij ) and ( jk ) to a -cycle ( ijk ) . However, if [Ω ij , Ω kl ] = 0 , then { i, j } ∩ { k, l } = ∅ ,and ι ( { Ω ij , Ω kl } ) = ( ij )( kl ) is a product of two disjoint cycles. The phenomenon thatelements in B with non-vanishing Lie brackets relating to a cycle with increased lengthextends inductively to larger subsets of B . To be more specific, if Γ ⊂ B contains m elements such that the iterated Lie brackets of them are non-vanishing, then ι (Γ) isan ( m + 1) -cycle. This observation immediately motivates the use of cycle length toexamine controllability of systems on SO( n ) in (3.1). Before we state and prove ourmain theorem, let us first illustrate the symmetric group method by two examples. Example
SO(5) , given by(3.3) ˙ X ( t ) = (cid:16) X i =1 u i ( t )Ω i,i +1 (cid:17) X ( t ) , X (0) = I, and let Γ = { Ω i,i +1 : i = 1 , . . . , } denote the set of control vector fields. The corre-spondence between Lie brackets in Γ and permutation multiplications in S follows(3.4) [Ω , Ω ] = Ω ↔ (12)(23) = (123) , [Ω , Ω ] = Ω ↔ (23)(34) = (234) , [Ω , Ω ] = Ω ↔ (34)(45) = (345) , [Ω , [Ω , Ω ]] = Ω ↔ (12)(234) = (1234) , [Ω , [Ω , Ω ]] = Ω ↔ (23)(345) = (2345) , [Ω , [Ω , [Ω , Ω ]]] = Ω ↔ (12)(2345) = (12345) . Note that successively Lie bracketing elements in Γ results in Ω , Ω , Ω , Ω , Ω ,and Ω , together with the elements in Γ , we have linearly independent vectorfields. Because so (5) is a -dimensional Lie algebra, we conclude Lie (Γ) = so (5) ,which implies that the system in (3.3) is controllable on SO(5) by the LARC. On the ther hand, (3.4) also shows ι (Γ) = (12345) , a cycle of maximum length in S . Thissuggests that controllability of systems on SO( n ) can be characterized by cycles of maximum length in the corresponding symmetric group. Example
SO(5) driven by three con-trols, given by(3.5) ˙ X ( t ) = (cid:0) u ( t )Ω + u ( t )Ω + u ( t )Ω (cid:1) X ( t ) , X (0) = I. In this case, the single Lie brackets, [Ω , Ω ] = Ω ↔ (12)(23) = (123) , [Ω , Ω ] = 0 ↔ (12)(45) , [Ω , Ω ] = 0 ↔ (23)(45) , and the double Lie brackets, [Ω , Ω ] = [[Ω , Ω ] , Ω ] = Ω ↔ (12)(23)(12) = (13) , [Ω , Ω ] = [Ω , [Ω , Ω ]] = Ω ↔ (23)(12)(23) = (13) , [Ω , Ω ] = [[Ω , Ω ] , Ω ] = 0 ↔ (12)(23)(45) = (123)(45) , result in a Lie subalgebra of dimension . Therefore, this system is not controllableon SO(5) . On the other hand, for
Γ = { Ω , Ω , Ω } , the computations above alsoshow ι (Γ) = (123)(45) , which is not a single cycle of maximum length in S .Examples 3.1 and 3.2 together verify the observation that cycles with the maxi-mum length characterize controllability of bilinear systems on SO( n ) , which we willprove in the next section. Remark ι introduced in (3.2) is not a well-definedfunction, because, for a given Γ ⊆ B , ι (Γ) depends on the ordering of the elements in Γ . If, say, Γ = { Ω , Ω , Ω , Ω , Ω } , then different element orderings, { Ω , Ω , Ω , Ω , Ω } ↔ (12)(14)(23)(24)(34) = (14) { Ω , Ω , Ω , Ω , Ω } ↔ (14)(12)(24)(23)(34) = (1234) could result in different permutations. Nevertheless, we can verify that for any Γ ⊆ B ,there always exists a subset Σ ⊆ Γ such that ι relates Σ to permutations with thesame (maximal) orbits, albeit different orderings of the elements in Σ . For example,for the subset Σ = { Ω , Ω , Ω } of Γ , ι (Σ) is always a -cycle with its orbit being { , , , } , regardless of its element orderings. The existence of such a subset will beclear once we develop a graph visualization of the permutations in Section 3.2. Leveraging the technique of mapping Lie brackets to permutations developed inthe previous section, we are able to characterize controllability of systems on
SO( n ) in terms of permutation cycles as shown in the following theorem. Theorem
The control system defined on
SO( n ) of the form (3.6) ˙ X ( t ) = (cid:16) Ω i j + m X k =1 u k ( t )Ω i k j k (cid:17) X ( t ) , X (0) = I, (same system as (3.1)) where Γ := { Ω i k j k } ⊆ B for k = 0 , . . . , m , is controllable ifand only if there is a subset Σ ⊆ Γ such that ι (Σ) is an n -cycle, where ι is the relationdefined in (3.2) . roof. By the LARC, the system in (3.6) is controllable on
SO( n ) if and only if Lie (Γ) = so ( n ) . Therefore, it is equivalent to showing that Lie (Σ) = so ( n ) if andonly if ι (Σ) is an n -cycle for some Σ ⊆ Γ .(Sufficiency): Suppose there exists a subset Σ ⊆ Γ such that ι (Σ) is an n -cycle.Because an n -cycle can be decomposed into a product of at least n − transpositions,this implies m > n − . Hence, it suffices to assume that the cardinality of Σ is n − , and, without loss of generality, let Σ = { Ω i j , . . . , Ω i n − j n − } , Because ι (Σ) is an n -cycle, it follows that the index set { i , j , . . . , i n − , j n − } = { , . . . , n } . Notethat the set { i , j , . . . , i n − , j n − } contains repeated elements. Next, we prove thesufficiency by induction.When n = 3 , suppose there exists a subset Σ = { Ω ij , Ω kl } ⊂ Γ and that ι (Σ) =( ij )( kl ) is a 3-cycle, so we must have one of the following: i = k , j = k , i = l , or j = l . Consequently, [Ω ij , Ω kl ] ∈ B\ Σ , so { Ω ij , Ω kl , [Ω ij , Ω kl ] } spans so (3) . Therefore,the system in (3.6) is controllable on SO(3) .Now let us assume that for n > , a system defined on SO( n − in the form of(3.6) is controllable if there is Σ ⊆ Γ such that ι (Σ) is an ( n − -cycle. Let Σ ⊆ Γ bea set of n − elements such that ι (Σ) = ( i n − j n − )( i n − j n − ) · · · ( i j ) is a cycle oflength n , then for every integer k n − , there exists some l n − such that { i k , j k } ∩ { i l , j l } 6 = ∅ . Consequently, there are n − transpositions of the form ( i k j k ) , k = 1 , . . . , n − , such that their product is a cycle of length n − . Without loss of gen-erality, we may assume that ι (Σ \{ Ω i n − j n − } ) = ( i n − j n − ) · · · ( i j ) is a ( n − -cyclewith the nontrivial orbit { i , j , . . . , i n − , j n − } = { , . . . , n − } . By the inductionhypothesis, the system in (3.6) is controllable on SO( n − ⊂ SO( n ) . Equivalently,any Ω ij ∈ B such that i < j n − can be generated by iterated Lie bracketsof the elements in Σ \{ Ω i n − j n − } . Because ι (Σ) = ( i n − j n − ) ι (Σ \{ Ω i n − j n − } ) is a n -cycle, we must have i n − ∈ { , . . . , n − } and j n − = n . Therefore, Ω kn can begenerated by the Lie brackets [Ω ki n − , Ω i n − j n − ] for any k = 1 , . . . , n − . As a result,the system in (3.6) is controllable on SO( n ) .(Necessity): Because the system in (3.6) is controllable, Lie (Γ) = so ( n ) . Then,there exists a subset Σ of Γ such that Lie (Σ) = so ( n ) and Σ contains no redundantelements , i.e., the elements that can be generated by Lie brackets of the other elementsin Σ . Without loss of generality, we assume Σ = { Ω i j , . . . , Ω i l j l } , where l m .By Lemma 2.2, for any Ω ab , Ω cd ∈ Σ , if [Ω ab , Ω cd ] = 0 , then there must exist abridging index, i.e., we must have one of the following cases: a = c , a = d , b = c ,or b = d . This, together with Lie (Σ) = so ( n ) , implies that the index set J of Σ is J = { i , j , . . . , i l , j l } = { , . . . , n } , and that for any Ω i k j k ∈ Σ , there exists some Ω i s j s ∈ Σ with s = k such that { i k , j k }∩{ i s , j s } 6 = ∅ . Moreover, because Σ contains noredundant elements, ι (Σ) = ι (Ω i l j l ) · · · ι (Ω i j ) is a cycle whose orbit contains everyelement in { , . . . , n } , namely, it is a cycle of length n . In addition, the cardinality of Σ is n − . Remark n − controls for thesystem on SO( n ) in (3.6) to be fully controllable and, on the other hand, for ι (Σ) , Σ ⊆ Γ , to reach a cycle of length n .Similar to the case in Theorem 3.4 for controllable systems, the controllable sub-manifold for an uncontrollable system also depends on the permutation related toa subset of Γ . To be more specific, the cycle decomposition of such a permutationdetermines the involutive distribution of the submanifold. Corollary
Given a system evolving on
SO( n ) in the form of (3.1) , let Ξ be a minimal subset of Γ , such that Lie (Ξ) = Lie (Γ) . If ι (Ξ) = σ · σ · · · σ l so hat each σ k , k l , are pairwise disjoint cycles with the nontrivial orbits O k ,then the controllable submanifold of the system is the Lie subgroup of SO( n ) withthe Lie algebra Lie (Γ) = L lk =1 span { Ω ij : i, j ∈ O k } . Conversely, if Lie (Γ) = L lk =1 span { Ω ij : i, j ∈ O k } for some O k ⊂ { , , . . . , n } , then ι (Ξ) = σ · σ · · · σ l and σ k are that disjoint cycles with nontrivial orbits O k .Proof. Let Ξ be a minimal subset of Γ such that Lie (Ξ) = Lie (Γ) and Ξ doesnot contain redundant elements. First, let σ = ι (Ξ) ∈ S n be a cycle with nontrivialorbit O , then Theorem 3.4 implies Lie (Ξ) = span { Ω ij : i, j ∈ O , i < j } . Next,if σ = σ · · · σ l is a permutation as a product of disjoint cycles σ , . . . , σ l with l > , then there exists a partition { Ξ , . . . , Ξ l } of Ξ such that ι (Ξ k ) = σ k for each k = 1 , . . . , l . Let O k denotes the nontrivial orbit of σ k for each k = 1 , . . . , l , then Lie (Ξ k ) = { Ω ij : i, j ∈ O k , i < j } and the sets O , . . . , O l are pairwise disjoint subsetsof { , . . . , n } . Hence, Lie (Ξ i ) ∩ Lie (Ξ j ) = { } holds for all i = j , and consequently,we have Lie (Ξ) = Lie (Ξ ) ⊕ · · · ⊕ Lie (Ξ l ) , where ⊕ denotes the direct sum of vectorspaces. By the Frobenius Theorem [26], Lie (Ξ) is completely integrable, and thatthe set of all its maximal integral manifolds forms a foliation F of SO( n ) . Since theinitial condition of the system in (3.6) is the identity matrix I , the leaf of F passingthrough I is the controllable submanifold of the system in (3.6). The converse isobvious following a very similar argument.According to Theorem 3.4 and Corollary 3.6, mapping the control vector fields in Γ to permutations provides not only an alternative approach to effectively examinecontrollability of systems defined on SO( n ) , but also a systematic procedure to char-acterize the controllable submanifold when the system is not fully controllable. Letus now revisit a previous example and see how permutations help determine systemcontrollability. Example
Γ = { Ω , Ω , Ω } such that ι (Γ) is a -cycle. In addition, the controllable submanifold is the integral manifoldof the involutive distribution ∆ = Lie { Ω X, Ω X, Ω X, Ω X } = span { Ω ij X : i, j ∈ { , , } or i, j ∈ { , }} , which can be identified by the nontrivial orbits of ι (Γ) = (1 , , , . On the other hand, for each X ∈ SO(5) , the complement ∆ ⊥ X = span { Ω ij X : i = 1 , , , j = 4 , } of the distribution evaluated at X containsthe bridging elements required for full controllability of this system. Graphsappear naturally in the research of networked systems, especially in modeling multi-agent systems and analyzing structural controllability [23, 24, 25]. However, mostgraph-theoretic methods were dedicated to studying networked control systems inexisting literature and were not invented and applied for understanding fundamentalproperties of a single bilinear system. Here, we use graphs to represent the structureof Lie algebras and then characterize controllability of bilinear systems by graphconnectivity. In contrast to the symmetric group method presented in Section 3.1, thisgraph-theoretic method establishes a correspondence between Lie bracket operationsof vector fields and operations on the edges of graphs.
A graph G , conventionally denotedby a 2-tuple, G = ( V, E ) , consists of a vertex set V and an edge set E . For the purposeof analyzing controllability of the system on SO( n ) , we are particularly interested insimple graphs, i.e., undirected graphs with no loops or multiple edges, of n vertices. ere, we denote the collection of such graphs G . Without loss of generality, we furtherassume that every graph in G has the same vertex set V = { v , . . . , v n } . Followingthese notations, we define a map(3.7) τ : P ( B ) → G by τ (Γ) = ( V, E Γ ) := G Γ , where P ( B ) denotes the power set of B , i.e., the set consisting of all subsets of B and E Γ = { v i v j : Ω ij ∈ Γ } . Some basic properties of τ are summarized in the followingproposition. Proposition τ ).(i) The map τ defined in (3.7) is bijective. (ii) For any Γ ⊆ B , | Γ | = | E Γ | holds, where | · | denote the cardinality of a set. (iii) Let K n denote the complete graph of n vertices, i.e., the graph whose verticesare pairwise adjacent, then τ ( B ) = K n .Proof. Note that (i) and (ii) directly follow from the definition of τ . For (iii), theedge set of τ ( B ) satisfies E B = { v i v j : Ω ij ∈ B} = { v i v j : 1 i < j n } = { v i v j : i, j = 1 , . . . , n } , and hence we conclude τ ( B ) = K n .The property (i) in Proposition 3.8 reveals a one-to-one correspondence betweenthe subsets of B and the graphs in G , which enables the representation of Lie bracketoperations by graph operations as follows.Algebraically, for any Ω ij , Ω jk ∈ B , Lemma 2.2 implies [Ω ij , Ω jk ] = Ω ik = 0 , sothat Lie { Ω ij , Ω jk } = span { Ω ij , Ω jk , Ω ik } . Graphically, by the definition of τ , thetwo edges τ (Ω ij ) = v i v j and τ (Ω jk ) = v j v k share a common vertex v j , and the edge τ ([Ω ij , Ω jk ]) = τ (Ω ik ) = v i v k intersects with τ (Ω ij ) and τ (Ω jk ) at endpoints v i and v k , respectively. Therefore, the three edges τ (Ω ij ) , τ (Ω jk ) , and τ ([Ω ij , Ω jk ]) forma triangle, or equivalently, τ ( { Ω ij , Ω jk , [Ω ij , Ω jk ] } ) = { v i v j , v j v k , v i v k } = K . Thisobservation, as summarized in the following lemma, reveals the relationship betweenfirst-order Lie brackets and graph operations for three standard basis elements of so ( n ) , which lays the foundation for the graph-theoretic controllability analysis ofbilinear systems. Lemma If Ω ij , Ω kl ∈ B satisfy [Ω ij , Ω kl ] = 0 , then (i) the two edges τ (Ω ij ) and τ (Ω kl ) are incident (i.e., they share a commonvertex); (ii) the three edges τ (Ω ij ) , τ (Ω kj ) , and τ ([Ω ij , Ω kl ]) form a triangle. To graphically characterize higher-order Lie brackets among arbitrary collectionsof standard basis elements of so ( n ) , we introduce the notion of triangular closure forgraphs, which generalizes the action of “forming triangles” in Lemma 3.9. Definition
Let G = ( V, E ) be a graph, and { G m =( V, E m ) : m = 0 , , . . . } be an ascending chain of graphs, i.e., G m ⊆ G m +1 for any m = 0 , , . . . , satisfying (i) G = G , i.e., E = E . (ii) For any m > , v i v j ∈ E m +1 if and only if v i v j ∈ E m or there exists somevertex v k ∈ V such that v i v k , v k v j ∈ E m .Then the union of all G m , denoted ¯ G = S ∞ m =1 G m , or equivalently, ¯ G = ( V, ¯ E ) =( V, S ∞ m =1 E m ) , is called the triangular closure of G . Moreover, a graph G is called triangularly closed if G = ¯ G . Note that for a finite graph G , i.e., G has finitely many vertices and edges, theascending chain of graphs G = G ⊆ G ⊆ · · · in Definition 3.10 stabilizes in finite teps, that is, there exists a nonnegative integer m such that G m = G m +1 = · · · ,which then implies ¯ G = G m . In particular, for a graph with n vertices, since it has atmost n ( n − / edges, its triangular closure can be obtained in at most n ( n − / steps. Remark transitive closure , and the equiva-lence will become transparent in the proof of Theorem 3.18. The triangular closure weintroduce here imitates the computations of graded Lie brackets/algebras in a morenatural way, so that all orders of Lie brackets can be calculated in a graph.Recall from Lemma 3.9 that given a subset Γ ⊆ B and its associated graph G = τ (Γ) , taking first-order Lie brackets of the elements in Γ corresponds to addingedges that connect the endpoints of incident edges in G . Applying this procedure to G = G , as defined in Definition 3.10, exactly results in G . Inductively, successivelyLie bracketing the elements in Γ up to order m will generate the graph G m , as shownbelow. Theorem
Given a subset Γ ⊆ B , let Γ ⊆ Γ ⊆ · · · be an ascending chainof subsets of B such that Γ = Γ , Γ = [Γ , Γ ] S Γ , . . . , Γ m +1 = [Γ m , Γ m ] S Γ m , . . . ,where [Γ m , Γ m ] = { [ A, B ] :
A, B ∈ Γ m } . Then G m = τ (Γ m ) holds for all m = 0 , , . . . Proof.
This follows immediately from the definitions of G m and Γ m .Recall that for any finite G ∈ G , G m stabilizes to ¯ G in finite steps. Meanwhile,by Theorem 3.12, Γ m also stabilizes to a subset ˆΓ ⊆ B which must satisfy ¯ G = τ (ˆΓ) .Intuitively, ˆΓ is supposed to contain all the elements that can be generated by theiterated Lie brackets of the elements in Γ , because ¯ G is the largest graph generatedby G . This conclusion is then rigorously verified in the following corollary. Corollary
Let Γ be a subset of B and G = τ (Γ) be the graph associatedwith Γ . If ˆΓ ⊆ B satisfies τ (ˆΓ) = ¯ G , then Lie (Γ) = span (ˆΓ) .Proof.
Let m be a nonnegative integer satisfying G m = ¯ G , then Theorem 3.12implies that ˆΓ = Γ m , hence Γ r = ˆΓ holds for all r > m . Consequently, by thedefinition of Lie (Γ) , we have
Lie (Γ) = span ( S ∞ i =0 Γ i ) = span (Γ m ) = span (ˆΓ) .For the purpose of controllability analysis, the subsets of B generating the wholeLie algebra so ( n ) is of great interest. Therefore, we characterize such subsets by theirassociated graphs below, which is also a special case of Corollary 3.13. Corollary
Consider a subset Γ ⊆ B with the associated graph G = τ (Γ) ,then Lie (Γ) = so ( n ) if and only if ¯ G = K n .Proof. (Sufficiency): Let ˆΓ ⊆ B satisfy ¯ G = τ (ˆΓ) = K n , then the properties(i) and (iii) in Proposition 3.8 imply ˆΓ = B . Consequently, Lie (Γ) = span (ˆΓ) =span ( B ) = so ( n ) by Corollary 3.13.(Necessity): If Lie (Γ) = so ( n ) , then there exists some nonnegative integer m suchthat Γ m = B . By Theorem 3.12, we obtain ¯ G ⊇ G m = τ (Γ m ) = K n . On the otherhand, because of ¯ G ⊆ K n , we conclude ¯ G = K n .Furthermore, Corollary 3.14 sheds light on a graph representation of controllabil-ity, which in turn can be characterized in terms of graph connectivity. In the followingsection, we will rigorously investigate this observation. The relationship between Lie brackets and graph operations developed in Sec-tion 3.2.1 enables us to employ graph theory techniques to analyze controllability of ystems on SO( n ) as in (3.1). In particular, motivated by the connection between aLie subalgebra and its associated graph presented in Corollary 3.14, controllabilitycan be analyzed through the notion of triangular closure defined in Definition 3.10. Proposition
The bilinear system in (3.1) is controllable on
SO( n ) if andonly if τ (Γ) = K n , where τ is defined as in (3.7) , Γ = { Ω i j , . . . , Ω i m j m } , and K n isa complete graph of n vertices.Proof. By the LARC shown in Theorem 2.1, the system in (3.1) is controllableon
SO( n ) if and only if Lie (Γ) = so ( n ) , which is equivalent to τ (Γ) = K n by Corol-lary 3.14.Using the following two examples, we will verify Proposition 3.15 and draw aparallel between examining the LARC and generating triangular closure of the graphassociated with the considered system. This comparison in turn illuminates a graphicvisualization of the algebraic procedure of generating Lie algebras for the set of driftand control vector fields. Example
SO(4) given by(3.8) ˙ X ( t ) = ( u Ω + u Ω + u Ω + u Ω ) X ( t ) , X (0) = I. Applying τ to the set of the control vector fields Γ results in its associated graph G = ( V, E ) as follows, Γ = { Ω , Ω , Ω , Ω } τ ←→ { v v , v v , v v , v v } = E. Because the first order Lie brackets [Ω , Ω ] = Ω and [Ω , Ω ] = Ω are notin Γ , we have Γ = Γ ∪ { Ω , Ω } . Correspondingly, according to Corollary 3.13, G = ( V, E ) can be obtained by applying τ to Γ , i.e., Γ = Γ ∪ { Ω , Ω } τ ←→ { v v , v v } ∪ E = E . Notice that span (Γ ) = so (4) and simultaneously G = ¯ G = K , which concludescontrollability of the system in (3.8) from both algebraic and graph-theoretic per-spectives. The graphs G and G are shown in Figure 3.1. In particular, the two rededges in G , which are not in G , correspond to the elements in [Γ , Γ] . v v v v v v v v G G Fig. 3.1 . The graph G associated with the system (3.8) in Example and its triangularclosure G . Note that the red edges in G correspond to the vector fields generated by the first-orderLie brackets of the control vector fields in Γ . Example 3.16 presents a controllable system whose associated graph has a com-plete triangular closure, which in turn validates the sufficiency of Proposition 3.15.The necessity is illustrated using the following example through an uncontrollablesystem.
Example
SO(5) driven by three control inputs, givenby(3.9) ˙ X ( t ) = ( u Ω + u Ω + u Ω ) X ( t ) , X (0) = I, nd let Γ = { Ω , Ω , Ω } denote the set of control vector fields. Some straightfor-ward calculations yield the Lie algebra Lie (Γ) = span { Ω , Ω , Ω , Ω , Ω , Ω } ,which has dimension . Therefore, the system in (3.9) is not controllable, since dim so (5) = 10 . Using the graph-theoretic approach, Figure 3.2 shows the proce-dure of generating ¯ H from H = τ (Γ) . In particular, ¯ H = H shown in Figure 3.2 isnot complete, which verifies the necessity of Proposition 3.15. H H H v v v v v v v v v v v v v v v
99K 99K
Fig. 3.2 . The graph visualization of Lie bracketing control vector fields of the system in (3.9) in Example . Specifically, the graph H is associated with the set of control vector fields, H visualizes the first-order Lie brackets, and H is the triangular closure of H . Note that the red edgescorrespond to the vector fields in Γ generated by Lie brackets. It is worth noting that the graph G in Figure 3.1 associated with the controllablesystem in (3.8) is connected, but the graph H in Figure 3.2 associated with theuncontrollable system in (3.9) is not. This observation inspires the characterizationof controllability for systems on SO( n ) by graph connectivity. Theorem
The system in (3.1) is controllable on
SO( n ) if and only if τ (Γ) is connected, where Γ = { Ω i j , . . . , Ω i m j m } and τ (Γ) is the graph associated with Γ .Proof. Owing to Proposition 3.15, it suffices to prove that the triangular closureof τ (Γ) is complete if and only if τ (Γ) is connected.(Sufficiency): Suppose that G = τ (Γ) = ( V, E ) is connected, then there is a pathin G from v i to v j for any v i , v j ∈ V , say v i w w · · · w k v j with w , . . . , w k ∈ V .Therefore, we have v i w ∈ E , . . . , v i w k ∈ E k − and v i v j ∈ E k ⊆ ¯ E . Since v i , v j ∈ V are chosen arbitrarily, we conclude that the triangular closure ¯ G contains all edges v i v j ,hence ¯ G = K n . In addition, this process of generating ¯ G is illustrated in Figure 3.3with the case of k = 5 . v i w w w w w v j Fig. 3.3 . Illustration of the proof of sufficiency of Theorem . (Necessity): We assume that the triangular closure ¯ G of G = τ (Γ) is complete. Ifthere exists an edge v i v j not in G , since v i v j is in ¯ G = ( V, ¯ E ) , we may then assume v i v j ∈ E k and v i v j E k − for some positive integer k . Hence, by Definition 3.10,there is some vertex w such that v i w , w v j ∈ E k − , i.e., there exists a path v i w v j in G k − connecting v i and v j . Repeating this procedure results in a path in G connecting v i and v j , which implies the connectivity of G , and hence the proof is done. emark τ , that is, Lie (Γ) = so ( n ) for some Γ ⊆ B if andonly if τ (Γ) is connected , which is an equivalent formulation of Theorem 3.18.Because a connected graph with n vertices contains at least n − edges, The-orem 3.18 also identifies the minimum number of control inputs for the system in(3.1) to be controllable, as identified using the symmetric group method presented inTheorem 3.4 and Remark 3.5. Corollary
If a system on
SO( n ) in (3.1) is controllable, then the numberof control inputs m is at least n − , i.e., m > n − . Although Theorem 3.18 is developed to examine controllability, it also helps es-tablish some general facts in graph theory from the control systems perspective. Inthe following, we present one such result that is related to triangular closures. Thisproperty also plays an important role in characterizing controllable submanifolds foruncontrollable systems by connected component of the graph associated with thecontrol system.
Lemma
The triangular closure ¯ G of a graph G is a disjoint union of itscomplete components.Proof. The proof is a direct application of the proof of Theorem 3.18 to eachconnected component of G .By the above Lemma 3.21, we can adopt our main result in Theorem 3.18 to studyan uncontrollable system by taking the triangular closure of its associated graph,which is the union of the triangular closures of all connected components. Theorem
The controllable submanifold of the system in (3.1) is determinedby the connected components of its associated graph.Proof.
Let Γ ⊆ B be the set of vector fields governing the dynamics of the systemin (3.1), G = τ (Γ) be the graph representation of Γ , and ¯ G denote the triangularclosure of G . Since connected components of G determine the complete componentsof ¯ G , it suffices to show that the controllable submanifold of the system is determinedby the complete components of ¯ G .According to the Frobenius Theorem [26], the controllable submanifold of the sys-tem in (3.1) is the maximal integral submanifold of Lie (Γ) passing through the identitymatrix I . Hence, by Lemma 3.21, because of the completeness of each component of ¯ G ,the set τ − ( ¯ G ) ⊆ B is closed under Lie bracket, which implies span τ − ( ¯ G ) = Lie (Γ) . Therefore, we conclude that
Lie (Γ) , and thus its maximal integral submanifold, isdetermined by ¯ G .Theorem 3.22 further reveals a one-to-one correspondence between the Lie alge-bra generated by a subset of B and the triangular closure of its associated graph in G .Leveraging this one-to-one correspondence, we are able to give an explicit character-ization of controllable submanifolds for uncontrollable systems in terms of connectedcomponents of their associated graphs. Example
SO(6) in the form of (3.1) governed by the vector fields Γ = { Ω , Ω , Ω , Ω } and Γ = { Ω , Ω , Ω , Ω } , respectively. Figure 3.4 shows their associated graphs G = τ (Γ ) and G = τ (Γ ) , neither of which is connected. Therefore, by Theo-rem 3.12, both systems are not controllable on SO(6) . On the other hand, we noticethat G = G . So by Theorem 3.22, the two systems have the same controllable ubmanifold. Specifically, the controllable submanifold is the Lie subgroup of SO(6) with the Lie algebra
Lie (Γ ) = Lie (Γ ) = span { Ω ij : 1 i < j } ⊕ span { Ω ij : 4 i < j } . Moreover, both G and G contain two complete components with the vertex sets U = { v , v , v } and W = { v , v , v } , which are also the vertex sets of the connectedcomponents of G (or G ). It then follows that the Lie algebra of the controllablesubmanifold, span { Ω ij : v i , v j ∈ U } ⊕ span { Ω ij : v i , v j ∈ W } , can be explicitlycharacterized by the vertex sets of the complete components of G and G , as well asthe connected components of G and G . v v v v v v v v v v v v G G v v v v v v v v v v v v G G Fig. 3.4 . The graphs and their triangular closures associated with the systems in Example .Specifically, the graphs G and G on the left are associated with the systems governed by Γ and Γ , respectively, and their triangular closures G and G are on the right. Red edges correspond tovector fields generated by Lie brackets. Furthermore, the developed method of characterizing controllability in terms ofgraph connectivity is not constrained to systems defined on Lie groups. In particular,as shown in the following example, it can be applied to study formation control ofmulti-agent systems defined on graphs. From an algebraic perspective, it is equivalentto using the graph-theoretic method to analyze the Lie algebra generated by symmetricmatrices . Example N agents with the coupling topology given by the graph G = ( V, E ) , V = { v , . . . , v N } , is generally represented by(3.10) ˙ x i ( t ) = X j ∈ V ( i ) u ij ( x j − x i ) , i N, where x i ( t ) ∈ R n denotes the state of the i -th agent, V ( i ) = { j N : v i v j ∈ E } denotes the set of neighboring agents of i , and u ij = u ji are the external inputs thatcontrol the reciprocal interaction between the i -th and j -th agents [6]. e will first formulate the dynamic law in (3.10) into a matrix form, and thenapply our analysis on a Lie algebra associated with it. To do this, let A ij := E ii + E jj − E ij − E ji be an N -by- N symmetric matrix with zero row and column sums, and let X ∈ R N × n denote a matrix whose row vectors are the states of the agents: X = x ⊺ ... x ⊺ N . Then, we can rewrite (3.10) into the following matrix form,(3.11) ˙ X = X v i v j ∈ E u ij A ij X. The formation controllability of the multi-agent system in (3.11) is determined bythe LARC [6]. Thus, to study this system, we need to know the algebraic structureof matrices { A ij } . Observe that for B ijk = − (Ω ij + Ω jk + Ω ki ) and distinct indices i, j, k, l, m N , we have(3.12) [ A ij , A jk ] = B ijk , [ B ijk , A ij ] = 2( A ik − A jk ) , [ B ijk , A il ] = − A ij + A jl + A ik − A kl , [ B ijk , B ijl ] = B ikl + B jkl , [ B ijk , B ilm ] = B jlm + B kml = B lkj + B mjk . Therefore, the Lie algebra g := Lie { A ij } has a decomposition, g = g ⊕ g − , with g = span { B ijk } and g − = span { A ij } . As a consequence, by the LARC, con-trollability of system (3.11) depends on whether the set Γ := { A ij : ( i, j ) ∈ E } generates the Lie algebra g . Similar to bilinear systems on SO( n ) , we can adopta graph-theoretic method for g by associating one part of g , i.e., g − , to a graph,which in the case of this example, coincides with the graph on which the system isdefined. To be more specific, for a complete graph K N and its set of edges E , wemay define a map τ : Γ → E , which sends A ij ∈ Γ to v i v j ∈ E , so that the imageof Γ is exactly the graph G . Following the correspondence τ , for two adjacent edges v i v j and v j v k , since Lie { A ij , A jk } = Lie { A ij , A jk , A ki } = span { A ij , A jk , A ki , B ijk } ,the triangle with edges v i v j , v j v k , v k v i in G represents the Lie subalgebra spanned by { A ij , A jk , A ki , B ijk } . More generally, by the algebraic relations in (3.12), any trian-gularly closed subgraph of K N is associated with a subalgebra of g . Therefore, theLie (sub)algebra generated by Γ can be represented by the triangular closure of G ;and if G is connected, then its triangular closure is complete, which suggests that Lie (Γ) contains all A ij ’s, so we have Lie (Γ) = g . In conclusion, the controllability ofsystem (3.10), and equivalently, system (3.11), is determined by graph connectivityof G .By now, we have conducted a detailed investigation into controllability of bilinearsystems on SO( n ) governed by the standard basis elements of so ( n ) . Before we extendthe scope of our investigation to general bilinear systems, we show that, in contrastto Corollary 3.20, a driftless bilinear system on SO( n ) can be controllable using onlytwo control inputs, for all n > . xample SO( n ) with control vector fields in the standard basis of so ( n ) require at least n − inputs tobe controllable. However, this conclusion may not hold for general systems governedby vector fields not in the standard basis. For example, the following system with twocontrol inputs(3.13) ˙ X ( t ) = (cid:2) u ( t ) C + u ( t ) C (cid:3) X ( t ) , X (0) = I, where C = Ω and C = P n − i =1 Ω i,i +1 , is controllable on SO( n ) . To see this, wewill show Ω k ∈ Lie ( { C , C } ) for any k n by induction. At first, note that Ω = [ C , C ] ∈ Lie ( { C , C } ) . Next, we assume Ω , Ω , . . . , Ω k ∈ Lie ( { C , C } ) for some k < n , which is the induction hypothesis. Consequently, we have [Ω k , C ] = Ω k − Ω ,k − + Ω ,k +1 and [Ω k , C ] = Ω k , which implies Ω ,k +1 =[Ω k , C ] − [Ω k , C ] + Ω ,k − ∈ Lie ( { C , C } ) . By induction, we conclude Ω k ∈ Lie ( { C , C } ) for any k n . This result implies Lie (Σ) ⊆ Lie ( { C , C } ) ,where Σ = { Ω k : 1 k n } . Obviously τ (Σ) is a connected graph, and hence byTheorem 3.18, Lie (Σ) = so ( n ) , and the the system in (3.13) is thus controllable. In Sections 3.1 and 3.2, we developed two combinatorics-based methodsto analyze controllability of bilinear systems. Both methods connect the Lie brack-ets of vector fields to operations on combinatorial objects. We will show next thatan equivalence exists between the symmetric group and the graph-theoretic methodwhen systems on
SO( n ) are concerned. We first illustrate this equivalence through acontrollable system on SO(4) . Example
Γ = { Ω , Ω , Ω , Ω } . We have shown therein that this systemis controllable on SO(4) by using the graph-theoretic method; and for the symmetricgroup method, we may choose Σ = { Ω , Ω , Ω } ⊂ Γ so that ι (Σ ) = (1342) isa -cycle. However, Σ is not the only subset that is related to a -cycle, and, forexample, one can easily verify that ι also relates the subsets Σ = { Ω , Ω , Ω } and Σ = { Ω , Ω , Ω } to -cycles as ι (Σ ) = (13)(23)(34) = (1342) and ι (Σ ) =(12)(23)(34) = (1234) . Moreover, it is worth noting that Σ , Σ , and Σ are theonly subsets of Γ that are related to -cycles. Meanwhile, and more importantly,their graph representations τ (Σ ) , τ (Σ ) , and τ (Σ ) coincide with all three spanningtrees of the graph τ (Γ) associated with the system (see Table 3.1). On the otherhand, from the aspect of Lie algebra, we observe that Σ i is a minimal subset of Γ generating Lie (Γ) for each i = 1 , , , that is, Σ ′ = Σ i for any Σ ′ ⊆ Σ i satisfying Lie (Σ ′ ) = Lie (Γ) . This observation sheds light on the general result: given a systemon SO( n ) governed by the set of vector fields Γ , if Σ is a minimal subset of Γ with Lie (Σ) = Lie (Γ) , then ι (Σ) is an n -cycle if and only if τ (Σ) is a spanning tree of τ (Γ) . Theorem
Consider a bilinear system on
SO( n ) as in (3.1) and let Γ ⊆ B denote the set of vector fields governing the system dynamics. Suppose Σ ⊆ Γ is a minimal subset such that ι (Σ) = σ ∈ S n is an n -cycle (i.e., Σ has no proper subset thatis also related to an n -cycle via ι ), then its associated graph τ (Σ) is a spanning tree of τ (Γ) , and the system is therefore controllable. Conversely, for a controllable system,any spanning tree T of the connected graph τ (Γ) corresponds to a subset Σ ′ = τ − ( T ) ,such that Σ ′ ⊆ Γ is minimal and that ι (Σ ′ ) is an n -cycle in S n .Proof. From group theory we know that a minimal Σ with ι (Σ) being an n -cycleshould consist of n − transpositions, and that the union of the orbits of all n − et of control vector fields Graph Permutation in S Γ = { Ω , Ω , Ω , Ω } v v v v ι (Γ) = (12)(23)(13)(34) = (234)Σ = { Ω , Ω , Ω } v v v v ι (Σ ) = (12)(13)(34) = (1342)Σ = { Ω , Ω , Ω } v v v v ι (Σ ) = (13)(23)(34) = (1342)Σ = { Ω , Ω , Ω } v v v v ι (Σ ) = (12)(23)(34) = (1234) Table 3.1
A comparison between two methods analyzing controllability: the symmetric groups method andthe graph-theoretic method. Note that the graphs associated with Σ , Σ and Σ are spanning trees of the associated graph of Γ , and that any tree is related to a -cycle in the symmetric group S . transpositions is the orbit of σ . This means the graph τ (Σ) has n vertices and n − edges. Since a graph with n vertices and n − edges is both connected and acyclic ,and since τ (Σ) covers all n vertices of τ (Γ) , we conclude that τ (Σ) is the spanningtree of τ (Γ) .On the other hand, for a subset Σ ′ ⊆ Γ satisfying that τ (Σ ′ ) is a spanning treeof τ (Γ) , we must have | Σ ′ | = n − . Since a decomposition of an n -cycle needs atleast n − transpositions, if ι (Σ ′ ) is an n -cycle, then Σ ′ is obviously minimal. Thefollowing claim shows that ι (Σ ′ ) is indeed an n -cycle, regardless of the ordering ofelements in Σ ′ . Claim.
A tree consisting of k edges in the connected graph τ (Γ) in Theorem is related to a ( k + 1) -cycle via ι , regardless of the ordering of transpositions.Proof of Claim. Let us consider a tree T with k edges in τ (Γ) , and prove the claimby induction. It is trivial for k = 1 ; and for k = 2 , say T = v j v j v j , then ι sends T to either ( j j j ) or ( j j j ) , depending on the orderings of ( j j ) and ( j j ) . Assumethe claim is true for k = l − for some l ∈ Z + ; and for a tree T with k = l edges, wecan choose a subtree T ′ of T that consists of l − edges. Without loss of generality,we may assume T ′ has vertices { v , . . . , v l } and that T has an additional vertex v l +1 and an additional edge v v l +1 . Let { σ , . . . , σ l − } be a set of transpositions such thateach σ i is related to a distinct edge in T ′ by ι , and let ρ = (1 , l + 1) denote thetransposition related to the additional edge v v l +1 in T . Our goal is to show that thepermutation(3.14) σ i · · · σ i t ρσ i t +1 · · · σ i l − is an ( l + 1) -cycle. Note that for any i, j , j l , we have the following law of ommutation:(3.15) ( i, l + 1)( j j ) = ( ( j j )( i, l + 1) if neither j or j equals to i ; ( j j )( j , l + 1) if j = i .Therefore, we can rewrite the permutation (3.14) as σ i · · · σ i l − ρ ′ , where ρ ′ = ( j p , l +1) for some j p l . By our assumption, σ i · · · σ i l − is an l -cycle: σ i · · · σ i l − =( j j · · · j l ) , so finally we have σ i · · · σ i t ρσ i t +1 · · · σ i l − = σ i · · · σ i l − ρ ′ = ( j j · · · j l )( j p , l + 1)= ( j · · · j p , l + 1 , j p +1 · · · j l ) , which is an ( l + 1) -cycle. It is clear that the ordering of transpositions is irrelevant inour proof. (cid:7) Therefore, a spanning tree τ (Σ ′ ) consisting of n − edges is related to an n -cyclein S n via ι , which finishes our proof.Given a controllable system on SO( n ) , Theorem 3.27 reveals the relation between n -cycles and spanning trees of the associated graph. In particular, for such a systemgoverned by the set Γ ⊆ B of vector fields, this theorem supplements Theorem 3.18by explicitly describing the subsets of Γ that are related to n -cycles using graphs.The following corollary then summarizes all the symmetric group and graph-theoreticcharacterizations of controllability for systems on SO( n ) . Corollary
Consider a bilinear system defined on
SO( n ) as in (3.1) , andlet Γ denote the set of vector fields governing the system dynamics. The following areequivalent: (1) The system is controllable on
SO( n ) . (2) τ (Γ) is a connected graph. (3) For any minimal subset Σ ⊆ Γ generating so ( n ) , ι (Σ) is an n -cycle and τ (Σ) is a spanning tree of τ (Γ) . In the remainder of this section, we will focus on uncontrollable systems. Re-call Theorem 3.22 that the controllable submanifold for an uncontrollable system on
SO( n ) is determined by the connected components of its associated graph. Mean-while, according to Corollary 3.6, by applying the method of symmetric groups, thecontrollable submanifold can also be characterized by the nontrivial orbits of ι (Ξ) for a minimal subset Ξ ⊆ Γ generating Lie (Γ) . To see that the two methods areequivalent and to extend Theorem 3.27 to uncontrollable cases, we first introduce theconcept of spanning forests , which generalizes the notion of spanning trees to dis-connected graphs. Given a (disconnected) graph, its spanning forest is a maximalacyclic subgraph, or equivalently, a subgraph consisting of a spanning tree in eachconnected component of the graph [1]. Following this definition, we will show thatthe minimal subset Ξ ⊆ Γ in Corollary 3.6 corresponds to a spanning forest of τ (Γ) ,so that the controllable submanifold can also be equivalently described by the con-nected components of the spanning forest. This result is illuminated in the followingexample. Example
SO(6) in the form of (3.1) governedby the set of vector fields
Γ = { Ω , Ω , Ω , Ω , Ω , Ω } . As shown in Table 3.2, ι (Γ) is disconnected with two components, and hence this system is not controllable on SO(6) . To describe its controllable submanifold, we choose a spanning forest τ (Ξ ) of et of control vector fields Graph Permutation in S andits nontrivial orbits Γ = n Ω , Ω , Ω , Ω , Ω , Ω o v v v v v v ι (Γ) = (12)(14)(23)(24)(34)(56)= (14)(56) Orbits = { , } , { , } Ξ = { Ω , Ω , Ω , Ω } v v v v v v ι (Ξ ) = (14)(24)(34)(56)= (1432)(56) Orbits = { , , , } , { , } Ξ = { Ω , Ω , Ω , Ω } v v v v v v ι (Ξ ) = (12)(24)(34)(56)= (1243)(56) Orbits = { , , , } , { , } Table 3.2
A comparison between the symmetric group method and the graph-theoretic method for anuncontrollable system on
SO(6) . Both graphs associated with subsets Ξ and Ξ are spanning forest of the associated graph of Γ . the associated graph τ (Γ) with Ξ = { Ω , Ω , Ω , Ω } . Note that the permutation ι (Ξ ) = (14)(24)(34)(56) = (1432)(56) has two nontrivial orbits: O = { , , , } and O = { , } , each corresponds to a connected component of the graph τ (Γ) , orequivalently, a summand in the decomposition of the Lie algebra of the controllablesubmanifold: Lie (Γ) = span { Ω ij : i, j ∈ O } ⊕ span { Ω ij : i, j ∈ O } . Now suppose we choose a different spanning forest τ (Ξ ) which corresponds toanother subset Ξ = { Ω , Ω , Ω , Ω } ⊆ Γ . Note that the permutation ι (Ξ ) =(1243)(56) is different from ι (Ξ ) , but both have the same orbits. The graphs andpermutations associated with Γ and its subsets Ξ and Ξ are also listed in Table 3.2.In general, for a spanning forest F of τ (Γ) , we know by Theorem 3.27 that eachtree T i consisting of n i vertices in F is related to an n i -cycle via ι , which characterizesa summand of the decomposition of Lie (Γ) . So by applying Theorem 3.27 to each(maximal) tree in the forest F , we have the following Corollary 3.30, which describesthe relation between the associated graphs and permutations for an uncontrollablebilinear system. Corollary
Given an uncontrollable bilinear system defined on
SO( n ) inthe form of (3.1) governed by the set of vector fields Γ . Let F be a spanning forestof τ (Γ) and if we denote Ξ = τ − ( F ) , then Ξ is a minimal subset of Γ with the samegenerating Lie algebra and the controllable submanifold of the system is determinedby the nontrivial orbits of ι (Ξ) . roof. For a spanning forest F of τ (Γ) , let T , . . . , T l be (maximal) trees in F s.t. F = T ⊔ · · · ⊔ T l , where T i = ( V i , E i ) with | V i | = n i , | E i | = n i − . By Theorem 3.27,each T i is related to an n i -cycle ι (Ξ i ) ∈ S n i for Ξ i = τ − ( T i ) , and the orbit of ι (Ξ i ) determines the Lie (sub)algebra g i := Lie (Ξ i ) . Therefore, distinct orbits of ι (Ξ) consist of the orbits of each ι (Ξ ) , . . . , ι (Ξ l ) , which determines the Lie algebragenerated by Γ : Lie (Γ) = Lie (Ξ) = g i ⊕ · · · ⊕ g l . Since the controllable submanifoldis determined by the Lie subalgebra Lie (Γ) = Lie (Ξ) , we conclude that it is alsodetermined by distinct orbits of ι (Ξ) , for Ξ = τ − ( F ) .
4. Combinatorics-Based Controllability Analysis via Lie Algebra De-compositions.
Utilizing the algebraic structure of so ( n ) , we have developed combi-natorial methods that identified vector fields in the standard basis of so ( n ) , as well asvector fields generating structured Lie algebras, e.g., the multi-agent system describedin Example 3.24, with transpositions in S n and edges of n -vertices graphs. It was alsoshown that such identifications lead to an equivalence between the two methods foranalyzing controllability of systems on SO( n ) as defined in (3.1).However, in many cases, the system Lie algebra may be too complicated to asso-ciate each of its elements to a permutation or a graph edge, so that the combinatorialmethods cannot be directly applied. This dilemma can be resolved through the de-composition of the Lie algebra into components with simpler algebraic structuressuch that the combinatorial methods can be applied to each component. This ideaallows us to generalize the combinatorial framework to bilinear systems defined onboarder classes of Lie groups. To this end, we adopt techniques in representationtheory, including the Cartan and non-intertwining decomposition. Some basics ofrepresentation theory can be found in Appendix B. The Cartandecomposition, named after the influential French mathematician Élie Cartan, pro-vides a major tool for understanding the algebraic structures of semisimple Lie groupsand Lie algebras. Its generalized form, the root space decomposition, decomposes aLie algebra into a direct sum of vector subspaces, called the root spaces, as introducedin Appendix B. However, each root space is not necessarily a Lie subalgebra, i.e., Liebracket operations may not be closed in the root spaces. This nature of the Cartan(root space) decomposition then disables the use of the graph-theoretic method sinceit violates the “triangle rule” shown in Lemma 3.9 (ii). As a result, here we pursueand generalize the symmetric group method to analyze controllability of systems withits vector fields living in the root spaces of semisimple Lie algebras.In representation theory, the Lie algebra sl (3 , C ) , which consists of × complexmatrices with vanishing trace, serves as a primary example to illustrate the Cartandecomposition of semisimple Lie algebras. Therefore, to illustrate our idea, we con-sider the driftless bilinear system evolving on the Lie group SL(3 , C ) consisting of × complex matrices with determinant , given by(4.1) ˙ Z ( t ) = (cid:16) m X j =1 u j ( t ) B j (cid:17) Z ( t ) , where the state Z ( t ) ∈ SL(3 , C ) , the control vector fields B j ∈ Γ ⊆ B ′′ := { H k , X l , Y l : k = 1 , l = 1 , , } , the basis of sl (3 , C ) with H = − , H = − , = , X = , X = ,Y = , Y = , Y = , and the control inputs u j ( t ) ∈ C .One can easily check that the two Lie subalgebras k = Lie { H , X , Y } and k = Lie { H , X , Y } , when considered as Lie algebras over R , are isomorphic to so (3) .As discussed in Section 3.1, controllability of systems on SO(3) can be characterizedby permutation cycles in S . This suggests that we can characterize controllabilityof systems on SL(3 , C ) by two copies of S . Formally, we want to establish a map ι : P ( B ′′ ) → S ⊕ S , where ⊕ denotes the direct sum of groups, so that non-vanishingLie brackets correspond to cycles with increased length. In this case, we define anelement σ = ( σ , σ ) in S ⊕ S to be a cycle if both σ and σ are cycles in S ,and the length of σ is defined to be the sum of the length of σ and σ . Here is onepossible definition of ι : H ( e, e ) , H ( e, e ) ,X ((12) , e ) , X ( e, (12)) , X ((12) , (12)) ,Y ((23) , e ) , Y ( e, (23)) , Y ((23) , (23)) , where e denotes the identity of S . Following this definition of ι , we can check thatif B , B ∈ B ′′ satisfy [ B , B ] = 0 , then the length of ι ([ B , B ]) is greater than orequal to the length of both ι ( B ) and ι ( B ) . Moreover, if neither B nor B is equalto H or H , then the length of ι ([ B , B ]) is strictly greater than the length of both ι ( B ) and ι ( B ) . This relation between Lie brackets of elements in B ′′ and length ofcycles in S ⊕ S allows us to draw the following conclusion: Proposition
The system in (4.1) is controllable on
SL(3 , C ) if and only ifthere exists a subset Σ of Γ = { B , . . . , B m } such that ι (Σ) is a 6-cycle in S ⊕ S . From the perspective of representation theory, the basis B ′′ induces the Cartandecomposition of the Lie algebra sl (3 , C ) , in which the -dimensional Cartan subal-gebra is spanned by H and H . Moreover, the Weyl group of sl (3 , C ) is S . Theabove facts provide another explanation for requiring two copies of S in the charac-terization of controllability for systems on SL(3 , C ) governed by vector fields in B ′′ .Notice that the concepts of Cartan subalgebras and Weyl groups are well-defined forall semisimple Lie algebras, not only for SL(3 , C ) . Also, Weyl groups are all finitegroups and thus subgroups of some symmetric groups. As a result, it is possibleto extend the symmetric-group characterization of controllability to systems definedon general semisimple Lie groups. To be more specific, consider the bilinear systemdefined on a semisimple Lie group G of the form,(4.2) ˙ X = (cid:16) m X i =1 u i B i (cid:17) X, X (0) = I, where B i are elements in the Lie algebra g of G . Moreover, let g = h ⊕ k be theCartan decomposition of g with h being the Cartan subalgebra and W be the Weylgroup of g . We further assume that B i ∈ h or B i ∈ k for every i = 1 , . . . , m , then the bove discussion leads to the following conjecture for systems defined on semisimpleLie groups. Conjecture
The system in (4.2) is controllable on G if and only if there ex-its Σ ⊆ Γ such that ι (Σ) is a cycle of maximal length in W h , where Γ = { B , . . . , B m } is the set of control vector fields, h = dim h , and W h denotes the direct sum of h copiesof W . Recall that the central idea of the symmetric group approach to controllabilityanalysis is to map elements with non-vanishing Lie brackets to cycles with increasedlength. However, all elements in the Cartan subalgebra have vanishing Lie brackets.The intuition behind the above conjecture comes from the need of appropriately rep-resenting these elements using permutations by mapping elements in different rootspaces to permutation cycles in different components of the direct sum of h copies ofsymmetric groups, where h denotes the dimension of the Cartan subalgebra. More-over, because the interaction between elements in and outside the Cartan subalgebrais characterized by the Weyl group, which is a subgroup of a symmetric group, thesymmetric group method applies directly. Inthe case that the Lie algebra generated by drift and control vector fields of a bilinearsystem can be decomposed into components that are Lie subalgebras, we will seethat the graph-theoretic method applies more naturally for controllability analysis.One decomposition of this type is the non-intertwining decomposition , through whicha Lie algebra is decomposed into a direct sum of Lie subalgebras so that elementsfrom different Lie subalgebras have vanishing Lie brackets. The non-intertwiningdecomposition generalizes the notion of block diagonalization for matrices.
Definition
For a given Lie algebra g , we call a decomposition g = g ⊕ · · ·⊕ g m non-intertwining if [ g i , g j ] = 0 for any Lie subalgebras g i , g j , i = j m . For example, every reductive Lie algebra admits a non-intertwining decomposi-tion, and many familiar Lie algebras are reductive, such as the algebra of n × n complexmatrices gl ( n, C ) and the algebra of n × n skew-symmetric complex matrices so ( n, C ) [17]. If a Lie algebra admits a non-intertwining decomposition, then we will be ableto associate each of its components with a graph. The subsequent question is whethergraph representation developed in Section 3.2 remains valid to characterize controlla-bility. The answer to this question can be illustrated by a system defined on SO(4) whose Lie algebra so (4) can be decomposed into a direct sum of two non-intertwiningcopies of so (3) , as shown in the following example. Example B ′ = { A , A , A , B , B , B } be a non-standard basis of so (4) ,where(4.3) A = Ω + Ω , A = Ω − Ω , A = Ω + Ω ,B = Ω + Ω , B = Ω − Ω , B = Ω − Ω . The Lie brackets of the elements in B ′ satisfy [ A i , A j ] = A k , [ B i , B j ] = B k for anyordered 3-tuple ( i, j, k ) = (1 , , , (2 , , or (3 , , , and [ A i , B j ] = 0 for any i, j . As a result, so (4) admits a non-intertwining decomposition as so (4) =Lie { A , A , A } ⊕ Lie { B , B , B } .We note that the Lie bracket relations among elements in { A , A , A } , as wellas { B , B , B } , are the same as the Lie bracket relations among elements in the tandard basis of so (3) . In other words, both Lie { A , A , A } and Lie { B , B , B } are isomorphic to so (3) , so K becomes the suitable graph representation for each set.Moreover, because [ A i , B j ] = 0 for any i, j = 1 , , , the graph representation for thenon-standard basis B ′ = { A , A , A } ⊔ { B , B , B } is a disjoint union of two copiesof K , as shown in Figure 4.1, instead of the complete graph K associated with thestandard basis of so (4) . v v v τ ′ ( A ) τ ′ ( A ) τ ′ ( A ) w w w τ ′ ( B ) τ ′ ( B ) τ ′ ( B ) Fig. 4.1 . The graphs associated with the sets { A , A , A } and { B , B , B } in Example . This example illuminates how the graph representation of controllability devel-oped in Section 3.2 can be extended to the bilinear system governed by vector fieldsgenerating a non-intertwining Lie algebra, after modifying the definition of τ in (3.7)accordingly. Proposition
Consider a bilinear system on
SO(4) governed by the vectorfields in B ′ , given by (4.4) ˙ X ( t ) = (cid:16) m X i =1 u i C i (cid:17) X ( t ) , X (0) = I, with Γ = { C , . . . , C m } ⊆ B ′ . Given a graph map τ ′ : P ( B ′ ) → G ′ , where G ′ denotesthe collection of subgraphs of K ⊔ K , satisfying τ ′ ( A i ) = v i v i +1 and τ ′ ( B i ) = w i w i +1 , with the index taken modulo , the system in (4.4) is controllable if and only if τ ′ (Γ) = K ⊔ K , or equivalently, if and only if each component of τ ′ (Γ) is connected in K .Proof. The above result becomes obvious once we verify the following propertiesof τ ′ (c.f. Lemma 3.9), which are straightforward.(1) τ ′ ( B ′ ) = K ⊔ K ;(2) For distinct C , C ∈ B ′ , their Lie bracket [ C , C ] = 0 if and only if the twoedges τ ′ ( C ) and τ ′ ( C ) have a common vertex;(3) The edges τ ′ ( C ) , τ ′ ( C ) and τ ′ ([ C , C ]) form a triangle if [ C , C ] = 0 , orequivalently, τ ′ ( { C , C , [ C , C ] } ) = K , for any C , C ∈ B ′ such that [ C , C ] = 0 .In addition, recall from Corollary 3.20 that three control inputs are enough tohave a controllable driftless system on SO(4) governed by the vector fields in thestandard basis; or equivalently, three edges can form a connected graph with fourvertices. However, for systems in the form of (4.4), they require at least four controlinputs to be controllable on
SO(4) . From the graph aspect, this is because bothcomponents of τ (Γ) require at least two edges to be connected. xample 4.4 further illustrates that for bilinear systems evolving on SO( n ) gov-erned by non-standard basis vector fields, i.e., vector fields that are not in the form ofstandard basis elements, in so ( n ) , controllability may not be characterized by usingone complete graph K n . Taking the system in (4.4) as an example, because the Liealgebra of its state-space can be decomposed into a direct sum of two non-intertwiningcomponents, its graph representation also requires two components. This finding elu-cidates that the number of components of the graph associated with a bilinear systemis determined by the number of summands in the non-intertwining decomposition ofthe underlying Lie algebra of the system. Theorem
Given a bilinear system (4.5) ˙ X ( t ) = (cid:16) m X i =1 n i X j =1 u ij B ij (cid:17) X ( t ) , X (0) = I, defined on a Lie group G whose Lie algebra g admits a non-intertwining decompositionas g = g ⊕ · · · ⊕ g m , where B ij ∈ B i and B i is a basis of g i for each i . Suppose each B i is associated with a connected graph G i such that a subset Σ i ⊆ B i generates g i ifand only if its associated graph τ (Σ i ) is a connected subgraph of G i , then the systemin (4.5) is controllable on G if and only if τ (Γ i ) is connected for every i = 1 , . . . , m ,where Γ i = { B ij : j = 1 , . . . , n i } .Proof. By the assumption, τ (Γ i ) is connected if and only if Lie (Γ i ) = g i for each i = 1 , . . . , m . Together with the non-intertwining property between each pair of g i and g j , the connectivity of τ (Γ i ) for all i is equivalent to Lie (Γ) = m M i =1 Lie (Γ i ) = m M i =1 g i = g , where Γ = S mi =1 Γ i . The proof is then concluded by applying the LARC. Remark ι . For instance, in Exam-ple 4.4, since both { A i } and { B i } in (4.3) are isomorphic to the standard basis in so (3) , the symmetric group method extends to the systems in (4.4) as well, by associ-ating each component in the decomposition to a copy of S and defining ι ( A i , B j ) = (cid:0) ( i, i + 1) , ( j, j + 1) (cid:1) , with the index taken modulo . Consequently, the system in(4.4) is controllable if and only if ι relates Γ to two disjoint -cycles in S ⊕ S .
5. Summary.
In this paper, we develop a combinatorics-based framework tocharacterize controllability of bilinear systems evolving on Lie groups, in which Liebracket operations of vector fields are represented by operations on permutations in asymmetric group and edges in a graph. Through such representations, we obtain thetractable and transparent combinatorial characterizations of controllability in termsof permutation cycles and graph connectivity. This framework is established by firstconsidering bilinear systems on
SO( n ) , and we show that, in this case, the permutationand graph representations are equivalent. Then, by exploiting techniques in repre-sentation theory, we extend our investigation into a more general category of bilinearsystems via proper decompositions of the underlying Lie algebras of the systems. Inparticular, we illustrate the application of the developed combinatorial methods to ilinear systems whose underlying Lie algebras admit the Cartan or non-intertwiningdecomposition. The presented methodology not only provides an alternative to theLARC, but also advances geometric control theory by integrating it with techniquesin combinatorics and representation theory. As a final remark, compared to knowngraph-theoretic methods mostly developed for networked or multi-agent systems, ourframework proposes novel applications of graphs to the study of bilinear control sys-tems. Appendix A. Symmetric Groups and Permutations.
In this appendix, we give a brief review of the symmetric group theory. For athorough discussion on symmetric groups, the reader can refer to any standard algebratextbook, for example [18]. Let X n be a finite set of n elements, and without loss ofgenerality, we may assume X n = { , · · · , n } . A permutation σ of X n is a bijectionfrom X n onto itself, and is denoted by σ = (cid:18) · · · ni i · · · i n (cid:19) if σ (1) = i , . . . , σ ( n ) = i n for distinct i , . . . , i n ∈ X n . A permutation that switchesonly two elements is called a transposition , and is denoted by σ = ( i i ) if i = i and σ fixes all other indices except for σ ( i ) = i and σ ( i ) = i . More generally, an r -cycle denoted by σ = ( i i · · · i r ) is a permutation that satisfies σ ( i ) = i , σ ( i ) = i ,. . . , σ ( i r ) = i and fixes all other indices. It can be shown that any permutation canbe decomposed uniquely into disjoint cycles (cycles that have no common indices).For example, when n = 4 , the permutation (cid:0) (cid:1) can be represented by a single -cycle (1234) ; while the permutation (cid:0) (cid:1) is the composition of two transpositions( -cycles): (13)(24) . Given a permutation σ of X n and an integer i , i n , the orbit of i is formed under the cyclic group generated by σ . So for σ = (1234) , the orbitof is { σ i (2) : i ∈ N } = { , σ (2) , σ (2) , σ (2) } = { , , , } ; and for σ = (13)(24) , theorbit of is { σ i (2) : i ∈ N } = { , σ (2) } = { , } . The symmetric group S n is definedas the group of permutations on X n , with its group operation being the compositionof bijections. Appendix B. Basics of Representation Theory.
Representation theory is a branch of algebra which studies structure theory byrepresenting elements in an algebraic object, such as a group, a module, or an algebra,using linear transformations of vector spaces. In this appendix, we will review somebasic concepts and results in the representation theory of Lie algebras that are used inthis paper. Detailed discussions of Lie representation theory can be found in [9, 17].To study the algebraic structure of a Lie algebra, let us introduce some relateddefinitions.
Definition
B.1. • A Lie algebra g is said to be abelian if [ g , g ] := span { [ X, Y ] :
X, Y ∈ g } = 0 . • A subspace h of g is a Lie subalgebra of g if [ h , h ] ⊆ h . In other words, h isa Lie algebra itself w.r.t. [ · , · ] . • A Lie subalgebra h g is an ideal in g if [ h , g ] ⊆ h . • The Lie algebra g is said to be simple if it is nonabelian and has no propernonzero ideals, and semisimple if it has no nonzero abelian ideals. t can be shown that every semisimple Lie algebra g can be decomposed into adirect sum of simple Lie algebras which are ideals in g . Moreover, this decomposition isunique, and the only ideals of g are the direct sums of some of these simple Lie algebras.For example, each special orthogonal Lie algebra so ( n ) = { Ω ∈ R n × n : Ω+Ω ⊺ = 0 } , aswe use extensively in this paper, is simple except for n = 4 , while so (4) is semisimplebut not simple: as shown in Example 4.4, so (4) = so (3) ⊕ so (3) .The study of algebraic structures of semisimple Lie algebras plays a central rolein representation theory. One of the most dominant results is the Cartan decompo-sition that traces back to the work of Élie Cartan and Wilhelm Killing in the 1880s,which generalizes the notion of singular value decomposition for matrices. Given asemisimple Lie algebra g , its Cartan subalgebra h is a maximal abelian subalgebra of g such that ad H is diagonalizable for all H ∈ h , where ad X Y = [ X, Y ] for all X, Y ∈ g .Moreover, the dimension of h is called the rank of g . Let h ∗ denote the dual space of h , i.e., the space of linear functionals on h , then a nonzero element α ∈ h is called a root of g if there exists some X ∈ g such that ad H X = α ( H ) X for all H ∈ h ∗ , and g α := { X ∈ g : ad H X = α ( H ) X, ∀ H ∈ h } is a vector space called the root space of g ,which can be shown to be one-dimensional. Let R denote the set of roots of g , then R is finite and spans h ∗ . With the above notations, the root space decomposition , whichgeneralizes the classical Cartan decomposition , is defined as g = h ⊕ (cid:16)M α ∈ R g α (cid:17) . A major tool to study the properties of R is the Weyl group, which is defined asfollows: Let α ∈ R be a root and s α : h ∗ → h ∗ denote the reflection about thehyperplane in h ∗ orthogonal to α , i.e., s α ( β ) = β − h β,α ih α,α i α for all β ∈ h ∗ , where h· , ·i is an inner product on h , then the Weyl group W of R is the subgroup of theorthogonal group O( h ∗ ) of h ∗ generated by all s α for α ∈ R . It can be shown that W is a finite group and hence a subgroup of a symmetric group by Cayley’s theorem. REFERENCES[1]
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