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(CBNs) and networks so BNs and of Some [19]–[21] kinds control BCNs. stabilizat optimal and and [18], stability 10]–[13], BNs controllabilit [6, of observability analysis analyzing product [9], the include for semi-tensor STP of called matrices, applications technique, of seminal (STP) a proposed aua usinoemyaki htfragvnpi fstates of pair given integer a some for exist that there is ask does may one question natural hoe nscinI) oee,fraspecific a is for However, which II). section in 2 Theorem htthe that xdtm otolblt yapyn h Perron-Frobeni the applying by controllability fixed-time nti ae,w rps n nwrtefloigques- following the answer and propose we paper, this In h oiaino u td stofl.Frt ent that note we First, two-fold. is study our of motivation The nteps eae hn n i olaus[]have [5] colleagues his and Cheng decade, past the In ebrIEEE Member k k ieses h eut n[]rltsthe relates [8] in results The steps. time pair ssc httepi snot is pair the that such ’s k xdtm otolbe ti o eesrl true necessarily not is it controllable, fixed-time is k p ssc httegvnpi fsae is states of pair given the that such ’s p xdtm otolbefrany for controllable fixed-time xdtm otolbefrany for controllable fixed-time ehaGui Weihua , k k xdtm otolbei vr ordered every if controllable fixed-time xdtm otolbe respectively), controllable, fixed-time p ≥ k k k k xdtm otolbe twas It controllable. fixed-time ftease sys esay we yes, is answer the If . uhta h aris pair the that such xdtm otolbe then controllable, fixed-time k k primitive ssc httepi is pair the that such ’s xdtm controllable? fixed-time aeoy fthe If category. aro states of pair k p p fixed-time o [14]– ion ≥ ≥ p k k study e k [6]– y fixed- fixed- fixed- k ome (see ives A . ity ce us of as ]– k o e e 1 , . 2 answer is no, we further classify those pairs into three other are the logical functions. With vector form expression, i.e., we 1 2 categories. The detailed formulation is provided in section II. use δ2 to represent state 1 and δ2 to represent state 0, one has A second motivation of our research comes from potential xi,ui ∈ ∆2. Then as in [5], (1) can be transformed into the biological applications. The goal of interest may be to drive algebraic form: a system from one state to another, assuming that the former x(t +1)= L ⋉ u(t) ⋉ x(t), (2) is undesired and the latter is desired. Additionally, one may ⋉n n encounter the situation that a biological system consists of sev- where x(t) = i=1xi(t) ∈ ∆N with N := 2 , u(t) = ⋉m m eral identical subsystems with no couplings among them, and j=1uj (t) ∈ ∆M with M := 2 , and L ∈ LN×NM . Let each subsystem is modeled by the same BCN. For example, M N ⋉ j (i) a multi-cellular organism has identical BCNs, each modeling M := B L δM and F := B M , (3) a cell-cycle [8]. We may be interested in finding a control Xj=1 Xi=1 law with respect to each subsystem to drive each subsystem where F is called the controllability matrix [7]. We define the from different initial states to the same desired state at some controllability of a BCN as follows. fixed time. Our results in this paper characterize all possible values of such fixed times efficiently, without checking each Definition 1 (Controllability [7, 8]). The BCN (1) is positive integer. If such a fixed time exists, all subsystems can 1) controllable from x0 to xd, if there are a T > 0 and a be applied with the same control law afterwards, resulting in sequence of control u(0),...,u(T − 1), such that driven a complete synchronization of the states of these subsystems by these controls the trajectory can go from x(0) = x0 in the following dynamical evolutions. to x(T )= xd; The remainder of this paper is organized as follows. Sec- 2) controllable at x0, if it is controllable from x0 to tion II introduces some preliminaries on algebraic graph destination xd = x, ∀x; theory and the existing controllability results of BCNs. In 3) controllable, if it is controllable at any x. section III we present the categorization problem on control- We also define the k fixed-time controllability of a BCN. lability of BCNs and establish our main result. An illustrative example is provided in section IV . Finally, we conclude the Definition 2 (k fixed-time controlllability [8]). Given a pair paper in section V. of states (x0, xd), the pair is called k fixed-time controllable if Before ending this section, we present the following nota- there exists a sequence of control u(0),...,u(k − 1) that steers tions that will be used throughout the paper: Z+ – the set of the BCN (1) from x(0) = x0 to x(k) = xd. The BCN (1) is the positive integers; [a,b] – the integer set {a,a +1, ..., b} k fixed-time controllable if all pairs (x0, xd) are k fixed-time controllable. with a ≤ b; Coli(A) – the ith column of the matrix A; i i i j ∆k := {δk | i = 1, 2, ··· , k}, where δk is the ith column For each ordered pair of states (δN ,δN ), we define two of the identity matrix Ik; D := {T = 1, F = 0} – the sets ρ(i, j) and σ(i, j) as follows: for each positive integer k, logic field; An m × n matrix A with Coli(A) ∈ ∆m for if there is a sequence of control u(0),...,u(k − 1) that steers i j all i – the logical matrix; Lm×n – the set of all m × n the BCN from x(0) = δN to x(k) = δN , then k ∈ ρ(i, j); logical matrices; A = δm[i1,i2, ..., in] – the simplified ex- otherwise, k ∈ σ(i, j). It should be clear that ρ(i, j)⊔σ(i, j)= i1 i2 in B + pression for A = [δm,δm, ..., δm ] ∈ Lm×n; n×n – the Z . As a reference, we call ρ(i, j) the set of reachable time set of n × n Boolean matrices, i.e., all entries are 0 or 1; steps and σ(i, j) the set of unreachable time steps. B(A) — Boolean form of nonnegative matrix A, which is a With the above definitions, we present the categorization Boolean matrix with the ijth entrie 1 if Aij > 0, and the problem as follows. ijth entrie 0 if Aij = 0. A +B B = (Aij ∨ Bij ) (resp. n Problem 1. Consider the BCN (1). The goal is to classify all i j A×BB := B(Aik ∧ Bkj ) ) – the Boolean addition pairs (δ ,δ ) into the four categories:   N N kP=1 ij 1) unreachable: |ρ(i, j)| =0; B (k) (resp. product) of A, B ∈ n×n; A := A×B ···×BA; 2) transient: 0 < |ρ(i, j)| < ∞ and |σ(i, j)| = ∞; k 3) primitive: |ρ(i, j)| = ∞ and |σ(i, j)| < ∞; A matrix means its entries are positive; – the A > 0 | |{zC| } 4) imprimitive: |ρ(i, j)| = ∞ and |σ(i, j)| = ∞. cardinal number of the set C. ⋉ – semi-tensor product (STP) of matrices. Equivalently, one wishes to obtain the controllability cate- gorization matrix C = (Cji), Cji ∈ [0, 3], where Cji is defined i j II. PROBLEM FORMULATION AND BACKGROUNDS to be k if the pair (δN , δN ) belongs to the category (k + 1). A. Problem formulation B. Backgrounds In this subsection, we formally introduce the categorization We note that, as the number of state pairs in Problem 1 problem. We first need the following definitions. is huge, and the BCN (1) can have complicated structures, A BCN with n state variables can be described as follows: solving Problem 1 requires nontrivial techniques. We will develop a method via algebraic graph theoretic approach. Prior x (t +1)= f (x (t), ..., x (t); u (t), ..., u (t)), (1) i i 1 n 1 m to that, we introduce in this subsection some preliminary m where xi ∈ D,i ∈ [1,n] are the state variables, ui ∈ D ,i ∈ results on digraphs and matrices, as well as the results on n+m [1,m] are the input variables, and fi : D → D, i ∈ [1,n] controllability of BCNs. 3

1) Directed graphs: Let G = (V, E) be a digraph with Lemma 1 ( [27]). Let G be a strongly connected digraph of the set of nodes (vertices) V and the set of directed edges order n with index of imprimitivity η. Then, for each pair of E ⊆ V × V . The order of a graph G is the number of nodes nodes vi and vj , the lengths of the directed walks from vi to in V . We denote by vi → vj a directed edge from vi to vj in vj are congruent modulo η. G, and if i = j, the edge is called the self-loop of the node i. 2) Matrices: We first define the reducibility of a matrix. The adjacency matrix A ∈ Bn×n of G is defined as follows: Aij =1 (resp. 0) if and only if vi → vj ∈ (resp.∈ /) E. For Definition 5 (Reducible matrix [27]). A matrix A of order n is simplicity, the digraph (i.e., G) of A is denoted by G(A). called reducible if there exists a permutation matrix P ∈ Ln×n Assumed that vi and vj are two nodes of G. A walk from such that ⊤ B C vi to vj , denoted by wij , is a sequence of nodes vi0 → vi1 → P AP = (4)  0 D  ···→ vim in which each vij → vij+1 , for j =0,...,m − 1, is an edge. If vi0 = vim , the walk is called a closed walk. A where B and D are square matrices of order at least one. A cycle is a closed walk with no repetition of nodes other than matrix is said to be irreducible if it is not reducible. the starting- and the ending- node. A walk is said to be a path For the rest of this paper, we let A be a Boolean matrix, i.e., if all the nodes in the walk are pairwise distinct. Let pij be all entries of A are either 0 or 1. It should be clear that there a path from vi to vj . We denote by Pij the set of all paths is an one-to-one correspondence between the set of Boolean from vi to vj . The length of a walk (resp. path, cycle) is the number of edges in that walk (resp. path, cycle). matrices of order n and the set of digraphs of order n. We then have the following lemmas. Two nodes vi and vj of G are called strongly connected if there exists a directed walk from vi to vj , and a directed Lemma 2 ( [27]). The matrix A of order n is irreducible if walk from vj to vi. A graph G is strongly connected if and only if its digraph G(A) is strongly connected. any two nodes vi and vj are strongly connected. A single node with self-loop is regarded as trivially strongly connected Lemma 3 ( [27]). Let A be a matrix of order n. Then there to itself. Evidently, strong connectivity between nodes is exists a permutation matrix P of order n and an integer S ≥ 1 reflexive, symmetric, and transitive, resulting in an equivalence such that the Frobenius normal form of (4) can be written as relation on the nodes of G and simultaneously yielding a A1,1 A1,2 ··· A1,S , ,S partition, V1 ⊔ V2 ⊔···⊔ VS, with Vi = V . Let Ei be  0 A2 2 ··· A2  ⊤ (5) the set of edges vij → vik such thatS vij , vik ∈ Vi. Then P AP = . . . . .  . . .. .  Gi = (Vi, Ei),i ∈ [1,S] are the induced subgraphs of G.    0 0 ··· AS,S  We also call each induced subgraph a strongly connected   component (SCC) of G. Specifically, a single node without If A in Lemma 3 is the adjacency matrix of digraph G, self-loop is an SCC by itself. In this paper, we call such a then A1,1, A2,2,...,AS,S are adjacency matrices of the SCCs i,i single node the Type 1 SCC (T1SCC), and all other SCCs the of G. Specifically, if the SCC Gi is a T2SCC, then A is Type 2 SCC (T2SCC). a square irreducible matrix; if the SCC Gi is a T1SCC, then We next present the following definition on condensation Ai,i is a 1-by-1 zero matrix. digraphs. We next define the primitivity of a matrix. Definition 3 (Condensation digraph [27]). Let G be a digraph Definition 6 (Primitive matrix [27]). A nonnegative matrix and A be its adjacency matrix. Assume that G has S SCCs: A ∈ Rn×n is primitive if there exits an integer j ≥ 1 such ∗ j j G1,...,GS, where Gi = (Vi, Ei). Let G (A) be the conden- that A > 0. If A is primitive, the smallest j such that A > 0 ∗ sation digraph of G(A), and A be the adjacency matrix of is called the exponent of A, denoted by γ(A). G∗(A). G∗(A) and A∗ are constructed as follows: We note that if matrix A is primitive, then it is also 1) The set of nodes of G∗(A) is obtained by identifying irreducible. Further, we also have the following results on each SCC as a node, primitive matrices. 2) If there exists a directed edge in G(A) from a node in ∗ ∗ 2 Vi to a node in Vj , then Aij =1; otherwise, Aij =0. Lemma 4 ( [27]). If A is primitive, then γ(A) ≤ (n−1) +1. ∗ The constructed condensation digraph G (A) has no closed Proposition 1 ( [27]). A digraph G is primitive if and only if directed walks. its adjacency matrix A is primitive. We then define the primitivity of a digraph. 3) Controllability and k fixed-time controllability: Given a Definition 4 (Primitive digraph [27]). Let G be a strongly BCN (1), we can compute the matrices M and F as in (3). connected digraph of order n. Let η = η(G) be the greatest The controllability and k fixed-time controllability of the BCN common divisor of the lengths of the cycles of G. The digraph can then be determined by the following theorems. G is primitive if η =1 and imprimitive if η > 1. The integer Theorem 1 ( [7]). The BCN (1) is η is called the index of imprimitivity of G. The index of j i 1) controllable from δN to δN , if and only if, Fij =1; imprimitivity η is also referred to as the loop number [25]. j 2) controllable at δN , if and only if, Colj (F) > 0; With Definition 4, we have the following result. 3) controllable, if and only if, F > 0. 4

Theorem 2 ( [8]). Consider the BCN (1). that M¯ 1,1 ∈ Bq1×q1 , M¯ 2,2 ∈ Bq2×q2 , ..., M¯ S,S ∈ BqS ×qS S 1) The BCN (1) is controllable, if and only if, M is where i=1 qi = N. The BCN (1) thereby has S SCCs, S irreducible. denotedP by, X1, ..., XS, with i=1 Xi = ∆N . Additionally, we i 2) The BCN (1) is k fixed-time controllable, if and only if, denote by Id(i) the index of theS SCC that the state δN belongs M(k) > 0. to. From the Definition 3, we can construct the condensation ∗ 3) If the matrix M is primitive, then γ(M) ≤ N 2 −2N +2 digraph G (M¯ ) of the BCN (1) as well as its adjacency matrix ∗ and the BCN (1) is γ(M) fixed-time controllable. If M¯ . i j M is not primitive, then the BCN is not k fixed-time Given a pair of states δN and δN , which are two nodes controllable for any k. in the state transition graph G, let Pij be the set of paths i j ∗ 4) If the BCN (1) is k fixed-time controllable, then it is p from δN to δN , and PId(i),Id(j) be the set of paths from fixed-time controllable for any p ≥ k. Id(i) to Id(j) in the condensation graph G∗(M¯ ). Let K := |P ∗ |. We denote these paths in the condensation graph Remark 1. We note that the controllability of the BCN can Id(i),Id(j) by p∗1 ,p∗2 ,...,p∗K . Then, we use the be determined with matrix F by Theorem 1, while whether Id(i),Id(j) Id(i),Id(j) Id(i),Id(j) following method to partition the set Pij into K subsets the BCN is the k fixed-time controllable or not can only be 1 K determined with matrix by Theorem 2. Specifically, if Pij ,...,Pij . M M l is primitive, then BCN (1) is k fixed-time controllable for any For any path pij ∈ Pij , we replace every node δN ∈ pij k ≥ γ(M). When M is reducible, although the BCN (1) is with the node Id(l), if the resulting path, ignoring self-loops, is ∗k , then k . not k fixed-time controllable, we may still have some pairs pId(i),Id(j) pij ∈ Pij With the above partitions, we further define k to be the (x0, xd) that are k fixed-time controllable. ηij greatest common divisor of the indexes of primitivity of the For convenience, we call BCN (1) P-controllable if it is ∗k T2SCC along path pId(i),Id(j). If there is no T2SCC along the k fixed-time controllable for some integer k > 0. We call ∗k k k path pId(i),Id(j), we let ηij = 0 and πij = ∅. Otherwise, we BCN (1) NP-controllable if it is controllable, but not k fixed- define time controllable for any k > 0. Equivalently, BCN (1) is πk = l(pk ) mod ηk | pk ∈ P k , (6) P-controllable if M is primitive; BCN (1) is NP-controllable ij ij ij ij ij  1 K if M is irreducible and imprimitive. let η¯ij be the least common multiple of {ηij ,...,ηij }, and k k k η¯ij π˜ij := a + b · ηij | a ∈ πij , b ∈ 0, k − 1 . Then, let ηij III. MAIN RESULTS n k h io π¯ij := π˜ij . With these definitions, we are in a position Recall that in Problem 1, we aim to classify all state pairs to presentS our main theorem. of BCN (1) into four categories. Equivalently, one wishes to Theorem 3. Considering the BCN , we have obtain the controllability categorization matrix C. We note that (1) the Boolean form of C is exactly F in (3), i.e., B(C) ≡F. 1) Cji =0, if and only if, Fji =0. Evidently, the form of the controllability categorization 2) Cji =1, if and only if, π¯ij = ∅ and Fji =1. matrix C is trivial in the following situation. If the BCN (1) is 3) Cji =2, if and only if, π¯ij = [0, η¯ij − 1]. 4) , if and only if, and . P-controllable (resp. NP-controllable), then, C = 2N×N (resp. Cji =3 π¯ij 6= ∅ π¯ij 6= [0, η¯ij − 1] C = 3N×N ). In other words, C is trivial if M is irreducible, We first provide a proof for the bulletins (1) and (2) of as it follows from the definitions that Theorem 3. In the next subsection, we will provide a complete 1) If the BCN (1) is P-controllable, then for any pair of proof of bulletins (3) and (4). We will also provide a follow-up i j states δN and δN , we have |ρ(i, j)| = ∞ and |σ(i, j)| < result in section III-C. ∞. 2) If the BCN (1) is NP-controllable, then for any pair of Proof of Theorem 3, part I. We now prove the first two bul- states δi and δj , we have |ρ(i, j)| = ∞ and |σ(i, j)| = letins of Theorem 3. N N j ∞. 1) We note that by definition, Cji = 0 if and only if δN i In the rest of this section, we investigate the case when M is unreachable from δN , or equivalently, BCN (1) is i j is reducible. uncontrollable from δN to δN , which, by Theorem 1, holds if and only if Fji =0. 2) We show that if π¯ij = ∅ and Fji =1, then Cji =1. The A. Main theorem other direction can be similarly shown. By definition, Let G = (V, E) be the state transition digraph of BCN (1), Cji =1 if and only if 0 < |ρ(i, j)| < ∞ and |σ(i, j)| = i k where V is the set of states, i.e., V := ∆N , and E := {δN → ∞. Note that π¯ij = ∅ if and only if π˜ij = ∅ (i.e., j j i i j k k δN | δN = LuδN for some u ∈ ∆M }, i.e., an edge δN → δN πij = ∅) for all k ∈ [1,K]. This implies that ηij =0 for ∗k exists in E if there exists some control u which drives the all k and there is no T2SCC along the path pId(i),Id(j) i j ¯ system from state δN to state δN in one step. Let M be for all k. Therefore, there is no T2SCC along any path ⊤ the adjacency matrix of G, then, M¯ := M , where M is pij ∈ Pij . We thus have that ρ(i, j) = {l(pij) | pij ∈ defined as in (3). Then, by Lemma 3, we can write M¯ in Pij }. Since |Pij | is finite, we have that |ρ(i, j)| < ∞. the Frobenius normal form as in (5) in a similar manner, Since Fji =1, there exists at least one path pij ∈ Pij , with the replacement of A in (5) with M¯ . Then we have which implies that |Pij | > 0 and |ρ(i, j)| > 0. Lastly, 5

for any positive integer k∈ / {l(pij) | pij ∈ Pij }, we Proof. (Sufficiency). When πij = [0, ηij − 1], by the argu- have that k ∈ σ(i, j). This implies that |σ(i, j)| = ∞. ments before Lemma 6, there exists an integer ′ ′ n := φ (l(c1), ..., l(ck)) ηij + max l(pij ) > 0 (8) pij ∈Pij

B. Analysis and proof of Theorem 3 such that one can construct a walk wij with the length n, for any integer n ≥ n′. In other words, |ρ(i, j)| = ∞ and In this subsection, we show the last two bulletins of The- |σ(i, j)| < ∞. This implies Cji =2. orem 3. We first consider the case that ∗ , |PId(i),Id(j)| = 1 (Necessity). When Cji =2, we have that |ρ(i, j)| = ∞ and i.e., there is only one path from Id(i) to Id(j) in the |σ(i, j)| < ∞. Suppose that to the contrary πij 6= [0, ηij − 1]. condensation graph. Later we will extend to the general case Then there exists an integer k ∈ [0, ηij − 1] such that k∈ / πij , ∗ ′ where |PId(i),Id(j)| = K for any positive integer K. which implies that a walk of length ηij n+k, ∀n ≥ n , cannot ∗ For ease of notation, we now use pId(i),Id(j) to denote the be constructed. This contradicts with |σ(i, j)| < ∞. path from Id(i) to Id(j) in the condensation graph, and let ηij Note that Lemma 6 essentially proves the third bulletin of be the greatest common divisor of the indexes of imprimitivity ∗ ∗ Theorem 3 for the case when . We next of the T2SCCs along the path p . Again, if there is |PId(i),Id(j)| = 1 Id(i),Id(j) consider the general case where |P ∗ | = K, with K no T2SCC along the path, let ηij =0. Id(i),Id(j) Let the cycles in the T2SCCs along path p∗ be being any positive integer, and prove the last two bulletins of Id(i),Id(j) Theorem 3. c1, ..., ck with the lengths equal to l(c1), ..., l(ck), respectively. Then any walk wij of G(M¯ ) has length of the form Proof of Theorem 3, part II. (3) (Sufficiency). Recall the k definitions before Theorem 3. Each ηij is the greatest l(wij )= l(pij )+ a1 · l(c1)+ ··· + ak · l(ck), common divisor of the index of primitivities of the ∗k T2SCCs along path pId(i),Id(j). Similar to the arguments where a1,...,ak are nonnegative integers. Note that, ∗ from [27], we have the following lemma. for the case |PId(i),Id(j)| = 1, it can be shown that k if s ∈ πij , we can construct a walk wij of length Lemma 5 ( [27]). Let Ψ= {l1,l2, ..., lk} be a nonempty set k ′ nkηij + s for any nk ≥ nk for some positive integer of positive integers and η be the greatest common divisor of ′ ′ nk. Here, we can pick nk as in (8). We perform the the integers in Ψ. Then there exists a smallest positive integer ∗k same implementation for each path pId(i),Id(j). Then, φ(l1,l2, ..., lk), called the Frobenius-Schur index of Ψ, such ′′ with the definition of η¯ij , for some nk, we can rewrite that for any integer n ≥ φ(l1,l2, ..., lk), nη can be expressed k ′ the set {nkηij + s | nk ≥ nk} as {nkη¯ij + s,nkη¯ij + as a nonnegative linear combination of these integers, i.e., as k k η¯ij ′′ ηij + s, ..., nkη¯ij + ηij ηk + s | nk ≥ nk}. Note that a sum, nη = a1l1 + a2l2 + ··· + aklk, where a1,a2, ..., ak are ij nonnegative integers. this can be done for each k ∈ [1,K]. Therefore, for ∗ ∗ k each s ∈ π¯ij , we have that s ∈ π˜ij for some k, From the Lemma 5 and the definition of η , there exists a ∗ ′′ ij and a walk of length nkη¯ij + s , nk ≥ nk, can be φ (l(c1), ..., l(ck)) such that for any n ≥ φ (l(c1), ..., l(ck)), we constructed. If π¯ij = [0, η¯ij − 1], then there exists ∗ ′′ ∗ have that nηij = a1l(c1)+ ··· + akl(ck). Similarly, we can some n = maxk∈ ,K n such that for any n ≥ n , ′ [1 ] k find some φ (l(c1), ..., l(ck)) ≥ φ (l(c1), ..., l(ck)) such that we can construct a walk of length n. This implies ′ for any integer n ≥ φ (l(c1), ..., l(ck)), we have that nηij = that |ρ(i, j)| = ∞ and |σ(i, j)| < ∞, or equivalently, ′ ′ ′ ′ a1l(c1)+ ··· + akl(ck) where a1,...,ak are positive integers. Cji =2. Therefore, by connecting these cycles with a path pij , we can (Necessity). Suppose that to the contrary π¯ij 6= [0, η¯ij − obtain a walk wij whose length can be l(pij )+ nηij for all 1], then there exists some s ∈ [0, η¯ij − 1] such that n ≥ φ′ (l(c ), ..., l(c )). k 1 k s∈ / π˜ij for any k ∈ [1,K], which implies that we cannot + As in (6), we define a set of integers, construct a walk of length nη¯ij + s, ∀n ∈ Z . This implies that |σ(i, j)| = ∞, which is a contradiction. (7) πij = {l(pij ) mod ηij | pij ∈ Pij } . (4) Since we have shown (1), (2), (3), the result of (4) follows directly. Evidently, if πij = [0, ηij − 1], i.e., πij is a complete residue system modulo ηij , then for any integer n ≥ ′ φ (l(c1), ..., l(ck)) ηij +maxpij ∈Pij l(pij ), there exists a walk wij of length n. In the special case that ηij = 0, then πij is C. Connection of categorization results to graph structure not well defined. We thereby redefine πij = ∅ for such a case. In this subsection, we provide a further result on the cate- Based on the above definitions, we have the following result. gorization of state pairs. In particular, we show the following i j fact which relates the categorization to the structure of the Lemma 6. Let δN and δN be two nodes of the state transition digraph G(M¯ ) of the BCN (1). Consider the case that there state transition digraph. is only one path ∗ from node to node i j pId(i),Id(j) Id(i) Id(j) Theorem 4. Let (δN ,δN ) be a pair of states of BCN (1). ∗ ¯ i j i j in the condensation digraph G (M). Then the pair (δN ,δN ) Suppose that δN ∈ Xα and δN ∈ Xβ, where Xα and Xβ are two SCCs of the state transition digraph. Then, for any states belongs to the primitive category, i.e., Cji = 2, if and only if ′ ′ j i ′ ′ πij = [0, ηij − 1]. δN ∈ Xα and δN ∈ Xβ, we have that Cj i = Cji. 6

We now prove the above theorem. To proceed, we first Remark 2. We note that our definition of condensation i j C recall some notations. Let δN and δN be two nodes of the controllability categorization matrix is a generalization state transition digraph G(M¯ ) of the BCN (1). Denote by of the so-called reduced controllability matrix B(C ) in [9]. α := Id(i) and β = Id(j). Let ηα (resp. ηβ) be the index of Notably, the dimension of C may be much smaller than the imprimitivity of the SCC Xα (resp. Xβ). In the special case one of C, if the number of nodes of the condensation digraph that Xα (resp. Xβ) is a T1SCC, we redefine ηα := 0 (resp. of the BCN (1) is much smaller than the number of states ηβ := 0). of the BCN (1), i.e., S ≪ N. In other words, to save the With the above notations, the proof of Theorem 4 is shown controllability information of BCNs, C is much better and as follows. more economical than C. Proof of Theorem 4. It should be clear that if there is no path IV. EXAMPLE from any node in Xα to any node in Xβ, then, we have that j In this section, we provide an example BCN as in [9] to C =0, ∀δi ∈ X , ∀δ ∈ X . We now restrict our discussion ji N α N β illustrate our results. In particular, the algebraic form of the to the situation that there exists a path from some node in X α BCN is to some node in Xβ. First, consider the case that α = β. If there is only one x(t +1)= δ8[2, 5, 3, 5, 6, 4, 8, 7, 4, 5, 4, 5, 6, 7, 8, 7]u(t)x(t), node in Xα (i.e., T1SCC or T2SCC), then the theorem trivially (9) holds. If Xα is a T2SCC with at least two nodes, one can where x(t) ∈ ∆8, u(t) ∈ ∆2. The state transition digraph prove the theorem as follows. (1) If ηα =1, then we have that of the BCN (9) and its condensation digraph are shown in i j Cji =2, ∀δN ,δN ∈ Xα. (2) If ηα > 1, then by the Lemma 1, the Fig. 1(a)-1(b). In particular, the state transition digraph i j 1 2 3 for any pair δN ,δN ∈ Xα, there exists an integer k ∈ [0, ηα] has 5 SCCs, X1 = {δ8}, X2 = {δ8}, X3 = {δ8}, X4 = 4 5 6 7 8 such that πij = {k}. In other words, πij 6= ∅ and πij 6= [0, ηα]. {δ8,δ8,δ8}, X5 = {δ8,δ8}, and Id(i)= i, i ∈ [1, 3], Id(j)= i j By Theorem 3, this implies that Cji =3, ∀δN ,δN ∈ Xα. 4, j ∈ [4, 6], Id(k)=5, k ∈ [7, 8]. Notably, X1 and X2 are i j Next, consider the case that α 6= β. For any pair (δN ,δN ) T1SCCs, whereas X3, X4 and X5 are T2SCCs with the indexes i j ∗ with δN ∈ Xα and δN ∈ Xβ, we have that η¯ij = ηαβ for of imprimitivity equal to 1, 3 and 2, respectively. ∗ 1 4 some ηαβ . Here, we consider the pair (δ8,δ8). From the Fig. 1(a), the ∗ ∗ 1 4 path set P14 from δ to δ has only two paths with the lengths 1) If ηαβ =1, then, π¯ij = {0} = [0, ηαβ −1]. This implies 8 8 i j 1 and 4, respectively. In the the condensation digraph, i..e, that Cji =2, ∀δ ∈ Xα,δ ∈ Xβ. N N′ ∗ j Fig. 1(b), there are also two paths from the node Id(1) = 1 2) If ηαβ > 1, then for any δN ∈ Xβ, from Lemma 1, one to node Id(4) = 4. So the path set P14 can be partitioned can conclude that there exists some s1 such that 1 1 2 2 into two distinct sets, P14 = {p14} and P14 = {p14}, where 1 1 4 2 1 2 5 6 4 π¯ij + s1 := {s1 + y modη ¯ij | y ∈ π¯ij }≡ π¯ij′ . p14 : δ8 → δ8 and p14 : δ8 → δ8 → δ8 → δ8 → δ8. ′ Since there is only one cycle with the length 3 in the X4, we i For any δ ∈ Xα, we can also conclude that there exists 1 2 N have η14 = 3 and η14 = 3. According to (6), we have that some s2 such that 1 1 2 2 π˜14 = π14 = {1} and π˜14 = π14 = {2}. Hence, η¯14 = 3 1 2 ′ and π¯14 =π ˜14 ⊔ π˜14 = {1, 2}= 6 [0, 2]. This, together with the π¯ij + s2 := {s2 + y modη ¯ij | y ∈ π¯ij }≡ π¯i j . 1 4 Theorem 3, implies that C41 =3, i.e., the pair (δ8,δ8) belongs Then we have that to the imprimitive category. Akin to the analysis above, the controllability categories of π¯ij + s := {s + y modη ¯ij | y ∈ πij }≡ π¯i′j′ , the other pairs can be obtained. Indeed, one can obtain the controllability categorization matrix. Then, based on the ma- where s := s − s . This implies that |π¯i′j′ | = |π¯ij |, 1′ 2 ′ j trix C, the condensation controllability categorization matrix ∀δi ∈ X , ∀δ ∈ X . Then by Theorem 3, we conclude N α N β can be induced. Both matrices are presented as follows, that Cj′i′ = Cji. 0 0 0 0 0 0 0 0  1 0 0 0 0 0 0 0  00000 Based on the Theorem 4, we can define an S × S matrix 0 0 2 0 0 0 0 0   C C C C i   10000 = ( βα), βα ∈ [0, 3], where βα := Cji with δN ∈ Xα  3 3 1 3 3 3 0 0  j C C =   , C =  00200 . and δN ∈ Xβ. We call the condensation controllability  3 3 1 3 3 3 0 0        categorization matrix of BCN (1). Then Theorem 4 has the  3 3 1 3 3 3 0 0   33130     following equivalent expression.  2 2 1 2 2 2 3 3   22123     Theorem 4 (An alternative version). Let Xα and Xβ be two  2 2 1 2 2 2 3 3  SCCs of the state transition digraph of the BCN (1). Then, the   i j i j pair (δN ,δN ) with δN ∈ Xα and δN ∈ Xβ belongs to the V. CONCLUSIONS 1) unreachable category, if and only if, Cα,β =0; In this paper, we have established a detailed analysis on 2) transient category, if and only if, Cα,β =1; the k fixed-time controllability of all state pairs of a BCN. 3) primitive category, if and only if, Cα,β =2; The definition of the controllability categorozation matrix 4) imprimitive category, if and only if, Cα,β =3. is first proposed, extending the conventional controllability 7

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