Reachability properties of continuous-time positive systems Maria Elena Valcher
Abstract In this paper, reachability properties of continuous-time positive systems are introduced and characterized. Specifically, first it is shown that reachability and strong reachability are equivalent properties, and thus the characterization of strong reachability derived in [8] is extended to the weaker notion of reachability. In the second part of the paper, essential reachability is introduced, and necessary and sufficient conditions for this property to hold are provided.
I. I NTRODUCTION A positive system is a linear state-space model in which the state variables always take positive, or at least nonnegative, values. The interest in these systems is motivated by the large number of contexts (like bioengineering, economic modelling, biology and behavioral science) where the physical meaning of the describing variables naturally constrains them to take nonnegative values. In particular, mathematical modeling in pharmacokinetics strongly relies on the class of internally stable continuous-time positive systems (compartmental models), which take into account both the nonnegative constraint on the system variables and the conservation laws that govern the system dynamics [12]. More recently, the context where positive systems have been fruitfully applied has significantly broadened, and positive systems have been used for modeling chargerouting networks [2], fiber-optic filters [3], rendez-vous problems [22], TCP-like congestion control problems [20], etc.. Research interests in positive systems [10] have mainly focused on the discrete-time case. Specifically, the positive realization problem (see the survey [4] for the state of the art and a M.E.Valcher is with the Dipartimento di Ingegneria dell’Informazione, Universit`a di Padova (I),
[email protected]
complete list of references) turned out to be significantly simpler in the continuous-time case with respect to the discrete-time one, as a consequence of the spectral properties of Metzler matrices, and hence received less attention [1]. On the other hand, structural properties of continuous-time systems, investigated in detail in the discrete-time case [7], [9], [23], have been surprisingly neglected and their investigation has produced, until recently, only partial results [10], [15]. A noteworthy exception is represented by the recent paper by Commault and Alamir [8] where strong reachability of continuous-time positive systems has been fully characterized. In that nice paper it was shown that strongly reachable positive systems must be endowed with a very peculiar structure. In this contribution we will show that strong reachability is not stronger than reachability and, indeed, the same characterization holds for reachable positive systems. On the other hand, we will introduce and address essential reachability (also known as almost reachability) and show that it represents a weaker concept with respect to almost strong reachability, characterized in [8]. In detail, the paper is organized as follows: section II introduces reachability and strong reachability for continuous-time positive systems, and proves that they are equivalent properties. In section III, essential reachability is introduced and characterized. Before proceeding, we introduce some basic notation. For every k ∈ N, we set hki := {1, 2, . . . , k}. In the sequel, the (i, j)th entry of a matrix A is denoted by [A]ij . If A is block partitioned, we denote its (i, j)th block either by block(i,j) [A] or by Aij . In the special case of a vector v, its ith entry is [v]i and its ith block is blocki [v]. We let ei denote the ith vector of the canonical basis in Rn (where n is always clear from the context), whose entries are all zero P except for the ith one which is unitary. If S ⊆ hni, we let eS denote the vector i∈S ei . When S = hni, we use the standard notation 1n .
Given a matrix A ∈ Rq×r , by the nonzero pattern of A we mean the set of index pairs corresponding to its nonzero entries, namely ZP(A) := {(i, j) : [A]ij 6= 0}. For a vector v, the nonzero pattern of v is ZP(v) := {i : [v]i 6= 0}. The symbol R+ denotes the semiring of nonnegative real numbers. A matrix A with entries in R+ is a nonnegative matrix (A ≥ 0); if A ≥ 0 and at least one entry is positive, A is a positive matrix (A > 0), while if all its entries are positive it is a strictly positive matrix (A ≫ 0). The same notation is adopted for nonnegative, positive and strictly positive vectors.
A Metzler matrix is a real square matrix, whose off-diagonal entries are nonnegative. If A is an n × n Metzler matrix, then [21] λmax (A) := max{Re(λ) : λ ∈ σ(A)}, σ(A) denoting the spectrum of A, is a (real) eigenvalue, and it is dominant, by this meaning that λmax (A) > Re(λ), ∀ λ ∈ σ(A), λ 6= λmax (A). To every n × n Metzler matrix A we associate [11], [19] a directed graph G(A) of order n, with vertices indexed by 1, 2, . . . , n. There is an arc (j, i) from j to i if and only if [A]ij 6= 0. We say that vertex i is accessible from j if there exists a path (i.e., a sequence of adjacent arcs (j, i1 ), (i1 , i2 ), . . . , (ik−1 , i)) in G(A) from j to i (equivalently, ∃ k ∈ N such that [Ak ]ij 6= 0). Two distinct vertices i and j are said to communicate if each of them is accessible from the other. Each vertex is assumed to communicate with itself. The concept of communicating vertices allows to partition the set of vertices hni into communicating classes, say C1 , C2 , . . . , Cℓ . The reduced graph R(A) [11], [19] associated with A (with G(A)) is the (acyclic) graph having the classes C1 , C2 , . . . , Cℓ as vertices. There is an arc (j, i) in R(A) from Cj to Ci if and only if there is a path in G(A) from some vertex in Cj to some vertex in Ci . With any class Ci we associate two index sets1 : A(Ci ) := {j : the class Cj has access to the class Ci } D(Ci ) := {j : the class Cj is accessible from the class Ci }. Each class Ci is assumed to have access to itself. A class Ci is final if it accesses no other class, namely D(Ci ) = {i}. Any path (i1 , i2 ), (i2 , i3 ), . . . , (ik−1 , ik ) in R(A) identifies a chain of classes (Ci1 , Ci2 , . . . , Cik ), from class Ci1 to class Cik . An n × n Metzler matrix A is reducible if there exists a permutation matrix P such that " # A A 11 12 P T AP = , 0 A22 where A11 and A22 are square (nonvacuous) matrices, otherwise it is irreducible. It follows that 1 ×1 matrices are always irreducible. In general, given a square Metzler matrix A, a permutation matrix P can be found such that
P AP = T
1
A11
A12
. . . A1ℓ
A22
. . . A2ℓ .. .. , . .
Aℓℓ
The symbols A and D are used to recall the words “arrival” and “departure”, respectively.
(1)
where each Aii is irreducible. (1) is usually known as Frobenius normal form of A [14]. Clearly, the directed graphs G(A) and G(P T AP ) are isomorphic and the irreducible matrices A11 , A22 , . . . , Aℓℓ correspond to the communicating classes C1 , C2 , . . . , Cℓ of G(P T AP ) (coinciding with those of G(A), after a suitable relabelling). When dealing with the graph of a matrix in Frobenius normal form (1), for every i ∈ hℓi, A(Ci ) ⊆ {i, i + 1, . . . , ℓ}, while D(Ci ) ⊆ {1, 2, . . . , i} = hii, so that A(Ci )∩D(Ci ) = {i}. On the other hand, if i > j then A(Ci )∩D(Cj ) = ∅, while if i < j the following conditions are equivalent A(Ci ) ∩ D(Cj ) 6= ∅
⇔
i ∈ D(Cj )
⇔
j ∈ A(Ci ).
Basic definitions and results about cones may be found, for instance, in [6]. We recall here only those facts that will be used within this paper. A set K ⊂ Rn is said to be a cone if αK ⊂ K for all α ≥ 0. A cone K is said to be polyhedral if it can be expressed as the set of nonnegative linear combinations of a finite set of generating vectors. A polyhedral cone K in Rn is said to be simplicial if it admits n linearly independent generating vectors. When so, v is an internal point of K if and only if v = Cu for some u ≫ 0. II. C ONTINUOUS - TIME
POSITIVE SYSTEMS AND THEIR REACHABILITY PROPERTIES
A continuous-time positive system is a continuous-time state-space model, described by the following equation: ˙ x(t) = Ax(t) + Bu(t),
t ∈ R+ ,
(2)
where x(t) and u(t) denote the n-dimensional state variable and the m-dimensional input, respectively, at the time instant t, A is an n × n Metzler matrix, and B is an n × m positive matrix. The state at any time instant t ∈ R+ , starting from the nonnegative initial condition x(0), and under the nonnegative soliciting input u(τ ), τ ∈ [0, t), is given by Z t At x(t) = e x(0) + eA(t−τ ) Bu(τ )dτ. 0
Since A is a Metzler matrix, the associated exponential matrix eAt is positive at any time t ≥ 0 [5], [18], thus ensuring the nonnegativity of the free state evolution; on the other hand, condition B ≥ 0 ensures that eA(t−τ ) Bu(τ ) ≥ 0 for every τ ∈ [0, t), and hence the forced evolution is nonnegative, too.
When dealing with positive systems, reachability properties always refer to the possibility of reaching positive final states by means of nonnegative input signals. Definition 1: [8], [10], [15] A state xf ∈ Rn+ is reachable at time T ∈ R+ , T > 0, if there exists a nonnegative piece-wise continuous input signal u : R+ → Rm + that leads the state trajectory from x(0) = 0 to x(T ) = xf . System (2) is reachable if every state xf ∈ Rn+ is reachable at some time instant T = T (xf ) > 0. System (2) is strongly reachable if for every T > 0, and every state xf ∈ Rn+ , the state xf is reachable at time T . In [8] it was proved (see Theorem 2) that system (2) is strongly reachable if and only if after a possible reordering of the inputs the matrix A is diagonal, the matrix B can be written as B = (D, B1 ) where D is an order n diagonal positive matrix with positive diagonal entries and B1 , which exists only if m > n, is an arbitrary n × (m − n) positive matrix. This amounts to saying that there exists an m × n selection matrix (a submatrix of some permutation matrix) S such that α1 α2 A=
..
. αn
BS =
β1
β2 ..
. βn
(3)
where αi ∈ R, while βi > 0 for every index i ∈ hni. This implies, in particular, that m ≥ n. In this contribution we aim to prove that strong reachability is equivalent to reachability also for continuous-time positive systems. This is possible by relating both properties to the (strong/standard) reachability of monomial vectors (see, also, Proposition 1 in [8]). This will allow us to provide for reachability the same characterization given in [8] for strong reachability.
Proposition 1: Given a continuous-time positive system (2), the following facts are equivalent: i) the system is reachable; ii) the system is monomially reachable [16], [17], which amounts to saying that every vector ei , i ∈ hni, of the canonical basis is reachable at some time instant Ti > 0; iii) the system is strongly reachable. Proof:
i) ⇒ ii) and iii) ⇒ i) are obvious.
ii) ⇒ iii) If ei is reachable at time Ti > 0 this means that ei =
R Ti 0
eA(Ti −t) Bui (t)dt, for some
nonnegative piece-wise continuous function ui : R+ → Rm + . Now, let T > 0 be an arbitrary positive number. If T ≥ Ti , then by simply using ( 0, ¯ i (t) := u ui (t − T + Ti ),
for t ∈ [0, T − Ti ); for t ∈ [T − Ti , T ],
we can surely reach ei at time T . On the other hand, if 0 < T < Ti , notice that for every t ∈ [0, Ti − T ] ZP
Z
t+T A(Ti −τ )
e
Bui (τ )dτ
t
and for some t¯ ∈ [0, Ti − T ] ! Z t¯+T ¯ ZP eA(Ti −τ ) Bui (τ )dτ = ZP eA(Ti −T −t) · t¯
Z
⊆ {i},
t¯+T
A(T +t¯−τ )
e
t¯
Bui (τ )dτ
!!
= {i}.
In fact, as t¯ varies from 0 to Ti − T , the window [t, t + T ] spans the whole integration interval. Consequently, there must be some interval of length T where the integral is nonzero, and when so the integral is necessarily a vector whose nonzero pattern coincides with i. Since ZP(eAs v) ⊇ R ¯ t+T ZP(v) ) ∅, for every s ≥ 0 and every v > 0 [18], it follows that ZP t¯ eA(T +t¯−τ ) Bui (τ )dτ =
{i}, and hence ei can be reached at time T by using in [0, T ] the input signal ui (t + t¯) R ¯ . ¯ i (t) = u t+T A(T +t¯−τ ) T ei · t¯ e Bui (τ )dτ
So, we have shown2 that if ei is reachable at some time instant Ti > 0, then it is reachable at
any time instant T > 0. So, if the system is monomially reachable, then for every T > 0 and every i ∈ hni the vector ei is reachable at time T . Consequently, for every T > 0, every strictly P ¯ i (t)[xf ]i , thus ensuring positive vector xf ∈ Rn+ is reachable at time T by using u(t) = ni=1 u
strong reachability.
So, once we resort to Theorem 2 in [8], we get the following result. Theorem 1: For the continuous-time positive system (2) the following facts are equivalent: i) the system is reachable; 2
It is worthwhile remarking that the reasoning here adopted does not extend to arbitrary positive states. So, if a state xf > 0
can be reached at finite time, it cannot always be reached at any time. This means that the concepts of reachability and strong reachability coincide, but when the system is not reachable what one can reach in finite time is generally different from what one can reach at any time.
ii) the system is strongly reachable; iii) m ≥ n and there exists an m × n selection matrix S such that the pair (A, B) is described as in (3), for some αi ∈ R and some βi > 0. Remark 1: In [8] it is erroneously stated, in the Introduction, that reachability and strong reachability are not equivalent. To support this claim, the Authors mention Example 13 in [10]. However, Example 13 provides evidence of a slightly different fact, namely that if a continuoustime positive system is not reachable, then the set of states that can be reached at time T > 0 may increase with T (this is the case, in particular, of essentially reachable systems, that we will address in the following section). As a matter of fact, as it has been now proved, when a continuous-time positive system is reachable, the set of states that can be reached at time T > 0 is always Rn+ , n being the system dimension, independently of T . It is worth noticing that the aforementioned situation is similar to what happens with discretetime positive systems as, indeed, under the reachability assumption every positive state can be reached in no more than n steps, while once reachability is lost there is no upper bound on the minimum number of steps required to reach every state belonging to the reachable set [10], [15]. III. E SSENTIAL
REACHABILITY
In this section we address a weaker form of reachability, which has been referred to in the literature either as “K-controllability” [13] or as “almost complete reachability” [10] or as “essential reachability” [7], [9], [23]. This property has been fully investigated in the discretetime case, first for single-input systems [13] and later for multi-input systems [7], [23]. We introduce here the definition and then provide a characterization for this property to hold in the continuous-time case. Definition 2: System (2) is essentially reachable if every strictly positive state xf ∈ Rn+ is reachable (by means of a nonnegative piece-wise continuous input) at some time T = T (xf ) > 0. We notice, first, that also in the case of essential reachability a characterization like the one provided in Proposition 1, which relates essential reachability to the possibility of going arbitrarily close (in finite time) to the vectors of the canonical basis, can be provided. Proposition 2: The continuous-time positive system (2) is essentially reachable if and only if it is essentially monomially reachable, meaning that for every vector ei , i ∈ hni, of the canonical
basis and every ε > 0 there exists a positive vector xi = xi (ε) ∈ Rn+ which is reachable at some finite time instant Ti > 0 and that satisfies kxi − ei k∞ < ε.
(4)
Proof: Necessity is obvious, since every strictly positive vector is reachable in finite time, and hence this must be true, in particular, for all strictly positive vectors which satisfy condition (4), for assigned i ∈ hni and ε > 0 (and for every ε > 0 there are infinitely many). Conversely, assume that the system is essentially monomially reachable and let xf be any strictly positive vector in Rn+ . Then there exists a sufficiently small ε > 0 such that xf is an internal point of the polyhedral cone generated by the strictly positive vectors ei,ε := ei + ε1n . For every i ∈ hni, let xi > 0 be a reachable state which satisfies (4) for the previously assigned ε, and suppose that xi is reached at time Ti > 0 by means of the nonnegative input ui (t), t ∈ [0, Ti ]. Clearly, xf is an internal point also of the polyhedral cone generated by the vectors xi , i ∈ hni. P Since this is a simplicial cone, it follows that xf = ni=1 xi ci , ∃ ci > 0. Set T := maxi Ti . By applying the same type of reasonings we resorted to within the first part
of the proof of Proposition 1, we easily prove that, for every i ∈ hni, there exists a nonnegative ¯ i (t) which allows to reach xi at time T . Consequently, xf can be reached at time T by input u P ¯ i (t)ci . This completes the proof. means of the nonnegative input signal u(t) = ni=1 u
Clearly, if If (A, B) denotes the set of indices i ∈ hni for which ei can be reached in finite
time (and hence, by the previous part of the paper, at any time instant), essential (monomial) reachability requires that, for every i 6∈ If (A, B), the corresponding canonical vector ei is reached at least approximately in finite time. Differently from the discrete-time case (and somehow counter-intuitively), approximately does not necessarily mean asymptotically [23]. Indeed, some canonical vectors may be reached with an arbitrarily good approximation by resorting to control intervals [0, T ] arbitrarily small, meanwhile other canonical vectors may be reached with an arbitrarily good approximation by resorting to intervals [0, T ] arbitrarily large. Dealing with the RT limit behaviors (for T → 0+ and for T → +∞) of the expression 0 eA(T −τ ) Bu(τ )dτ for suitable choices of u does not seem to be a feasible path to the problem solution. So, in order
to solve this problem it is convenient to resort to a different approach, which makes use of a technical lemma obtained by Ohta et al. in [15].
For any T > 0, let Xr (T ) denote the set of states that can be reached at time T . Then, the set of states that can be reached in finite time is Xr = ∪T >0 Xr (T ). So, while reachability amounts to requiring that Xr = Rn+ , essential reachability corresponds to condition Cl(Xr ) = Rn+ , Cl(·) denoting the topological closure. So, we search, now, for necessary and sufficient conditions for condition Cl(Xr ) = Rn+ to hold. As it has been proved in [15] (for single input systems, but the result can be easily extended to the multi-input case), Cl(Xr ) = K, where K := Cl Cone(x : x = eAt Bej , ∃ j ∈ hmi, t ∈ R+ ) .
(5)
So, Cl(Xr ) = Rn+ if and only if all canonical vectors ei , i ∈ hni, belong to K. Clearly, if there exist k ∈ N, j1 , j2 , . . . , jk ∈ hmi, τ1 , τ2 , . . . , τk ∈ R+ and c1 , c2 , . . . , ck ∈ R+ such P that ei = kr=1 eAτr Bejr cr , then it must necessarily be ZP(eAτr Bejr ) = {i} for some index r.
Consequently, if τr > 0 then [18] Aei = αi ei and Bejr = βi ei , for some αi ∈ R and some βi > 0, while if τr = 0 then Bejr = βi ei , for some βi > 0. So, a necessary and sufficient condition for the existence of a finite number of vectors in Cone(x : x = eAt Bej , ∃ j ∈ hmi, t ∈ R+ ) whose positive combination gives ei is that i belongs to IB := {i ∈ hni : ∃ j ∈ hmi, βi > 0 s.t. Bej = βi ei }. If ei ∈ K, but i 6∈ IB , then ei must belong to K \ Cone(x : x = eAt Bej , ∃ j ∈ hmi, t ∈ R+ ), which implies that it must be obtained as a limit. Since K is a closed cone in Rn+ and ei lies on the boundary of the positive orthant, this condition occurs if and only if there exists j ∈ hmi such that
eAt Bej = ei . lim t→+∞ keAt Bej k∞
(6)
So, we are remained with searching for conditions ensuring that, for every i 6∈ IB , (6) holds for some index j ∈ hmi. In order to investigate this condition, we assume throughout the rest of the paper w.l.o.g. that A is an n × n Metzler matrix in Frobenius normal form3 (1), with irreducible diagonal blocks Aii ∈ Rni ×ni , i = 1, 2, . . . , ℓ. Correspondingly, we will let Ci = {(n1 + n2 + . . . + ni−1 ) + 1, (n1 + . . . + ni−1 ) + 2, . . . , (n1 + . . . + ni−1 ) + ni } (n0 := 0, by 3
Of course, in the general case, we can reduce A to Frobenius normal form (1) by resorting to a suitable permutation matrix
P , meaning that P T AP is in Frobenius normal form. When so, we must correspondingly replace B with P T B.
definition) denote the ith communicating class of G(A), associated with Aii . As a consequence, the exponential matrix of A at t > 0 takes the following form [18] A11 (t) A12 (t) . . . A1ℓ (t) A22 (t) . . . A2ℓ (t) At e =: A(t) = .. .. , . .
(7)
Aℓℓ (t)
where
Aij (t) =
(
0,
if A(Ci ) ∩ D(Cj ) = ∅;
≫ 0, if A(Ci ) ∩ D(Cj ) 6= ∅.
Even more, we can refer to the following result (obtained by merging Theorem 5.4 and Proposition 6.1 in [18]), which enlightens the dominant modes of each block of the exponential matrix eAt and, consequently, the column dominant modes of eAt , when A is Metzler. Theorem 2: Let A ∈ Rn×n be a Metzler matrix in Frobenius normal form (1). For any pair of indices i and j in hℓi, we have: •
if A(Ci ) ∩ D(Cj ) = ∅, then Aij (t) = 0;
•
if A(Ci ) ∩ D(Cj ) 6= ∅, then the dominant mode of each entry of Aij (t) is eλi,j t
tm¯ i,j (and it m ¯ i,j ! := max{λmax (Akk ) : k ∈ A(Ci ) ∩ D(Cj )}, ∗
is weighted by a positive coefficient), where λ∗i,j
and m ¯ i,j + 1 is the maximum number of classes Ck with λmax (Akk ) = λ∗i,j that lie in a single chain from Cj to Ci in R(A). Consequently, there exist (not necessarily distinct) positive (eigen)vectors of A, vj ∈ Rn+ , with tm¯ j λ∗j t e , with λ∗j ∈ R and m ¯ j ∈ Z+ , j ∈ hℓi, and strictly kvj k∞ = 1, and real modes mj (t) = m ¯ j! 1×ni positive row vectors cTi ∈ R+ such that T c1 m1 (t) .. .. . . (8) A(t) = eAt = [ v1 . . . vℓ ] + Alc (t), mℓ (t)
cTℓ
and for every i ∈ hni, if we let Cj be the class of vertex i, Alc (t)ei = 0. t→+∞ mj (t) lim
Moreover, λ∗j ≡ max{λmax (Akk ) : k ∈ D(Cj )}, and m ¯ j + 1 is the maximum number of classes Ck with λmax (Akk ) = λ∗j that lie in a single chain from Cj in R(A).
We are now in a position for deriving necessary and sufficient conditions ensuring that (6) holds for some index j ∈ hmi. Proposition 3: Consider a continuous-time positive system (2). Assume w.l.o.g. that A is in Frobenius normal form (1) and refer to the notation and decomposition of Theorem 2. For any given i ∈ hni, condition (6) holds for some given index j ∈ hmi if and only if there exists k ∈ hni such that a) limt→+∞
eAt ek keAt ek k∞
= ei ;
b) k ∈ ZP(Bej ) and for every r ∈ ZP(Bej ), r 6= k, either limt→+∞ limt→+∞
mq (t) mp¯(t)
eAt er keAt er k∞
= ei or
= 0, where Cp¯ denotes the class of k and Cq the class of r.
Proof: For any j ∈ hmi, partition the jth column of B, Bej , according to the block . . . bTℓj ]T . Also, set Ij := {p ∈ hℓi : bpj 6= 0} and m (t) s < +∞}. Ij∗ := {p ∈ Ij : ∀ s ∈ Ij , lim t→+∞ mp (t)
partition (1) of A, i.e. as Bej = [ bT1j
From the decomposition (8), one easily gets
P T p∈Ij∗ (cp bpj )vp eAt Bej
. = lim
P
t→+∞ keAt Bej k∞
p∈Ij∗ (cTp bpj )vp ∞
As for every p ∈
Ij∗ ,
cTp bpj
> 0, the second term of the previous expression coincides with ei
if and only if, for every p ∈ Ij∗ , vp = ei . This condition is equivalent to a) and b), together. By making use of the previous result and of the graph theoretic interpretation of the previous conditions, we may obtain the following characterization of essential reachability. Theorem 3: Consider a continuous-time positive system (2), and assume w.l.o.g. that A is in Frobenius normal form (1). System (2) is essentially reachable if and only if, for every i 6∈ IB , there exist p ∈ hℓi and j ∈ hmi such that i) Cp is a final class consisting of the vertex i alone; ii) the set I(p, j) := {q ∈ hℓi : q ∈ A(Cp ) and blockq [Bej ] 6= 0} is not empty; iii) if we denote by d + 1 the maximum number of classes Ck with λmax (Akk ) = λmax (App ) that lie in a single chain from Cq to Cp in R(A), as q varies within I(p, j), then the following conditions hold: if blockq [Bej ] 6= 0 then λmax (Ass ) ≤ λmax (App ) for every s ∈ D(Cq ), and if there exists s ∈ D(Cq ), s 6= p, such that λmax (Ass ) = λmax (App ), then the number of classes Ck with λmax (Akk ) = λmax (App ) in the chain from Cq to Cs is not greater than d.
Proof: By resorting to the previous proposition, we are remained to showing that for every i 6∈ IB , the set of conditions i)÷iii) is equivalent to the pair of conditions a) and b). Specifically, we first prove that if conditions a) and b) hold for some indices i, j and k, then i)÷iii) hold true for a suitable index p. Let Cp be the communicating class vertex i belongs to. If condition a) holds, namely there exists k ∈ hni such that the kth column of eAt asymptotically aligns with ei , this means that the mode of the ith entry of eAt ek dominates all the other modes of eAt ek . But then, by the first part of Theorem 2, the class Cp must consist of a single element (so that App = [A]ii ), because if there would be h 6= i, h ∈ Cp , then hth entry of eAt ek would have the same dominant mode as the ith entry, and the limit vector would have at least two positive entries. Also, Cp must be final. If not, for every q such that Apq (t) 6= 0 it would be Asq (t) 6= 0 for every s ∈ D(Cp ), s 6= p, and the dominant mode of Asq (t), by Theorem 2, would either coincide with or dominate the dominant mode of Apq (t). So, if Cq is the class k belongs to, this would rule out the possibility that eAt ek asymptotically align with ei . This way we have proved i). If there exists k ∈ hni such that k ∈ ZP(Bej ) and eAt ek asymptotically aligns with ei , this implies that ∃ q¯ ∈ hℓi such that blockq¯[Bej ] 6= 0 and q¯ ∈ A(Cp ), namely ii) holds. Let now m∗ (t) be the dominant mode among the dominant modes of all blocks Apq (t), q ∈ I(p, j). Clearly, if the second part of condition b) holds, then m∗ (t) dominates the dominant modes of all nonzero blocks Asq (t), for every q such that blockq [Bej ] 6= 0 and every s ∈ D(Cq ), s 6= p. Even more, it must be m∗ (t) =
td [A]ii t e . d!
all the columns of the column block [ A1q (t)T
This implies that, when blockq [Bej ] 6= 0,
. . . Apq (t)T
. . . Aqq (t)T
...
0 ]T ei-
ther have dominant mode mq (t) dominated by m∗ (t) or have dominant mode m∗ (t) and they asymptotically align with ei . In both cases, this implies that λmax (Ass ) ≤ λmax (App ) for every s ∈ D(Cq ), s 6= p, and if there exists s ∈ D(Cq ), s 6= p, such that λmax (Ass ) = λmax (App ), then the chain in R(A) from Cq to Cs includes less than d + 1 classes Ck with λmax (Akk ) = λmax (App ). In fact, in this situation, the block Asq (t) exhibits a dominant mode associated with λmax (App ) = [A]ii but dominated by m∗ (t), i.e. a dominant mode like
tr [A]ii t e , r!
with r < d.
The fact that conditions i)÷iii) imply a) and b) can be proved by reversing the previous reasonings. Remark 2: It is worthwhile noticing that essential reachability (also known, as previously
reminded, as almost reachability) is not equivalent to almost strong reachability, characterized in [8] (see Proposition 7), which represents the possibility of reaching, at any time instant T > 0 and with arbitrarily good approximation ε, any positive state. Indeed, almost strong reachability is equivalent to condition IB = hni and hence trivially implies essential reachability. Remark 3: As an additional remark, it is worth noticing that once A is in Frobenius normal norm (1), we immediately determine the size nj of each class Cj . By Theorem 3, a necessary condition for vector ei , i 6∈ IB , to be obtained as a limit is that there is a class Cp with Cp = {i} and hence np = 1. This clearly implies that a necessary condition for essential reachability is that all indices i which correspond to classes Cj with nj > 1 belong to the set IB . Example 1: (Example 13 in [10]) Consider the continuous-time positive system which represents the Streeter-Phelps model for describing river pollution: " # " # −k1 0 1 ˙ x(t) = Ax(t) + Bu(t) = x(t) + u(t), k1 −k2 0 and assume k1 > k2 > 0. By resorting to the only nontrivial 2 × 2 permutation matrix P we get " # " # −k k 0 2 1 P T AP = , PTB = . 0 −k1 1 As IB = {2}, the second column of eAt asymptotically aligns with e1 and {2} = ZP(B), the system is essentially reachable.
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