On the positivity, excitability and transparency properties of a class of time-varying bilinear dynamic systems under multiple point internal and external delays

Abstract: This paper investigates formally the internal and external positivity, together with its excitability and transparency properties, of a class of bilinear time-varying continuous-time dynamic systems subject to (in general, non-commensurate) multiple internal and external point delays. The evolution operator is calculated in a closed form and the mentioned properties can be checked through direct testable expressions. The bilinear system class under consideration is driven by two inputs, so-called, the control input and the bilinear action input, which are not necessarily coincident, the second one taking account of the coupling stateinput defining the bilinear terms.


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Positive systems are characterized by the property that the state and output components are nonnegative for all time under non-negative inputs and non-negative initial conditions. In the context of differential, difference or hybrid equations, the solution is non-negative for non-negative forcing terms. Positivity is a relevant property inherent to biological and epidemic models and also to predator-prey-related problems, population evolution models, etc., often combined with the presence of delays either in the dynamics and/ or in the forcing terms or controls. On the other hand, the bilinear dynamic systems are a special class of non-linear systems where the nonlinearity consists of products between state and input components. An example is the regulation of electrical motors when neither the current nor the voltage are constant for all time so that the electric torque has a bilinear nature. It seems important to mention that positivity is a property related to the time behavior of the various signals rather than to the frequency responses.

Introduction
The positive systems are characterized by the property that the state and output components are non-negative for all time under non-negative inputs and non-negative initial conditions. Positivity is a relevant property in biological models (De la Sen, 2008a;Mailleret, 2004;Stevic, 2006) epidemic models (De la Sen, Agarwal, Ibeas, & Alonso-Quesada, 2011;McCluskey, Conell, & Yang, 2015;Monteiro, Gonçalves, & Piqueira, 2000), predator-prey-related problems (Al-Omari, 2015;Yagi & Ton, 2011), population evolution models, etc. The background literature in the field is abundant. See, for instance, Al-Omari (2015), De La Sen (2002), De la Sen (2007), De la Sen et al. (2011), Ebihara (2015), Farina and Rinaldi (2000), Haldar, Chakraborty, and Kar (2015), Kaczorek (2001), Kao (2014), Mailleret (2004), McCluskey et al. (2015), Monteiro et al. (2000), Nalbant et al. (2012), Nickel and Rhandi (2005), Sau, Niamsup, and Phat (2016), de la Sen (2008b), Shenand and Zheng (2015), Tingting, Baowei, and Yunxu (2015), Yagi and Ton (2011), and Zames (1966) and some references therein. Two classical books or recommended reading for those interested in the subject are Kaczorek (2001) and Farina and Rinaldi (2000) while more recent extensions have been discussed in Mailleret (2004). Bilinear systems are a special class of non-linear systems where the non-linearity consists of products between state and input components. In particular, the positivity theory has been extended from the continuous-time and discrete-time linear cases to mathematical models including hybrid continuous-time and discrete-time differential equations and hybrid dynamic systems (De la Sen, 2007) as well as to those eventually possessing internal (i.e. in the state) and external (i.e. in the inputs and/or outputs) delays. See, for instance, De la Sen (2007), De la Sen et al. (2011), Kao (2014), Mailleret (2004), Nickel and Rhandi (2005), Sau et al. (2016), de la Sen (2008b), Shenand and Zheng (2015), Tingting et al. (2015), Zames (1966), and De la Sen and Ibeas (2008) and references therein. In particular, stability results for positive systems are given in De la Sen (2007), Kao (2014), Nickel and Rhandi (2005), Sau et al. (2016), Shenand and Zheng (2015), Tingting et al. (2015) and Zames (1966), including the problem of hybrid systems with mixed coupled continuous-time and either discrete-time (arising from continuous-time discretization) or digital (arising from purely digital unrelated-to-continuous) dynamics, and some references therein while classical stability results are detailed in Kaczorek (2001). It is convenient to point out so as to avoiding potential confusion that positivity is a property related to the time-behavior of the various signals (input, state, and output components) through time (De La Sen, 2002), rather than to the respective frequency. Two properties of interest in positive systems are the "excitability" and the "transparency". The first one is related to the capability of all the state components to reach positive values in finite time from injection of some positive control with the system initially at rest. The second one is related to the capability of all the output components of the homogeneous system to reach positive values in finite time for any given positive initial condition. Another research subject of increasing interest is that of time-delay differential/difference systems and related dynamic systems. The main motivation is that the subjects related to time-delayed dynamics are of interest to mathematicians since such a dynamics is described by functional equations of difficult analysis while they are also of relevant interest to engineers since a number of physical models have inherent delays (sunflower daily motion equation, ship maneuver dynamics, war-peace models, transmission signal problems, biological problems, etc.). In particular, time-delayed systems with internal delays, i.e. those being present in the state-dynamics) possess infinitely many poles. The related background literature is exhaustive. See, for instance, Al-Omari (2015), De La Sen (2004), Ebihara (2015), Haldar et al. (2015), McCluskey et al. (2015), Yi, Ulsoy, and Nelson (2006), Domoshnitsky and Volinsky (2015) and De la Sen and Ibeas (2008) and references therein. It turns out that some systems which are positive, by nature, have also delayed dynamics. See, for instance, Al-Omari (2015), Haldar et al. (2015), andMcCluskey et al. (2015). On the other hand, sums and differences of positive operators on separate Hilbert spaces have been investigated in Kittaneh (2004) through the derivation of norm inequalities. This paper investigates formally the properties of positivity, excitability and transparency of a class of generalized bilinear dynamic systems with multiple internal delays (i.e. in the state dynamics) and two inputs, namely, the control input and the, so-called, bilinear input action which is coupled to the state implying the presence of bilinear terms in the dynamics. In some particular cases of interest in applications, the bilinear action can be identical to the control action itself (Feng, Chen, Sun, & Zhu, PAC qn (Dom ; Ran) is the set of nth vector functions of class q of domain Dom and range Ran, whose qth derivative (or the function itself if q = 0) is everywhere piecewise continuous.

Problem formulation: Dynamic bilinear system with delays and positivity conditions
Consider the bilinear time-varying system: where x(t) ∈ R n , u(t) ∈ R m and y(t) ∈ R p are the state, piecewise-continuous control input and output vectors, respectively, and v j (t) ∈ R n are the bilinear piecewise-continuous action inputs; ∀j ∈q ∪ 0 , are piecewise-continuous matrices of bounded entries, where q = 1 , 2 , .... , q and q � = 1 , 2 , .... , q � , h j ; ∀j ∈q � ∪ 0 and h � j ; ∀j ∈q ∪ 0 are internal and external point delays A system of similar structure of (1)-(2) parameterized by A (G) i (t); ∀i ∈q ∪ 0 , E (G) (t), B (G) j (t); ∀j ∈q � ∪ 0 , C (G) (t) and D (G) (t); ∀t ∈ R 0+ whose entries are zero if the corresponding ones of (1)-(2) are zero and unity otherwise is said to be the associated system to (1)-(2). The inputs u (G) (t) and v (G) j (t) are defined with unity and zero components, if the corresponding ones of u(t)and v j (t) are non-zero or zero, respectively, for each given t ∈ R + . The above matrices are said to be the associated matrices to the corresponding ones of (1)-(2). Theorem 1. Assume that A j (t) = A 0j +Ã j (t), where A 0j is constant; ∀j ∈q ∪ 0 . Then, the following properties hold: (i) The solution of (1) is given explicitly by: and x(t) = (t) for t ∪ −h, 0 , which is unique in R + ∪ −h, 0 and continuously differentiable in R + , for each given piecewise continuous vector function : −h, 0 → R n continuous at t = 0 of initial conditions with x 0 = x 0 = 0 and for each given piecewise-continuous inputs u : R + ∪ −h � , 0 → R m , with u(−t) = 0; ∀t ∈ R + , and v j : R 0+ → R n ; ∀j ∈q ∪ 0 such that the fundamental matrix : −h, 0 ∪ R + → R n×n satisfies the homogeneous auxiliary time-invariant delayed system: where U(t) is the Heaviside function. (1) The fundamental matrix satisfying (4) is continuously differentiable in R + and defined by: which becomes (t) = e A 00 t for t ∈ 0 , h 1 .
(iii) The fundamental matrix satisfies the following constraint for any t 1 , t 2 ≥ t 1 ∈ R 0+ : Proof By convenience, one first prove Property (ii). One gets directly from (6) that (t) = 0 for t ∈ −h, 0 and 0 = I n . By taking derivatives with respect to time, one gets: so that (5) holds. Thus, the first expression of (5) is a solution of (4) for all t ∈ R 0+ subject to initial con- The proof of Property (i) is as follows. Take time derivatives in (3) using (4) and then (3) again to recover the values x t − h j for t ≥ h j with x(t) = (t) for t ∈ −h, 0 to yield: which coincides with (1) so that (3) is a solution of (1) for the function of initial conditions : −h, 0 → R n . Such a solution is unique as it follows under close arguments to those of Property (i) to prove the uniqueness of the fundamental matrix being based on Picard-Lindelöff theorem since the matrix functions parameterizing (1) and the control and bilinear action inputs are piecewise continuous, and the function of initial conditions is piecewise continuous.
To prove Property (iii), take the homogeneous time-invariant auxiliary delay system: with initial conditions (t) = 0 for t ∈ −h, 0 and z 0 = z 0 = 0 ≠ 0. For any t 2 ≥ t 1 ≥ 0, one has from (3): since the solution z : R 0+ ∪ −h, 0 → R n is unique, Property (iii) follows directly. □ The following result, whose proof is similar to that of Theorem 1 and then omitted, is a consequence of Theorem 1 using alternative auxiliary differential systems to obtain corresponding fundamental matrices.
Corollary 1. The unique solution of (1) obtained in Theorem 1 can be equivalently expressed as follows: with any fundamental matrix L : −h L , 0 ∪ R + → R n×n defined for any given subset L such that 0 ⊆ L ⊆q ∪ 0 , so that L = q ∪ 0 �L and h L = max j∈L h j as follows: satisfying the homogeneous auxiliary time-invariant dynamics: □ Note that all the dynamics which is not present in the selected homogeneous auxiliary system is incorporated as a forced term in the solution using the superposition principle. Note that if L = 0 then the fundamental matrix {0} (t) = e A 00 t is a C 0 -semigroup of infinitesimal generator A 00 and if L =q ∪ 0 then L (t) = (t) as addressed in Theorem 1. In general, the associated evolution operators T : R 0+ → R n which defines the unique state-trajectory solution of the homogeneous system (1) are not strongly continuous evolution operators since they do not satisfy the semigroup property.
The following result establishes that any fundamental matrix of the form (9) is positive for all time if A 00 is a Metzler matrix and A 0j ≥ 0; ∀j ∈q.
Proof Since A 00 ∈ M n E , e A 00 t ≻ 0; ∀t ∈ R 0+ (Lemma A.2, Appendix A). Now, from (9), L (t) = e A 00 t ≻ 0; ∀t ∈ 0, h 1 . If L (t) = e A 00 t ≻ 0; ∀t ∈ R 0+ , the result is proved. Otherwise, and since all the entries of the fundamental matrix are everywhere continuous in R 0+ , assume that there is some t 1 > h 1 such that L (t) ≻ 0; ∀t ∈ 0, t 1 and at least one entry of the fundamental matrix L ij t 1 < 0 for some i, j ∈n. Then, there exists a connected real interval t 2 , t 1 ⊂ R + such that L ij : R + � t 2 , t 1 → R − in view of (9) since e A 00 t ≻ 0; ∀t ∈ R 0+ and A 0j ≻ 0; ∀j ∈q. But this contradicts L (t) ≻ 0; ∀t ∈ 0, t 1 . Thus, L (t) ≻ 0; ∀t ∈ R 0+ and the result is proved. □ It can be pointed out that in the internal delay-free case, i.e. q = 0, (t) = e A 00 t ≻ 0; ∀t ∈ R 0+ , since A 00 ∈ M n E , so that Lemma 1 still holds. This property also holds in the time-delay case if t ∈ 0, h 1 . Note that Lemma 1 is independent of the bilinear dynamics of (1). Some specific definitions are first needed and now established for the subsequent formal framework and note that the matrices of dynamics, functions of initial conditions and control input and bilinear input action are always assumed to satisfy the piecewise continuity and absolute piecewise continuity assumptions, without giving specific "ad hoc" indications, given when defining the dynamic system (1)-(2).
(2) The bilinear system (1)-(2) is said to be positive with respect to some C v such that R (q+1)n (3) The bilinear system (1)-(2) is said to be externally positive (respectively, C v -externally positive) if for every u : R 0+ → R n 0+ and every v ∊ C v , one has y : A set C v as defined in Definitions 1.2 and 1.3 is (in general) non-properly inclusive of R (q+1)n 0+ in the sense that C v -positivity (respectively, C v -external positivity) implies positivity (respectively, external positivity). Note that if the system is C v -positive then it is also C v -externally positive since C v -positivity implies output positivity for zero initial conditions if v ∊ C v from the above definitions. The converse is not true, in general. Note that the positivity properties of a system (1)-(2) are kept by its associated system.

Theorem 2. Assume that
Then, the bilinear system (1)-(2) is C v -positive and then also positive, where Proof Since A 00 ∈ M n E and A 0j ≻ 0 (Assumption A.1); ∀j ∈q (Assumption A.1) then (t) ≻ 0; ∀t ∈ R 0+ , irrespective of the delays, from Lemma 1. In addition, one has from (3) from Assumption A.2, that . This follows directly via a contradiction argument as follows. Assume that there is some t 1 > 0 such that x t 1 < 0 and is negative, such a first time t = t 1 has to exist for that since x(t) is everywhere continuously differentiable subject to non-negative initial conditions and non-negative controls. But, from the assumptions, this is impossible from (3) unlessx i (t) < 0 for some i ∈nand t ≺ t 1 , a contradiction. Furthermore,y(t) ⪰0 from (2) and (3) ; ∀t ∈ R 0+ since Assumption A.3 holds. As a result, the bilinear system (1) then it is also positive. Property (i) has been proved. □ Note that the spurious case C(t) = 0 for some t ∈ R 0+ is not considered in Theorem 2, although the result still holds in the case when C(t) ⪰ 0for t ∈ R 0+ . Note that a modified bilinear system. could be considered instead of (1)-(2) where the bilinear dynamics contribution results from a coupling in-between the output and the action input and the linear dynamics matrices are constant. In this case, the following result close to Theorem 2 holds: THEOREM 3. The following properties hold:

Proof Note that the solution (3) to (1) is replaced with
From (13) and (2), one gets: The proof of Property (i) is similar to the proof of Theorem 2 and then omitted. To prove Property (ii), take zero initial conditions in (13) to yield the "zero-state" output Thus, it follows from (14) under continuity arguments of the zero state output that the system is externally positive if Assumption A.5 holds. □ The subsequent result establishes that Assume also that h 1 ≥ h 10 with h 10 being sufficiently large. Then, a necessary and sufficient condition for the time-invariant resulting bilinear system to be positive if v : Proof The sufficiency part of the proof follows directly from Theorem 2 since the time-invariant bilinear system is a particular case of (1)-(2). To prove the necessity part, first note that the solution of (3) becomes in this particular case by incorporating the constant Ã j matrix to the evolution operator and B i are the kth columns of A 0j , Ã j and B i , respectively, for j ∈q, i ∈q � ∪ 0 , k ∈n and i ∈m and k (.), x k (.), E T j and u (⋅) are the kth and ℓth components of (⋅), x(⋅), E j and u(⋅), there is some t 1 ∈ R + such that e (A 00 +Ã 0 )t 1 is not positive and e (A 00 +Ã 0 )t ≻ 0 for t ∈ 0, t 1 . Furthermore, (t) = e (A 00 +Ã 0 )t for t ∈ 0, h 1 . Also, if the lower-bound h 10 of the smaller delay h 1 is large enough such that h 10 ≥ t 1 then x t 1 = t 1 x 0 = e (A 00 +Ã 0 )t 1 x 0 and there is some pair i, j ∈n ×n such that < 0 for some , s ∈n. Now, take x 0 = 0 and u(t) = 0, j (t) = 0; ∀j ∈q ∪ 0 , ∀t ∈ R 0+ and a piecewise-continuous function of initial conditions: Thus, one gets from (16.b) that the state-trajectory solution becomes: j for ∈n and j ∈qis the ℓth component of (t) = j for The expression (19) follows from the mean value theorem for integrals which is applicable here since the evolution operator is everywhere continuous on its definition domain. Since (t) is also positive for t ∈ R 0+ , 0 = I n , for each given ∈ 0, 1 , it always exists a sufficiently small for t ∈ h j − j , h j and any sufficiently small j ∈ h j−1 , h j if the ℓth component of (t) in −h j , −h j + j is a constant value j and ψ is small enough satisfying: A 0i +Ã i ⪰ 0; ∀i ∈q while B i ⪰0 fails for some i ∈q � ∪ 0 , the system (1)- (2) is not positive. Thus, B i ⪰0 ; ∀i ∈q � ∪ 0 is necessary for positivity of (1)-(2). The extended proof under negativity of more than one entry of the set B i : i ∈q � ∪ 0 could be obtained under a direct more cumbersome development via contradiction arguments.
To prove the necessity of E i ⪰0; ∀i ∈q ∪ 0 for positivity of the system, take u(t) = 0; ∀t ∈ R 0+ and some constant i (t) =̄i ⪰0; ∀i ∈q ∪ 0 , ∀t ∈ R 0+ . Then, one gets from (1): if some component of E 0 is negative, say E 0 i < 0, it always exists an off-diagonal i, j entry of A 0 +v 0 E T 0 which is negative for a sufficiently large ̄0 j > 0. Thus, A 0 +v 0 E T 0 is not Metzler so that the unique solution of (24) given, equivalently, by: ∀t ∈ R + with x(t) = (t)⪰ 0 for t ∈ −h, 0 is not positive for some t ∈ R + and x 0 = x 0 = 0 ≻ 0 for some given admissible vector function of initial conditions is not positive for some such a function if the first delay h 1 is sufficiently large. In the same way, if E j is not positive for some j ∈q, it also follows that A j +v j E T j is not positive for sufficiently large acting bilinear input ̄j so that the solution is not positive for some t ∈ R + . So, a necessary condition for positivity of the time-invariant version of (1)- (2) is that E j ≻ 0; ∀j ∈q ∪ 0 . Also, it turns out from (2) that if either C ≻ 0 or D⪰0 fails then there are x 0 ≻ 0 with u 0 = 0 or u 0 ≻ 0 with x 0 = 0 such that y 0 is not positive. □ Remark 1. Note that, more generally, and without essential changes in the proof of Corollary 2, the bilinear action input could be chosen constant belonging to the admissibility class ∀x ∈ X e ≻ 0 and X e ⊃ x e being the equilibrium set. Since it follows that X e ⊂ R n + , (De la Sen, 2007;Kao, 2014;Mailleret, 2004). On the other hand, since the system is positive Since such an inverse and u e are unique then X e = x e so that the equilibrium point is unique. Furthermore, is non-singular and positive it has at least one non-zero entry

Excitability, strong excitability, external excitability and transparency
The following excitability definitions extend directly the one given in Farina and Rinaldi (2000) for the concept of excitability of a time-invariant linear system then extended in de la Sen (2008b) for a time-invariant system under point delays.
Definition 2 (excitability). A positive bilinear system (1)-(2) is said to be excitable (respectively, externally excitable) if each state variable (respectively, output variable) can be made positive by applying an appropriate non-negative control input and bilinear input action to the system being initially at rest, i.e. for the case (t) = 0 for t ∈ −h, 0 .
Remark 2. It turns out that we could refer to excitability and external excitability of a particular state (respectively, output) variable it is individually excitable without the need for excitability of the complete state (respectively, output) vector. We could also refer in a natural way to asymptotic excitability if the excitability is an asymptotic property.
On the other hand, we can also refer to the above excitability properties with respect to any of the two inputs or, eventually, their components. Excitability, respectively, external excitability being achievable from any control input component or bilinear action component of some control input or bilinear input action will be referred to as strong excitability or strong external excitability, respectively. Note that in many applications the bilinear input action coincides with the control input and it can be of no free-choice.
Theorem 4 leads to a direct excitability consequent result as follows: Corollary 3. Assume that all the assumptions of Theorem 4 hold. Then, the resulting time-invariant (1) is asymptotically excitable and also excitable (in a finite time).
Proof The asymptotic excitability of is obvious from Theorem 4 by fixing u(t) = u e , v j (t) = v je for j ∈q ∪ 0 and t ∈ R 0+ since the state trajectory solution converges asymptotically to a strictly positive equilibrium point. On the other hand, (25) implies that that the strictly positive equilibrium point is strictly increasing with u e for any constant fixed v j (t) = v je for j ∈q ∪ 0 and t ∈ R 0+ . So, for any such a u e , it exists , 0 ∈ R + , and a finite time T = T u e , , 0 such that the state-trajectory solution is strictly positive, x e i > ; ∀i ∈n, and the system is excitable if u e (t) = u e = u e , v j (t) = v je for j ∈q ∪ 0 and all t ∈ R 0+ and any ∈ R + ≥ 0 verifying furthermore that 0 < | | | x e i − | | | ≤ x i (t) ≤ x e i − ; ∀i ∈n and any t ≥ T with x(t) → x e as t → ∞. Thus, the result is proved. □ Remark 3. Note that the excitability of Corollary 3 is guaranteed without requiring the need for conditions for the existence of a strictly positive equilibrium, just from the positivity of the system and the constraint which is a particular case guaranteeing the necessary and sufficient condition of excitability of the time-invariant version of the positive system.
Conditions of excitability of a positive system are now formalized in the subsequent result through the excitability of its associated system. For such purposes, one takes advantage of the fact that the successive powers of the matrix A (G) 00 are positive even if all those of the Metzler A 00 are not positive matrices.
Theorem 5. Consider the system (1) Assume that each entry of each of the matrices Ã i (t), E i (t), B j (t), C(t) and D(t); ∀i ∈q ∪ 0 , ∀j ∈ q � ∪ 0 is either null or non-zero for all time so that their associated matrices are constant. Assume also that the maximum internal and external delays h and h ′ are sufficiently small such that some real t exists which satisfies the constraint max h, h � = max h q , h � q < t < h 1 + with ɛ defined in Lemma A.1 (Appendix A). Then, the following properties hold: (i) A necessary and sufficient condition for the system to be strongly excitable, and then excitable, from the control input is: A sufficient condition for the ith state component to be strongly excitable, and then excitable, from the control input is: (ii) Assume, in addition, that ∑ q � j=0 B j is monomial. A sufficient condition for the system to be excitable from the control input is that A sufficient condition for the ith state component to be excitable from the control input is Weaker corresponding sufficient conditions if ∑ q � j=0 B j ≻≻ 0 is monomial are, respectively, Further weaker corresponding sufficient conditions, which are also necessary, are, respectively, (iii) A necessary and sufficient condition for the system to be strongly externally excitable from the control input is: A sufficient condition for the ith output component to be strongly excitable from the control input is: (iv) A necessary and sufficient condition for the system to be externally excitable from the control input is: A necessary and sufficient condition for the ith output component to be excitable from the control input is: Now, define for any ∈ R 0+ (note that C v0 = C v ) and, in addition, assume that v ∊ C vδ . Then, the following additional properties hold: (v) A sufficient condition for the system to be strongly excitable from the bilinear input action is A sufficient condition for the ith state component to be strongly excitable from the bilinear input action is that (vi) A sufficient condition for the system to be externally excitable from the bilinear input action is: A sufficient condition for the ith output component to be excitable from the bilinear input action is: (vii) Define e = (1, 1, … , 1) T = ∑ n i=1 e i ∈ R n and the subsets of n ; ∀j ∈n. Then, a sufficient condition for the system to be excitable from the combined control input and bilinear input action is that n = n u ∪ �⋃ n k=1 n vk � . A sufficient condition for strong combined excitability is quite similar by redefining the sets n u ℓ and n vjs ; ∀ ∈q � ∪ 0 , ∀ s ∈q ∪ 0 , by removing from the definitions of n u and n vj , the summations ∑ q � l=0 (⋅) and ∑ q =0 (⋅). Sufficient conditions for strong output excitability and output excitability are also direct under small "ad hoc" modifications.
Proof Assume with no loss of generality that Ã i (t)⪰ 0; ∀i ∈q ∪ 0 , ∀t ∈ R 0+ and that the fundamental matrix function (G) (⋅) is calculated from Theorem 1 by replacing the parameterization of the original system by that of its associated one. If this were not the case, since A i (t)⪰ 0; ∀i ∈q ∪ 0 , ∀t ∈ R 0+ by hypothesis, there are (non-unique) additive decompositions of the forms so that a valid fundamental matrix for the homogeneous associated system can be calculated via Theorem 1 with the replacements A 0i →Ā 0i , Ã i (t) →Ā i (t); ∀i ∈q ∪ 0 , ∀t ∈ R 0+ to be used in the subsequent formulas of the proof.
One has from Theorem 1 for u:R 0+ → R m and v ∊ C v that the state of the associated system, under corresponding normalized non-negative initial conditions, is described by: since each entry of each of the matrices Ã i (t), E i (t), B j (t), C(t) and D(t); ∀i ∈q ∪ 0 , ∀j ∈ q � ∪ 0 is either null or non-zero their corresponding matrices of the associate system are constant for all time and since the conditions on the parameters guarantee the positivity which lead to the inequalities (31.a)-(31.c) under any non-negative initial conditions and any non-negative control input as well as the time-invariance of the associated system, since v ∊ C v , with where ∈ + ≤ n is the degree of the minimal polynomial of A (G) 00 and k (t) : k ∈ − 1 ∪ 0 , t ∈ R 0+ is a linearly independent set of real functions of real domain which are everywhere infinitely time-differentiable with respect to time. The use of (32) in (31.c) leads to If h and h ′ are sufficiently small such that some real t exists which satisfies the constraint max h q , h � q = max h, h � < t < h 1 + for ɛ defined in Lemma A.1 so that the functions of the set k (t) , , k ∈ − 1 ∪ 0 , t ∈ R 0+ are positive on the real interval 0, . Thus, it follows from (32.b) and (33) that, if max h, h � < t < h 1 + , then x (G) (t) ≻≻ 0 (then the associated system and the system are both strongly excitable through any non-negative control having just one non-zero component), so that x(t) ≻≻ 0 (then strongly excitable as well), if u (G) (t) ≻ 0 then (with just any of their components being non-zero (G) 0j ⪰0; ∀i ∈q ∪ 0 , ∀j ∈q ∪ 0 and A k 00 (G) ;∀k ∈ 0+ . The sufficiency part of Property (i) concerning strong excitability (i.e. excitability of all the sate components from any control input component) and state component-wise excitability has been proved. On the other hand, note from (32.a) and (34) that, since for any positive control input with just one positive component for a non-zero time interval, what yields to the sufficient conditions for excitability and component-wise state excitability. This proves the sufficiency of the given constraint of non-strong excitability. On the other hand, if the corresponding parametrical constraint fails then the state of the

0+
= 0 for any given non-negative control input under zero initial conditions for all t ∈ R 0+ , where its parameterizing vector (G) 0+ ∈ R 0+ has positive and/or null components depending, respectively, of if the component to the entry of the corresponding parameterizing matrix is zero or unity, where Ω is the total number of entries accounting for all the parameterizing matrices. This implies, as a result by construction and the excitability constraint, that the state of any given system having the above associated one, by a vector 0+ ∈ R 0+ of either positive and/or null components which are either zero (if its corresponding entry of a parameterizing matrix is zero) or unity (if the corresponding matrix entry is positive), has a state-solution trajectory x (G) t, 0+ = 0; ∀t ∈ R 0+ . On the other hand, any other system of state x( , t), whose associated system state is still x (G) (⋅), being parameterized by any ∈ R (of eventual negative components of the off-diagonal entries of the Metzler delay-free dynamics), satisfies 0 = x( , t)≺ sup 0+ ∈R 0+ x 0+ , t = 0; ∀ ∈ R , ∀t ∈ R 0+ if the corresponding strong excitability constraints fails. Hence, the necessity of each of the given constraints for the corresponding excitability property.
The proof of the sufficiency part of Property (ii) follows directly from those considerations and Property (i) leading to the three given pairs of sufficiency-type conditions from the strongest one to the weakest one since if ∑ q � j=0 B j ≻ 0 and monomial then j is monomial and positive, so that it is non-singular with just a non-zero positive entry per row and per column, then its pre-multiplication by any strictly positive matrix, as those in the given conditions, yields a strictly positive matrix as a result. The necessity follows by a close argument as that used for the proof of necessity of Property (i) got from the further weakest sufficient conditions of excitability and component-wise excitability: The proofs of the sufficient parts of Properties (iii) and (iv) follow in a very close way as those of Properties (i)-(ii) via (2) and the corresponding output equation for its associated system since via (2) and the corresponding output equation for its associated system since C ≻ 0, D ⪰0. The proof of the necessary part follows from very close arguments to those used for the proof of the necessity part of Property (i).
To prove Property (v), note that if the initial conditions and control are zero, one gets from (31.a) for all t ∈ R + , since v ∊ C vδ for some ∈ R + , for t ∈ max h, h � , h 1 + . Note that (36) does not leads to excitability since the system initially at rest implies that the state is identically zero for identically zero control input Comparing with the parametrical constraint specifying Property (i) which guarantees (33)  for any t a ∈ R + and some : R 0+ → R 0+ such that (t) > 0 within some, non-necessarily connected, interval T a ⊂ t a , ∞ of non-zero measure and, in addition, assume that v ∊ C vδ . Note that Theorem 5 [(v)-(viii)] are also fulfilled under the weaker condition that v ∈ C v (t) t a for t a ∈ max h, h � , h 1 + .
Definition 3 (transparency). A positive bilinear system (1)-(2) is said to be transparent if each output component of the homogeneous system can be made positive for some given non-negative appropriate function of initial conditions, i.e. for the case u(t) = v j (t) = 0 for t ∈ R 0+ , j ∈q ∪ 0 .
Transparency being achievable from any function of initial conditions will be referred to as strong transparency.
The following result holds: Theorem 6. Consider the system (1) Assume that each entry of each of the matrices Ã 0i (t), E i (t), B j (t), C(t) and D(t); ∀i ∈q ∪ 0 , ∀j ∈ q � ∪ 0 is either null or non-zero for all time. Assume also that the maximum internal and external delays h and h ′ are sufficiently small such that some real t exists which satisfies the constraint Then, the following properties hold: (i) A sufficient condition for the system to be strongly transparent, and then transparent, is: A necessary and sufficient condition for the ith output component to be strongly transparent, and then transparent, is: (ii) A sufficient condition for the system to be transparent is: (iii) A necessary and sufficient condition for the system to be transparent is that Proof From (2), (31.a) and (32.b), one gets if u(t) ∈ R m 0+ and v ∊ C vd ; ∀t ∈ R 0+ , ∀ ∈ R 0+ that Since h and h ′ are sufficiently small such that some real t exists which satisfies the constraint max h q , h � q = max h, h � < t < h 1 + for ɛ defined in Lemma A. Thus, the real functions of the set k (t) , , k ∈ − 1 ∪ 0 , t ∈ R 0+ are positive on the real interval 0, . The proofs of the sufficiency part of Properties [(i)-(ii)] follow directly from (38)-(39). The proof of necessity of Property (i) is close to its counterpart of Theorem 5(i). On the other hand, one has from (37) that and, using (32.b) for the lower bound of the fundamental matrix, the sufficiency part of Property (iii) follows. The proof of necessity follows under close arguments to those used in the proof of the necessary part of Theorem 5(i). □ Note that, in the discrete-time case, the excitability and transparency conditions of a positive system can be equivalently got from the system itself, instead of from its associated one, since the matrix of dynamics has to be positive instead of simply Metzler. The subsequent example addresses these facts.
Example 1. Consider the following discrete bilinear system with a single one-step delay: under initial conditions x i = 0 for i = −1, 0, with the non-negative scalar bilinear action sequence v k k∈ 0+ ⊆ R 0+ , where C ∈ R p×n 0+ and i , B 2 ∈ R n×m 0+ , B i ∈ R n×n 0+ ; i = 0, 1 are non-zero. It follows that the system is not excitable from any bilinear action sequence since x k = 0; ∀k ∈ 0+ if u k = 0; ∀k ∈ 0+ . Equation (41) maybe described equivalently through an extended system of dimension 2n and Equation (44) establishes the following result: THEOREM 7 The following properties hold: (i) Assume that, in addition, rank ̄ k = 2n and that is monomial for a given bilinear action sequence v k ⊂ R 0+ for some k ∈ 0+ , where ̄ k is the controllability matrix with respect to the control input sequence u k defined by: and (43) yields for any k, j ∈ 0+ proceeding recursively that and then The following result holds: THEOREM 8. Assume that Ker ̄,̄̄, … ,̄2 n−1̄ ∩ R 2n 0+ ≠ 0 , i.e. the pair ̄,̄ is uncontrollable. Then, there is a subclass of bilinear control actions belonging to the admissibility class such that the system (41) is not excitable under zero control input for the bilinear input actions in such a subclass.
Proof Note that, equivalently, it is assumed that rank ̄,̄̄, … ,̄2 n−1̄ < 2n, i.e. Then, there are infinitely many non-zero sequences v k ⊂ R 2 , equivalently, infinitely many real non-negative sequences u k such that x 2n+k =̄2 n+kx 0 for any given x 0, −1 ∈ R n 0+ . If x 0 = x −1 = 0 then x k = 0 from (45) and X k = 0; ∀k ∈ 0+ for the class defined by such bilinear input action sequences. It is obvious that infinitely many of them are in the admissibility class for positivity while they do not excite the state. The result has been proved. □ Note that if the control input is identically zero then one gets from (47) that where C = C, 0 p×n and Thus, the following holds: THEOREM 9. The system (41) is observable on k, k + 2n if and only if rank k = 2n. This condition also guarantees that the system is transparent although it is not a necessary condition for it.
Proof The necessity and sufficiency for observability is a standard condition for the extended system of dimension 2n. Since the observability matrix is non-singular and positive, there is some positive x k ≻ 0 such that y k ≻ ≻ 0(it suffices to take anyx k ≻ ≻ 0) but, since k ≻ 0, the observability is not needed to guarantee transparency. □ Note that the system is not strongly transparent on k, k + 2n since O k ≻≻ 0 fails since C ≻≻ 0 fails because of its structure.
x(t) = Ax(t) + B 2 u k + B 0 v k x k + B 1 v k−1 x k−1 ; t ∈ kT, k + 1 T ; ∀k ∈ 0+ u k = u kT = Kx k + K 0 x k−1 subject to initial conditions x i ∈ R 0+ for i = −1, 0 for some given T ∈ R + , and some given control gains K, K 0 ∈ R m×n 0+ and ∈ R 0+ or ∈ R n×n 0+ , where A ∈ M n×n E , B 0 , B 1 ∈ R n×n 0+ and B 2 ∈ R n×m 0+ . Then, where and = ∫ T 0 e A(T− ) d . Since A ∈ M n×n E , e AT ≻ 0 and also non-singular, since it is a fundamental matrix, and then ≻ 0. Since, furthermore, B i ≻ 0; i = 0, 1, 2 and K 0 , K, ≻ 0, C ≻ 0; ̄k ≻ 0 and C k ≻ 0 for all k ∈ 0+ . So, the system is transparent for any given x 0 ≻≻ 0. The discussion of potential extensions of the above examples to the positivity of dynamic systems subject to switching in-between several parameterizations and to the discretization of continuous-time systems under non-periodic sampling can be addressed directly being supported by some technical results proved in De la Sen (1983), De la Sen, Paz, and Luo (1998) and Ibeas, De La Sen, and Alonso-Quesada (2004). In particular, note that if the continuous -time matrix of dynamics of a given parameterization is a Meltzer one, then its associate state-transition (or fundamental) matrix is positive for each transition in-between any two consecutive samples and non-singular irrespective of the used sampling period sequence.

Funding
This work was supported by the Spanish Government and to the European Fund of Regional Development FEDER for Grant DPI2015-64766-R and to UPV/EHU for Grant PGC 17/33.