# Small-gain stability theorems for positive Lur’e inclusions

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## Abstract

Stability results are presented for a class of differential and difference inclusions, so-called positive Lur’e inclusions which arise, for example, as the feedback interconnection of a linear positive system with a positive set-valued static nonlinearity. We formulate sufficient conditions in terms of weighted one-norms, reminiscent of the small-gain condition, which ensure that the zero equilibrium enjoys various global stability properties, including asymptotic and exponential stability. We also consider input-to-state stability, familiar from nonlinear control theory, in the context of forced positive Lur’e inclusions. Typical for the study of positive systems, our analysis benefits from comparison arguments and linear Lyapunov functions. The theory is illustrated with examples.

## Keywords

Differential inclusion Exponential stability Input-to-state stability Lur’e systems Population biology Positive systems## Mathematics Subject Classification

34A60 37N35 39A30 47B65 93D20## 1 Introduction

Positive dynamical systems, or simply positive systems, are dynamical systems where the evolution map leaves a positive cone invariant. The most natural positive cone is the nonnegative orthant of Euclidean space, equipped with the partial order of component wise inequality, noting also the often interchangeable use of the words “positive” and “nonnegative” in this context. The study of positive systems is motivated by numerous applications across a diverse range of scientific and engineering contexts, such as communications, logistics, economics, biology, chemistry, and ecology. Invariance of a positive cone captures the essential property that state-variables of positive systems, typically modelling abundances or concentrations, must take nonnegative values to be meaningful. Consequently, positive systems are well-studied objects, with textbooks which address the subject including [1, 2, 3, 4, 5, 6]. The study of positive systems described by linear dynamic equations is grounded in the seminal work by Perron and Frobenius in the early 1900s on irreducible and primitive matrices. Attention has more recently turned to generalisations of the Perron–Frobenius theorem to nonlinear maps, including [7, 8, 9], for instance. Related to the theory of positive systems is the theory of monotone dynamical systems, where solutions inherit the same ordering (that is, with respect to the partial order which defines the positive cone) throughout time as their initial states; see, for example [10, 11] or [12], and the references therein.

The breadth and depth of applications of positive and monotone systems has generated significant interest in their control [1, 13] which is currently an active research field. The analysis of positive control systems benefits from readily constructed and easily scalable Lyapunov functions [14]. In addition to numerous important examples, control of positive systems is motivated by the challenges which nonnegativity constraints place, since subtraction is not always well-defined in positive cones, for instance. As such, the following fundamental facets of linear control theory all require different treatments for positive systems: reachability and controllability [15, 16], observability [17], realisability [18, 19] and stabilisability [20, 21]. However, the additional structure afforded by positivity is often intuitive, mathematically helpful and thus simplifies matters. For example, in classical robust control, the complex and real stability radii, introduced in [22, 23] are, in general, different for linear systems, but are known to be equal for linear positive systems, see [24].

Differential inclusions arise as the appropriate mathematical framework for rigorously describing the solutions of differential equations specified by discontinuous functions and are now a classical subject addressed in many textbooks, including [25, 26, 27]. Their study has partly been motivated by optimal control theory [28], developed in economics and engineering, see also [29, 30]. Crucial to the mathematical underpinning of differential inclusions is the concept of a set-valued (or multi-valued) map [31], one of the building blocks of non-smooth analysis. In systems and control theory, differential inclusions provide a toolbox for modelling hybrid systems [32], robust control [33, 34] as well as hysteresis effects, such as backlash or play [35, 36].

*A*,

*B*and

*C*are appropriately sized matrices, with certain nonnegativity properties and

*F*is a nonnegative set-valued function. Differential inclusions of the form (1.1) often arise in closed-loop from the linear control system

*F*is singleton-valued.

In the usual situation that \(0 \in F(0)\), it follows that zero is an equilibrium of (1.1). We present sufficient conditions in Theorem 2.5 for the zero equilibrium to be globally stable, asymptotically stable, and exponentially stable. We note that in a dynamical systems context, exponential stability is in many respects a more important and natural notion of stability than asymptotic stability alone, see [37]. Our results are instances of so-called absolute stability theory; see, for example [36, 38, 39, 40, 41, 42] and the references therein, in that we formulate assumptions on the transfer function \(\mathbf{G}(s) = C(sI -A)^{-1}B\), where *s* is a complex-variable, which ensure the respective notions of stability for all set-valued functions *F* for which the “product” \(\mathbf{G}(0)F(y)\) satisfies certain norm bounds. The conditions may be interpreted as small-gain conditions in a weighted one-norm. In the spirit of absolute stability theory, stability of the zero equilibrium is determined by the norm-estimates and not by the individual nonlinearity *F* itself. The inherent robustness to uncertainty in *F* adds to the appeal and utility of absolute stability results. As corollaries we obtain stability results for certain time-varying Lur’e inclusions and to systems of positive Lur’e differential inequalities, both of which we describe. Our analysis crucially depends on the positive systems structure as we make extensive use of comparison arguments, and linear Lyapunov functions, which also go by the names of linear copositive Lyapunov functions [43] and are examples of sum-separable Lyapunov functions [44]. These techniques are not applicable in the general (non-positive) case, where, for example (see [47, Theorem 3.2]), quadratic Lyapunov functions are used which come from classical systems theoretic concepts such as the solutions of controller/observer Lyapunov equations or algebraic Riccati equations associated with the bounded real lemma or positive real lemma.

*D*is a set-valued function which models external forcing or disturbances. Clearly, when \(D(t) = \{0\}\) for all \(t \ge 0\), then (1.1) and (1.3) coincide. A sample forced Lur’e system is depicted in a block diagram arrangement in Fig. 1. We comment that (1.3) includes the situation wherein the forcing acts through

*B*, that is, \(D = B E\), for another set-valued function

*E*, in which case (1.3) is the feedback interconnection of (1.2) and

*x*is a solution of (1.3). Our Theorem 2.11 states that the same assumptions which ensure exponential stability of zero in (1.1) also ensure that the zero equilibrium pair of (1.3) enjoys the exponential ISS property, paralleling recent findings in [47, 48].

*x*under the left-shift operator, viz. \(x^+(t) = x(t+1)\), for all \(t \in \mathbb N_0\). In Theorem 3.1 we present analogous sufficient small-gain conditions for various global stability properties of (1.4).

The motivation for the current study is to derive stability results for multivariable (that is, multi-input multi-output, or MIMO) positive Lur’e inclusions—nonlinear positive control systems—which, as already stated, arise in a variety of applied scenarios. Of particular interest to the authors are models arising in biology and ecology, naturally positive systems, which are the subject of Example 4.2. In this context, Lur’e difference equations have been proposed as models in, for example [49, 50, 51, 52], and have also been proposed for models of plant species with seed banks [53]. Briefly, this connection has been made as Lur’e systems allow both vital or transition rates which are density-independent (that is, linear) and density-dependent (that is, nonlinear), the latter facilitating the modelling of Allee [54, 55, 56], competition or crowding effects. As inherently “noisy” systems, not described by classical, well-understood equations of motion, there is often much uncertainty in parametrising ecological models, with different qualitative trends being observed in different parametrisations [57]. The robustness to such uncertainty afforded by the treatment of dynamic inclusions and set-valued analysis is thus especially applicable and appealing. Although clearly existing results for the absolute stability of Lur’e inclusions such as those in [36, 48] do apply here, since these papers are not aimed at positive systems, their results are necessarily more conservative.

There is some partial overlap between the present work and [58, 59], where absolute stability results for continuous- and discrete-time positive dynamical systems (of differential or difference equations, respectively, including Lur’e systems) are presented, although for unforced systems only. These results also appear in the monograph [1, Chap. 5] by the same authors. In [1, 58, 59] a linear dissipativity theory approach is adopted and we compare results in Remarks 2.14 and 3.4. Our work in part extends that of [49, 60] which consider positive Lur’e difference and differential equations, respectively, both with scalar nonlinearities. It is possible in these specific situations to present “trichotomies of stability” explicitly in terms of the model data, which ensure that, under certain assumptions, solutions either converge to the zero equilibrium, or a unique non-zero equilibrium, or diverge. Similar limit set trichotomies have been established for various classes of monotone discrete-time dynamical systems in [2, 61, Chap. 6] for finite-dimensional systems and in [12, 50, 51] for infinite-dimensional systems. In Remark 3.5, we compare some of our findings with those from the monotone systems and control literature, namely [62, 63]. In the present paper we restrict attention to stability of the zero equilibrium alone and do not consider limit set trichotomies.

The paper is organised as follows. Sections 2 and 3 contain our main results, namely, global stability properties of the Lur’e differential inclusions (1.1) and (1.3), and the Lur’e difference inclusion (1.4), respectively. Section 4 contains three examples discussed in detail. We present some brief summarising remarks in Sect. 5.

**Notation**We collect notation and terminology used in the sequel. The symbols \(\mathbb N\) and \(\mathbb R\) denote the sets of positive integers and real numbers, respectively, and \(\mathbb N_0 = \{0\} \cup \mathbb N\). For \(n, m \in \mathbb N\), we let \(\underline{n}:= \{1,2,\ldots ,n\}\), \(\mathbb R^n\) and \(\mathbb R^{n \times m}\) denote usual

*n*-dimensional Euclidean space, and the set of \(n \times m\) matrices with real entries, respectively. The superscript \({}^T\) denotes both matrix and vector transposition. For \(M,N \in \mathbb R^{n \times m}\) with entries \(m_{ij}\) and \(n_{ij}\), respectively, we write

*M*is nonnegative. We call

*M*positive or strictly positive if \(0 <M\) or \(0 \ll M\), respectively, noting that there are different conventions present in the literature for the term

*positive matrix*. Given \(v \in \mathbb R^n_+\), \(0 \ll v\), we letwhere the

*i*-th component of \(\vert x \vert \in \mathbb R^n_+\) is defined to be \( \vert x_i \vert \). We note that \(\vert \cdot \vert _{v}\) is a norm on \(\mathbb R^n\) and, if \(x \in \mathbb R^n_+\), then \(\vert x \vert _{v} = v^T x\).

*i*,

*j*)-th entry of \(M^k\) is positive. Strictly positive matrices are evidently irreducible, and small irreducible matrices (which are not strictly positive) include

*r*(

*M*) and \(\alpha (M)\) denote the spectral radius and spectral abscissa of

*M*, respectively, which we recall are given by

*M*. For \(v \in \mathbb R^n\), \(\Vert v \Vert \) and \(\Vert M \Vert \) denote a (any) monotonic norm of

*v*and the corresponding induced operator norm of

*M*, respectively. Recall that a norm is monotonic if for all \(x, y \in \mathbb R^n\)

*p*-norm for \(1 \le p \le \infty \) is monotonic. We let \(\Vert \cdot \Vert _1\) denote both the one-norm and induced one-norm.

*I*and \(\mathbb {1}\) denote the identity matrix and vector with every component equal to one, respectively, the size of which shall be consistent with the context. We note that with the above definitions \(\Vert x \Vert _1 = \vert x \vert _{\mathbb {1}}\), for \(x \in \mathbb R^n\).

*X*, we let

*P*(

*X*) denote the power set of

*X*, \(P_0(X) = P(X){\setminus } \{\emptyset \}\) and, if

*X*is a subset of a normed-space,

*TX*are defined as

## 2 Continuous-time systems

In this section we consider continuous-time Lur’e inclusions, treating the unforced and forced cases in separate subsections. We first collect some terminology and results which are used in both subsections.

*x*is said to be a global solution. If \(x: [0, \omega ) \rightarrow \mathbb R^n\) is a solution of (2.1) for some \(0<\omega \le \infty \), then \(\dot{x}\) is a locally integrable selection of \(t \mapsto H(t, x(t))\). Therefore, for all \(0\le t_1 \le t_2 < \omega \), the set

*x*satisfies the integral inclusion

*G*is singleton-valued then (2.5) simplifies to a system of Lur’e differential equations; a special case which we consider in Sect. 2.3. We next state and prove a “variation of parameters” expression for solutions of forced Lur’e inclusions. No stability or positivity properties are required for this result.

### Lemma 2.1

*x*satisfies the inclusion

### Proof

*z*on [0,

*t*] by \(z(\tau ) := \mathrm {e}^{A(t-\tau )} x(\tau )\). Obviously, \(z(0) = \mathrm {e}^{At}x^0\) and \(z(t) = x(t)\), and a routine calculation shows that

*z*satisfies the differential inclusion

*x*is a solution of (2.5) if, and only if,

*x*is a solution of

*A*,

*B*,

*C*,

*D*and

*G*as in (2.4):

- (
**A1**) -
\((A,B,C) \in \mathbb R^{n \times n} \times \mathbb R^{n\times m}_+ \times \mathbb R^{p \times n}_+\),

*A*is Metzler, \(D : \mathbb R_+ \rightarrow P_0(\mathbb R^n_+)\) and \(G : \mathbb R_+ \times \mathbb R^p_+ \rightarrow P_0(\mathbb R^m_+)\); - (
**A2**) -
\(\alpha (A)<0\);

- (
**A3**) - there exists \(R>0\) such that$$\begin{aligned} \Vert w \Vert \le R \Vert y\Vert \quad \forall \; w \in G(t,y) \; \; \forall \, t \in \mathbb R_+ \; \; \forall \, y \in \mathbb R^p_+. \end{aligned}$$

**A1**) and (

**A3**) in the autonomous case as well, meaning that \(G(t,y) = F(y)\) for some \(F : \mathbb R^p \rightarrow P_0(\mathbb R^m)\).

**A1**). Indeed, under (

**A1**), the function

*G*is only defined on \(\mathbb R_+ \times \mathbb R^p_+\). Some existence theory (see, for example [25, Theorem 1, p. 97]) for (2.1) assumes that \(H : I \times \varOmega \rightarrow P_0(X)\) where \(I \subseteq \mathbb R\) and \(\varOmega \subseteq X \subseteq \mathbb R^n\) are open, which clearly fails with \(I = \mathbb R_+\) and \(\varOmega = \mathbb R^n_+\). Therefore, if \(J : \mathbb R_+ \times \mathbb R^n_+ \rightarrow P_0(\mathbb R^n_+)\) denotes the right hand side of the differential inclusion (2.5) under (

**A1**), then we extend

*J*to \(\mathbb R\times \mathbb R^n\), and denote the extension \(J_\mathrm{e}: \mathbb R\times \mathbb R^n \rightarrow P_0(\mathbb R^n_+)\), by setting \( J_\mathrm{e}(t,x) := J(\max \{0,t\},\mu (x))\) with \(\mu : \mathbb R^n \rightarrow \mathbb R^n_+\) defined by

*J*and \(J_\mathrm{e}\) coincide on \(\mathbb R^n_+\), we may seek solutions of \(\dot{x} \in J_\mathrm{e}(x)\), as an immediate consequence of the positivity assumptions in (

**A1**) is the following.

### Lemma 2.2

Under assumption (**A1**), every solution \(x:[0, \omega ) \rightarrow \mathbb R^n\) of (2.5) with \(x^0 \in \mathbb R^n_+\) takes values in \(\mathbb R^n_+\) for all \(t \in [0, \omega )\).

### Proof

The claim follows from the variation of parameters inclusion (2.6) combined with the nonnegativity assumptions in (**A1**). Crucially, we have used that *A* is Metzler if, and only if, \(\mathrm{e}^{At} >0\) for all \(t\ge 0\); see, for example [10, Section 3.1].\(\square \)

We conclude this section by commenting that the loop-shifting property, see (2.7), holds without the assumptions (**A1**)–(**A3**). It demonstrates that we may replace *A* and *G* in (1.3) by \(A+BKC\) and \((t,y) \mapsto G(t,y) - Ky\), respectively. In particular, *A* may not satisfy assumption (**A2**), but \(A+BKC\) may do so, for some \(K \in \mathbb R^{p\times m}\). In other words, this means that there is a so-called stabilising static output feedback \(u = Ky\) for the linear system (1.2), which is the approach we take in Example 4.1. Of course, in the current setting of positive Lur’e inclusions, the choice of *K* shall be constrained by the requirement that \(A+BKC\) and \((t,y) \mapsto G(t,y) - Ky\) satisfy (**A1**) as well, which may be infeasible in some cases.

### 2.1 Unforced systems

We consider the Lur’e inclusion (1.1) with assumptions (**A1**)–(**A3**). Recall from the previous section that this is a special case of (2.5). We note that the assumptions on *A*, *B*, *C* and *F* do not *a priori* guarantee that (1.1) admits solutions. Existence of solutions is not the focus of the present investigation, primarily as it is an extensively studied subject in the literature. That said, we do make some comments and provide some references regarding the existence of solutions. Recall the extension \(J_\mathrm{e}\) from Sect. 2, defined in terms of \(\mu \) from (2.8). The following lemma shows that \(J_\mathrm{e}\) inherits many useful properties from *J*. The proofs are readily established once it is noted that \(\mu \) is Lipschitz with Lipschitz constant equal to one.

### Lemma 2.3

Let \(J : \mathbb R_+ \times \mathbb R^n_+ \rightarrow P_0(\mathbb R^n_+)\) and define \(J_\mathrm{e}(t,x) := J(\max \{0,t\}, \mu (x))\) where \(\mu \) is given by (2.8). If *J* has any of the following properties: bounded, closed valued, convex valued, upper semi-continuous, lower semi-continuous, then \(J_\mathrm{e}\) has the corresponding property.

Examples of results ensuring existence of solutions of (1.1) include [25, Theorem 3, p. 98], [26, Proposition 6.1, p. 53] and [27, Theorem 7.5.1, p. 279]. The results [25, Theorems 1 and 4, p. 97, p. 101], [26, Lemma 5.1, p. 53] and [26, Theorem 6.1, p. 53] provide conditions under which global solutions of (1.1) exist, from which we obtain the following.

### Proposition 2.4

Given the Lur’e inclusion (1.1), assume that (**A1**) and (**A3**) hold and that *F* is upper semi-continuous with closed, convex values. Then, for all \(x^0 \in \mathbb R^n_+\), there is a global solution of (1.1), and every solution may be extended to a global solution. Further, every solution *x* satisfies \(x(t) \in \mathbb R^n_+\) for all *t* where *x*(*t*) is defined.

### Proof

The lemma follows from applications of [26, Lemma 5.1, p. 53] and [26, Corollary 5.2, p. 58], which establish existence of solutions, and extension to global solutions, respectively. Lemma 2.2 ensures that every solution is nonnegative. \(\square \)

We shall later briefly consider non-autonomous versions of (1.1), specifically (2.5). In this context [26, Theorem 5.2, p. 58] provides further assumptions on *G* which, combined with (**A1**)–(**A3**), guarantee the existence of global solutions. In fact, under (**A1**)–(**A3**), any result guaranteeing existence of local solutions which uses continuous (or Caratheodory) selections of \(x \mapsto Ax + BF(Cx)\) (or \((t,x) \mapsto Ax + BG(t,Cx)\)) will, in fact, ensure existence of global solutions by well-known theory of maximally defined solutions of ordinary differential equations (see the proof of [25, Theorem 1, p. 97]).

The main result of this section is presented next, and contains a suite of stability results for (1.1) formulated in terms of weighted one-norm inequalities and \(\mathbf{G}(0)\), assuming that global solutions exist. Here \(\mathbf{G}\) denotes the transfer function of the triple (*A*, *B*, *C*), that is, \(\mathbf{G}(s) = C(sI-A)^{-1}B\), where *s* is a complex variable. Recall that a matrix \(A \in \mathbb R^{n \times n}\) is Metzler with \(\alpha (A) <0\) if, and only if, \(-A^{-1} > 0\) (see, for example [5, Characterisation N\({}_{38}\) in Section 6.2] or [66, characterisation F\({}_{15}\)]). Consequently, under assumptions (**A1**) and (**A2**), it follows that \(\mathbf{G}(0) = -CA^{-1}B \in \mathbb R^{p \times m}_+\).

### Theorem 2.5

**A1**)–(

**A3**) hold.

- (i)If there exists a strictly positive \(v \in \mathbb R^p_+\) such thatthen there exists \(\varGamma >0\) such that, for all \(x^0 \in \mathbb R^n_+\), every global solution$$\begin{aligned} \vert \mathbf{G}(0) w \vert _{v} \le \vert y \vert _{v} \quad \forall \; w \in F(y) \; \; \forall \, y \in \mathbb R^p_+, \end{aligned}$$(2.9)
*x*of (1.1) satisfies$$\begin{aligned} \Vert x(t) \Vert \le \varGamma \Vert x^0\Vert \quad \forall \, t \in \mathbb R_+. \end{aligned}$$ - (ii)If there exist a strictly positive \(v \in \mathbb R^p_+\) and a lower semi-continuous function \(e : \mathbb R^p_+ \rightarrow \mathbb R_+\) such thatthen, for all \(x^0 \in \mathbb R^n_+\), every global solution$$\begin{aligned} e(y) \; >0 \quad \text {and} \quad \vert \mathbf{G}(0) w \vert _{v} +e(y) \; \le \vert y \vert _{v} \quad \forall \; w \in F(y) \; \; \forall \, y \in \mathbb R^p_+{\setminus }\{0\}, \end{aligned}$$(2.10)
*x*of (1.1) satisfies \(x(t) \rightarrow 0\) as \(t \rightarrow \infty \). - (iii)If there exist a strictly positive \(v \in \mathbb R^p_+\) and \(\rho \in (0,1)\) such thatthen there exist \(\varGamma , \gamma >0\) such that, for all \(x^0 \in \mathbb R^n_+\), every global solution$$\begin{aligned} \vert \mathbf{G}(0) w \vert _{v} \le \rho \vert y \vert _{v} \quad \forall \; w \in F(y) \; \; \forall \, y \in \mathbb R^p_+, \end{aligned}$$(2.11)
*x*of (1.1) satisfies$$\begin{aligned} \Vert x(t) \Vert \le \varGamma \mathrm{e}^{-\gamma t}\Vert x^0\Vert \quad \forall \, t \in \mathbb R_+. \end{aligned}$$

The notion of stability concluded in statement (i) is often called “stability in the large”, see [36, Definition 3], which, when combined with statements (ii) and (iii) yield that the zero equilibrium of (1.1) is globally asymptotically and globally exponentially stable, respectively. Before proving the above theorem, we provide some commentary on assumptions (2.9)–(2.11).

### Remark 2.6

*M*corresponding to the eigenvalue

*r*(

*M*), the existence of which is ensured by the Perron–Frobenius Theorem (see, for example [5, Theorem 1.4, p. 27]). Further, if there exists an irreducible matrix \(M \in \mathbb R^{p \times p}_+\) with \(r(M)\le 1\) and a lower semi-continuous function \(\zeta : \mathbb R^p_+ \rightarrow \mathbb R_+^p\) such that

*M*which satisfies (2.12) with the property that \(r(M) <1\), then (2.11) holds. \(\square \)

### Proof of Theorem 2.5

Throughout the proof, we let \(x:\mathbb R_+ \rightarrow \mathbb R^n_+\) denote a global solution of (1.1) for given \(x^0 \in \mathbb R^n_+\).

*A*is Metzler with \(\alpha (A)<0\), in light of (2.17) we have, for \(z \in \mathbb R^n_+\),

**A3**) and (2.18), we may estimate

**A2**), we conclude that statement (i) holds.

**A2**), (

**A3**) and (2.19) to estimate that \(\xi \) satisfying

*x*satisfies

**A2**) gives \(T_2> T_1\) such that for all \(t \ge T_2\)

*x*is bounded, and hence from (1.1) and (

**A3**) it follows that \(\dot{x}\) is bounded as well. Hence

*x*is uniformly continuous, and so is

*Cx*. Consequently, there exists \(\delta >0\) such that

*x*from statement (i). As \(t_k \nearrow \infty \) as \(k \rightarrow \infty \) we may assume that \(t_{k+1} > t_{k} + \delta \) for all \(k \in \mathbb N\) (by redefining the sequence \((t_k)_{k \in \mathbb N}\) if necessary). Define

*e*is a lower semi-continuous function which is positive-valued for positive arguments, and \(\mathcal {M}\) is a compact set which does not contain zero.

*f*is non-increasing and \(t_{k+1} \ge t_k + \delta \) for all \(k \in \mathbb N\)

*f*.

**A3**), (2.11), (2.24) and (2.25). Multiplying both sides of (2.26) by \(\mathrm {e}^{\gamma t}\), and taking \(w \in F(\mathrm {e}^{-\gamma t} \xi )\) with \(\xi \in \mathbb R^p_+\), we see that

*z*is a solution of

*Cz*and (

**A3**) we see that

In certain cases, the weighted one-norm estimate (2.9) itself implies that *F* must satisfy (**A3**), which we formulate as the next lemma.

### Lemma 2.7

*A*,

*B*,

*C*and

*F*satisfy (

**A1**) and (

**A2**), and let \(\mathbf{G}(s) = C(sI-A)^{-1}B\). If (2.9), and at least one of the two conditions

- (a)
\(m =p\) and \(\mathbf{G}(0)\) is irreducible;

- (b)
\(B,C \ne 0\),

*B*has no zero columns and there exists \(\varDelta \in \mathbb R^{m \times p}_+\) such that \(A + B\varDelta C\) is irreducible;

*F*satisfies (

**A3**).

### Proof

**A3**) holds with \(R : = c\theta _2/\theta _1\).

*A*is Metzler). As the integrand is continuous and nonnegative, (2.30) implies that

*i*-th row and

*j*-th column of

*C*and

*B*, respectively, where we have used that \(C \ne 0\). For each \(r \in \underline{m}\), we see from (2.31) that

The condition (2.10) is an intermediate between (2.9) and (2.11) which, as the next result shows, simplifies if more regularity assumptions are made on *F*.

### Corollary 2.8

**A1**)–(

**A3**) hold and that

*F*is upper semi-continuous with closed values. If there exists a strictly positive \(v \in \mathbb R^p_+\) such that

*x*of (1.1) satisfies \(x(t) \rightarrow 0\) as \(t \rightarrow \infty \).

### Proof

*F*means that we may choose another subsequence of \((y_k)_{k \in \mathbb N}\), again not relabelled, such that

*x*with radius \(r >0\). Assumption (

**A3**) implies that \(F(y_*)\) is bounded and so, by (2.34), \((w_k)_{k \in \mathbb N}\) is bounded, and hence has a convergent subsequence, not relabelled, with limit \(w_*\). Necessarily, from (2.34) we see that \(w_* \in \overline{F(y_*)} = F(y_*)\), as

*F*is assumed to be closed valued. We conclude that

Although formulated for autonomous problems, Theorem 2.5 readily extends to the non-autonomous differential inclusion (2.5) (still with \(D=\{0\}\)), provided the conditions in Theorem 2.5 hold uniformly in time. We formulate these claims in the next corollary.

### Corollary 2.9

*A*,

*B*,

*C*and

*G*satisfy (

**A1**)–(

**A3**). Define

*F*given by (2.35) satisfies the conditions (2.9)–(2.11) in Theorem 2.5, then the conclusions of Theorem 2.5 hold for every global solution of (2.5) with \(D = \{0\}\).

### 2.2 Forced systems

With regards to the existence of solutions of (1.3), we refer to, for example [26, Chap. 3, Sections 5–6] or [67, Theorems 1.1, 1.3, pp. 30–31]. The following result is based on [26, Theorem 5.2].

### Proposition 2.10

- (i)
(

**A1**)–(**A3**) hold; - (ii)
*F*has closed, convex values, \(F(t, \cdot )\) is upper semi-continuous and \(F(\cdot ,x)\) is measurable; - (iii)
*D*is measurable, locally bounded with closed, convex values.

*x*satisfies \(x(t) \in \mathbb R^n_+\) for all

*t*where

*x*(

*t*) is defined.

### Proof

Existence of global solutions follows from an application of [26, Theorem 5.2] and [26, Corollary 5.2, p. 58] ensures that solutions may be extended to global solutions. Lemma 2.2 ensures that every solution is nonnegative. \(\square \)

Assumption (**A3**) implies that \(F(0) = \{0\}\) and hence \(x=0\), \(D = \{0\}\) is a solution of (1.3) with \(x^0 = 0\), which we shall hereafter refer to as the zero equilibrium pair. We proceed to state and prove the main result of the present section, namely that the assumptions made in statement (iii) of Theorem 2.5 are sufficient for the zero equilibrium pair of (1.3) to be so-called exponentially ISS. The proof is similar to that of statement (iii) of Theorem 2.5, and uses a so-called exponential weighting argument, see [36].

As will become apparent in the proof of Corollary 2.12, it is convenient in the next result to impose no assumption on sign of the disturbance term *D*, but instead assume that the state *x* remains nonnegative.

### Theorem 2.11

**A1**)–(

**A3**) hold. If there exist a strictly positive \(v \in \mathbb R^p_+\) and \(\rho \in (0,1)\) such that (2.11) holds, then there exists \(\varGamma , \gamma >0\) such that, for all \(x^0 \in \mathbb R^n_+\), all locally bounded \(D : \mathbb R_+ \rightarrow P_0(\mathbb R^n)\) and every global nonnegative solution

*x*of (1.3),

If *D* is nonnegative-valued, that is \(D(t) \subseteq \mathbb R^n_+\) for almost all \(t \ge 0\), then the hypotheses of Theorem 2.11 ensure that (2.36) holds for every global solution *x* of (1.3). The inequality (2.36) is the definition of exponential ISS of the zero equilibrium pair (in the current set-valued setting).

### Proof of Theorem 2.11

*x*be a global, nonnegative solution of (1.3). Let \(\varepsilon , \gamma >0\) be such that \(\rho + \varepsilon <1\) and \(\alpha (A + \gamma I) <0 \). Defining \(z(t) := \mathrm {e}^{\gamma t}x(t)\) for \(t \ge 0\), which is also nonnegative, an elementary calculation using (1.3) shows that

*z*is a solution of

*a*and monotonicity of the integral

*a*and \(\xi \), it follows from (2.42) thatby (2.43). Now, as \(0 \le v^T C(-\gamma I - A)^{-1} z(t)\), appealing to (2.41) yields that

*t*,

*D*and \(x^0\).

**A3**) and \(\alpha (A+\gamma I) <0\), we estimate \(\Vert z(t) \Vert \) using the above inclusion as follows

*t*,

*D*and \(x^0\). Thus,

The next corollary is a so-called ISS *with bias* result (see [36, 48]) which states that if the condition (2.11) fails on a bounded set, then solutions of (1.3) (which include those of (1.1)) still admit uniform estimates of the form (2.36), but with an additional positive constant term. We note that ISS with bias is closely related to the concept of *input-to-state practical stability* (ISpS), see [68, 69]. Although ISpS applies to more general nonlinear forced control systems, a difference is that it does not typically specify the form of the additional constant, denoted \(\beta \) in the bounds below.

### Corollary 2.12

**A1**)–(

**A3**) hold. If there exist a strictly positive \(v \in \mathbb R^p_+\), \(\rho \in (0,1)\) and \(\varTheta \ge 0\) such that

*x*of (1.3) satisfies

The number \(\beta \) in (2.47) seeks to capture the extent to which the inequality in (2.45) is violated on the set \(\{ y \in \mathbb R^p_+ {:} \, \Vert y \Vert < \varTheta \}\). Observe that if \(\varTheta = 0\), then \(\beta =0\) and the conclusions of Theorem 2.11 and Corollary 2.12 coincide.

### Proof of Corollary 2.12

*S*(

*y*) is given by (2.48), so that

*H*satisfies (

**A3**). For given \(x^0 \in \mathbb R^n_+\) and \(D : \mathbb R_+ \rightarrow P_0(\mathbb R^n_+)\), let

*x*denote a global solution of (1.3). As \(D(t) \subseteq \mathbb R^n_+\) for almost all \(t \ge 0\), we have that \(x \ge 0\) and thus

*x*is also a global nonnegative solution of

*F*and

*D*in (1.3) replaced by

*H*and

*E*, respectively. Although it is possible that \(E(t) \not \subseteq \mathbb R^n_+\) for some \(t \ge 0\), since

*x*is nonnegative it suffices to verify that

*E*is locally bounded. We consider two exhaustive cases: if \(\Vert Cx(t) \Vert \ge \varTheta \), then

*x*satisfies the estimate (2.46). \(\square \)

### 2.3 Positive Lur’e differential equations and inequalities

*x*is a solution of (1.3) with

*F*satisfies assumption (

**A3**) if, and only if, there exists \(R>0\) such that

### Corollary 2.13

**A1**) and (

**A2**) hold and that

*g*satisfies (2.54).

- (i)If \(d=0\) and there exists a strictly positive \(v \in \mathbb R^p_+\) such thatthen there exists \(\varGamma >0\) such that, for all \(x^0 \in \mathbb R^n_+\), every global solution$$\begin{aligned} \vert \mathbf{G}(0) g(t,y) \vert _{v} \le \vert y \vert _{v} \quad \forall \, t \in \mathbb R_+, \; \; \forall \, y \in \mathbb R^p_+, \end{aligned}$$(2.55)
*x*of (2.53) satisfies$$\begin{aligned} \Vert x(t) \Vert \le \varGamma \Vert x^0\Vert \quad \forall \, t \in \mathbb R_+. \end{aligned}$$ - (ii)If \(d=0\) and there exist a strictly positive \(v \in \mathbb R^p_+\) and a lower semi-continuous function \(e : \mathbb R^p_+ \rightarrow \mathbb R_+\) such thatthen, for all \(x^0 \in \mathbb R^n_+\), every global solution$$\begin{aligned} e(y) \; >0 \quad \text {and} \quad \vert \mathbf{G}(0) g(t,y) \vert _{v} + e(y) \; \le \vert y \vert _{v} \quad \forall \, t \in \mathbb R_+, \; \; \forall \, y \in \mathbb R^p_+\backslash \{0\}\, \end{aligned}$$(2.56)
*x*of (2.53) satisfies \(x(t) \rightarrow 0\) as \(t \rightarrow \infty \). - (iii)If there exist a strictly positive \(v \in \mathbb R^p_+\), \(\rho \in (0,1)\) and \(\varTheta \ge 0\) such thatthen there exist \(\varGamma , \gamma >0\) such that, for all \(x^0 \in \mathbb R^n_+\) and all \(d \in L^\infty _\mathrm{loc}(\mathbb R_+ ; \mathbb R^n_+)\), every global solution$$\begin{aligned} \vert \mathbf{G}(0) g(t,y) \vert _{v} \le \rho \vert y \vert _{v} \quad \forall \, t \in \mathbb R_+, \; \; \forall \, y \in \mathbb R^p_+ \; \Vert y \Vert \ge \varTheta , \end{aligned}$$(2.57)
*x*of (2.53) satisfiesHere$$\begin{aligned} \Vert x(t) \Vert \le \varGamma \Big ( \mathrm {e}^{-\gamma t} \Vert x^0\Vert + \Vert d(\tau )\Vert _{L^\infty (0,t)} + \beta \Big )\quad \forall \, t \in \mathbb R_+. \end{aligned}$$and$$\begin{aligned} \beta = \beta (\varTheta ) = \Vert B \Vert \sup _{\Vert y \Vert \le \varTheta } \left( \sup _{t \ge 0} \Big (\mathrm{dist}{ }\big (g(t, y),T(t,y)\big )\Big ) \right) , \end{aligned}$$$$\begin{aligned} T(t,y) := \big \{ w \in [0, g(t,y)] \subseteq \mathbb R^m_+ {:} \, |\mathbf{G}(0)w|_v\le \rho |y|_v \big \}. \end{aligned}$$

The following remark provides some commentary on the above corollary.

### Remark 2.14

- (a)In the situation where \(m=p=1\) and
*g*is continuous, the conditions (2.55), (2.56) and (2.57) (the latter with \(\varTheta =0\)) simplify to:respectively.$$\begin{aligned} \mathbf{G}(0) g(y) \le y, \quad \mathbf{G}(0) g(y) < y \; (\text {for }y >0) \quad \text {and} \quad \mathbf{G}(0) g(y) \le \rho y \quad \forall \, y \in \mathbb R_+, \end{aligned}$$ - (b)When \(m=p\), a sufficient condition for (2.55) or (2.57) are the inequalitiesrespectively, where$$\begin{aligned} \Vert \mathbf{G}(0) \Vert _v \Vert g\Vert _v \le 1 \quad \text {or} \quad \Vert \mathbf{G}(0) \Vert _v \Vert g\Vert _v < 1, \end{aligned}$$(2.58)are the induced$$\begin{aligned} \Vert \mathbf{G}(0) \Vert _v := \sup _{\begin{array}{c} \xi \in \mathbb R^m_+ \\ \xi \ne 0 \end{array}} \frac{\vert \mathbf{G}(0)\xi \vert _{v}}{\vert \xi \vert _{v}} \quad \text {and} \quad \Vert g \Vert _v := \sup _{\begin{array}{c} \xi \in \mathbb R^m_+ \\ \xi \ne 0 \end{array}} \frac{\vert g(\xi ) \vert _{v}}{\vert \xi \vert _{v}}, \end{aligned}$$
*v*-norms. The inequalities in (2.58) are reminiscent of classical small-gain conditions, only here formed in the induced*v*-norm. We note that in general, \(\Vert \mathbf{G}(0) \Vert _v \ne \Vert \mathbf{G}(0) \Vert _2 = \Vert \mathbf{G}\Vert _{H^\infty }\), where the final equality is a property enjoyed by linear positive systems; see, for example [24, Theorem 5]. - (c)The conclusions of statements (i) and (ii) of Corollary 2.13 are similar to those in [58, Theorem 7.2] or [1, Theorem 5.6, p. 156], where a linear dissipativity theory approach to the absolute stability of positive Lur’e systems is taken. In [58, Theorem 7.2] the authors assume that the pair (
*C*,*A*) is observable, and that the nonlinearity*g*satisfies \(0 \le g(y) \; \le My\) for all \(y \in \mathbb R^p_+\), for some nonnegative matrix \(M \in \mathbb R^{m \times p}_+\). Further, it is assumed that the triple (*A*,*B*,*C*) is exponentially linearly dissipative with respect to the supply rate \(s(u,y) = \mathbb {1}^T u - \mathbb {1}^T My\), which is equivalent to the inequalityThe inequality (2.59) is itself equivalent to \(\vert M \mathbf{G}(0) \vert _{\mathbb {1}} = \Vert M \mathbf{G}(0) \Vert _1 < 1\) and may be interpreted as a small-gain condition in the induced the one-norm. Although not directly comparable, our norm conditions in (i)–(iii) are more general as they allow a small-gain condition in a weighted one-norm induced by any strictly positive vector, not just the usual one-norm, see Example 4.3. Finally, we remark that [58] focusses on unforced systems only. \(\square \)$$\begin{aligned} \mathbb {1}^T M\mathbf{G}(0) \ll \mathbb {1}^T. \end{aligned}$$(2.59)

## 3 Discrete-time systems

- (
**B1)** -
\((A,B,C) \in \mathbb R^{n \times n}_+ \times \mathbb R^{n\times m}_+ \times \mathbb R^{p \times n}_+\), \(D : \mathbb N_0 \rightarrow P_0(\mathbb R^n_+)\) and \(F : \mathbb R^p_+ \rightarrow P_0(\mathbb R^m_+)\);

- (
**B2)** -
\(r(A)<1\);

*F*is also assumed to satisfy (

**A3**). Given \(x^0 \in \mathbb R^n_+\) and \(D : \mathbb N_0 \rightarrow P_0(\mathbb R^n_+)\), a function \(x:\mathbb N_0 \rightarrow \mathbb R^n_+\) satisfying (1.4) is called a solution of (1.4), the guaranteed existence of which is not a concern in discrete-time. As with the continuous-time setting, assumption (

**A3**) implies that \(x = 0, D = \{0\}\) is a solution of (1.4) with \(x^0 = 0\) which we refer to as the zero equilibrium pair. We comment that “loop-shifting”, see (2.7), is also possible in discrete-time, particularly if (

**B2)**fails, by replacing

*A*and

*F*by \(A+BKC\) and \(y\mapsto F(y) -Ky\), respectively, for \(K \in \mathbb R^{m \times p}\).

Our main result of this section contains a series of global stability (when \(D=\{0\}\)) and ISS results for (1.4), formulated in terms of induced weighted one-norm constraints. As before, let \(\mathbf{G}\) denote the transfer function of the triple (*A*, *B*, *C*) with \(\mathbf{G}(1) = C(I-A)^{-1}B\). Assumption (**B2)** implies that \(\mathbf{G}(1)\) is well-defined and together with (**B1)** implies that \(\mathbf{G}(1) \in \mathbb R^{p \times m}_+\).

### Theorem 3.1

**B1)**, (

**B2)**and (

**A3**) hold.

- (i)If \(D=\{0\}\) and there exists a strictly positive \(v \in \mathbb R^p_+\) such thatthen there exists \(\varGamma >0\) such that, for all \(x^0 \in \mathbb R^n_+\), every solution$$\begin{aligned} \vert \mathbf{G}(1) w \vert _{v} \le \vert y \vert _{v} \quad \forall \; w \in F(y) \; \; \forall \, y \in \mathbb R^p_+, \end{aligned}$$(3.1)
*x*of (1.1) satisfies$$\begin{aligned} \Vert x(t) \Vert \le \varGamma \Vert x^0\Vert \quad \forall \, t \in \mathbb N_0. \end{aligned}$$ - (ii)If \(D=\{0\}\) and there exist a strictly positive \(v \in \mathbb R^p_+\) and a lower semi-continuous function \(e : \mathbb R^p_+ \rightarrow \mathbb R_+\) such thatthen, for all \(x^0 \in \mathbb R^n_+\), every solution$$\begin{aligned} e(y) \; >0 \quad \text {and} \quad \vert \mathbf{G}(1) w \vert _{v} +e(y) \; < \vert y \vert _{v} \quad \forall \; w \in F(y) \; \; \forall \, y \in \mathbb R^p_+{\setminus }\{0\}, \end{aligned}$$
*x*of (1.1) satisfies \(x(t) \rightarrow 0\) as \(t \rightarrow \infty \). - (iii)If there exist a strictly positive \(v \in \mathbb R^p_+\) and \(\rho \in (0,1)\) such thatthen there exist \(\varGamma >0\) and \(\gamma \in (0,1)\) such that, for all \(x^0 \in \mathbb R^n_+\) and all \(D: \mathbb N_0 \rightarrow P_0(\mathbb R^n_+)\), every solution$$\begin{aligned} \vert \mathbf{G}(1) w \vert _{v} \le \rho \vert y \vert _{v} \quad \forall \; w \in F(y) \; \; \forall \, y \in \mathbb R^p_+, \end{aligned}$$(3.2)
*x*of (1.1) satisfies$$\begin{aligned} \Vert x(t) \Vert \le \varGamma \big ( \gamma ^t \Vert x^0\Vert + \max _{\tau \in \underline{t-1}} {\left| \left| \left| D(\tau ) \right| \right| \right| }\big ) \quad \forall \, t \in \mathbb N. \end{aligned}$$(3.3)

Statements (i), (ii) and (iii) of Theorem 3.1 imply that, with \(D=\{0\}\), the zero equilibrium of (1.4) is stable in the large, globally asymptotically stable and globally exponentially stable, respectively. Moreover, statement (iii) implies that the zero equilibrium pair of (1.4) is exponentially ISS.

### Proof of Theorem 3.1

Throughout the proof let *x* denote a solution of (1.4) for given \(x^0 \in \mathbb R^n_+\).

*t*), the estimate (3.6) and (

**A3**).

**B2)**and (

**A3**). To that end, fix \(x^0 \in \mathbb R^n_+ {\setminus }\{0\}\) and, seeking a contradiction, suppose that \(Cx(t) \not \rightarrow 0\) as \(t \rightarrow \infty \), implying that there exists \((t_k)_{k \in \mathbb N_0} \subseteq \mathbb N\) with \(t_k \nearrow \infty \) as \(k \rightarrow \infty \) and \(\varepsilon > 0\) such that

**A3**), (2.24), (3.2) and (3.9). Multiplying both sides of the above inequality by \(\gamma ^{-t}\), and taking \(y = \gamma ^t \xi \) for \(\xi \in \mathbb R^p_+\), we see that

*E*and

*H*, respectively, such that

*t*and

*D*, such that

*z*satisfies the following inclusion

**A3**) and (3.9), may be estimated as follows:

*t*and

*D*. Inserting (3.16) into (3.17) establishes the claim with \(\varGamma := \max \big \{ K_3 + K_1 K_4, K_5 + K_2 K_4\big \}\), as required. \(\square \)

### Remark 3.2

In Sect. 2 we obtained two corollaries to Theorem 2.5—stability of certain non-autonomous differential inclusions and boundedness of solutions when the linear constraint (2.9) fails on a compact set, formulated as Corollary 2.9 and Corollary 2.12, respectively. The corresponding discrete-time versions of these results also hold *mutatis mutandis*, but, for the sake of brevity, are not formally stated. \(\square \)

### 3.1 Lur’e difference equations and inequalities

*F*defined analogously to those in Sect. 2.3.

### Corollary 3.3

**B1)**and (

**B2)**hold and that

*g*satisfies (3.20).

- (i)If \(d=0\) and there exists a strictly positive \(v \in \mathbb R^p_+\) such thatthen there exists \(\varGamma >0\) such that, for all \(x^0 \in \mathbb R^n_+\), every solution$$\begin{aligned} \vert \mathbf{G}(1) g(t,y) \vert _{v} \le \vert y \vert _{v} \quad \forall \, y \in \mathbb R^p_+, \; \forall \, t \in \mathbb N_0, \end{aligned}$$
*x*of (2.53) satisfies$$\begin{aligned} \Vert x(t) \Vert \le \varGamma \Vert x^0\Vert \quad \forall \, t \in \mathbb N_0. \end{aligned}$$ - (ii)If \(d=0\) and there exist a strictly positive \(v \in \mathbb R^p_+\) and a lower semi-continuous function \(e : \mathbb R^p_+ \rightarrow \mathbb R_+\) such thatthen, for all \(x^0 \in \mathbb R^n_+\), every solution$$\begin{aligned} e(y) \; >0 \quad \text {and} \quad \vert \mathbf{G}(1) g(t,y) \vert _{v} + e(y) \; \le \vert y \vert _{v} \quad \forall \, y \in \mathbb R^p_+{\setminus }\{0\}, \; t \in \mathbb N_0, \end{aligned}$$
*x*of (3.19) satisfies \(x(t) \rightarrow 0\) as \(t \rightarrow \infty \). - (iii)If there exist a strictly positive \(v \in \mathbb R^p_+\), \(\rho \in (0,1)\) and \(\varTheta \ge 0\) such thatthen there exist \(\varGamma >0\), \(\gamma \in (0,1)\) such that, for all \(x^0 \in \mathbb R^n_+\) and all \(d \in \ell ^\infty _\mathrm{loc}(\mathbb N_0 ; \mathbb R^n_+)\), every solution$$\begin{aligned} \vert \mathbf{G}(1) g(t,y) \vert _{v} \le \rho \vert y \vert _{v} \quad \forall \, y \in \mathbb R^p_+, \; \; \Vert y \Vert \ge \varTheta , \; \forall \, t \in \mathbb N_0, \end{aligned}$$
*x*of (3.19) satisfiesHere$$\begin{aligned} \Vert x(t) \Vert \le \varGamma \Big ( \gamma ^{t} \Vert x^0\Vert + \max _{\tau \in \underline{t-1}}\Vert d(\tau ) \Vert + \beta \Big )\quad \forall \, t \in \mathbb N. \end{aligned}$$and$$\begin{aligned} \beta = \beta (\varTheta ) = \Vert B \Vert \sup _{\Vert y \Vert \le \varTheta } \left( \sup _{t \in \mathbb N_0} \Big (\mathrm{dist}{ }\big (g(t, y),T(t,y)\big )\Big ) \right) , \end{aligned}$$$$\begin{aligned} T(t,y) := \big \{ w \in [0, g(t,y)] \subseteq \mathbb R^m_+ {:} \, |\mathbf{G}(1)w|_v\le \rho |y|_v \big \}. \end{aligned}$$

We conclude this section with some commentary on the discrete-time Lur’e system (3.18), and compare our results with others available in the literature.

### Remark 3.4

- (a)
Note that if \(\varTheta =0\) in statement (iii) of Corollary 3.3, then \(\beta (\varTheta ) = 0\) and the zero equilibrium pair of the Lur’e difference inequality (3.19) is exponentially ISS, that is, (3.3) holds. Consequently, if \(\varTheta =0\), then the zero equilibrium of the unforced (\(d=0\)) Lur’e difference inequality is globally exponentially stable.

- (b)
The comments in Remark 2.14 pertaining to the continuous-time result Corollary 2.13 have discrete-time analogues in the context of Corollary 3.3. To avoid repetition, we do not repeat them here, other than noting that the statements (i)–(iii) of Corollary 3.3 are similar to [59, Theorem 15], where different assumptions are made to those made here.

- (c)Corollary 3.3 demonstrates that the weighted one-norm constraints imposed in (i)–(iii) are sufficient for respective notions of absolute stability. We claim that when \(m=p=1\), the conclusions of statements (ii) and (iii) do not hold if their hypotheses are not satisfied. To that end, consider (3.18) with \(m=p=1\), \(d=0\) and a linear function \(y \mapsto g(y) \; = \beta y\), for some \(\beta >0\). Suppose further that \(\beta \) and \(\mathbf{G}(1)\) are such that (3.1) holds with equality (for any \(0 \ll v^T\)—when \(p=1\),
*v*appears as a multiplicative scalar on both sides), that is,Clearly, \(\beta \) and \(\mathbf{G}(1)\) must be related by$$\begin{aligned} \mathbf{G}(1) \beta y = y \quad \forall \, y \ge 0. \end{aligned}$$and it is straightforward to prove that \(r(A+\beta b c^T) = 1\) meaning that \(\beta \) is a destabilising perturbation. In particular, (3.18) becomes$$\begin{aligned} \beta = \frac{1}{\mathbf{G}(1)} = \frac{1}{\Vert \mathbf{G}\Vert _{H^\infty }}, \end{aligned}$$and so \(\Vert x(t) \Vert \not \rightarrow 0\) as \(t \rightarrow \infty \) in general (dependent on \(x^0\)). We note that statement (i) of Corollary 3.3 holds, but that statements (ii) and (iii) do not. More generally, for arbitrary \(g : \mathbb R_+ \rightarrow \mathbb R_+\), whenever \(y^* >0\) is such that$$\begin{aligned} x(t+1) = \left( A + \beta bc^T\right) x(t), \quad x(0) = x^0, \quad t\in \mathbb N_0, \end{aligned}$$in particular meaning that the estimates in both (ii) and (iii) do not hold, then \(x^* := (I-A)^{-1} B g(y^*)\) is a non-zero equilibrium of (3.18) and the zero equilibrium of (3.18) cannot then be globally asymptotically or exponentially stable. The papers [49, 64] consider attractivity and stability properties of the non-zero equilibrium \(x^*\) when \(m = p =1\) and under conditions on$$\begin{aligned} \mathbf{G}(1) g(y^*) = y^*, \end{aligned}$$(3.21)*A*,*B*,*C*and*g*which ensure that \(y^* >0\) in (3.21) is unique. \(\square \)

### Remark 3.5

## 4 Examples

We consider three worked examples.

### Example 4.1

*u*,

*x*,

*y*and

*d*denote the input, state, output and exogenous disturbance, respectively, suppose that \(A \in \mathbb R^{n \times n}\) is Metzler and \(b, c \in \mathbb R^n_+\) are nonnegative, that \(x^0 \in \mathbb R^n_+\), and that \(\alpha (A) \ge 0\), meaning that the uncontrolled linear system is unstable. The control objective is to stabilise (4.1) by static output feedback which is subject to quantisation. Specifically, for fixed \(\tau >0\), consider the upper semi-continuous quantisation function \(q :\mathbb R_+ \rightarrow \mathbb R_+\) defined by

*q*and

*k*is the set-valued function \(H_k : \mathbb R_+ \rightarrow P_0(\mathbb R_+)\), given by

*kq*with closed intervals and leads to the differential inclusion for

*x*

*A*replaced by \(A-kbc^T\) and \(F: \mathbb R_+ \rightarrow P_0(\mathbb R^n_+)\) given by \(F(y) := ky - H_k(y)\), which is plotted in Fig. 2b.

By our hypothesis on *k* and construction of \(H_k\), assumptions (**A1**)–(**A3**) are satisfied by the Lur’e inclusion (4.4). Further, *F* has compact values and is upper semi-continuous (which is readily verified or see, for example [26, Example 1.3, p. 5]) and *D* is locally bounded, measurable, with closed, convex values. Therefore, by Proposition 2.10, for every \(x^0 \in \mathbb R^n_+\), there is a global solution of (4.4) which satisfies \(x(t) \in \mathbb R^n_+\) for all \(t \in \mathbb R_+\).

^{1}We consider two exhaustive cases. If \(\mathbf{G}(0) \ge 0\), then \(\mathbf{G}_k(0) = \mathbf{G}(0)/(1+k\mathbf{G}(0)) = \rho /k\) for some \(\rho \in [0,1)\), and noting from Fig. 2b that

*A*is Hurwitz, which is false by hypothesis. We conclude that \(\mathbf{G}(0) < 0\), so that \(\mathbf{G}_k(0) > 1/k\) and by choosing \(\rho = \mathbf{G}_k(0)(k-\theta ) \in (0,1)\) for \(\theta \) sufficiently close to

*k*, in light of Fig. 2b it follows that

*d*given by

We conclude by summarising that “linear positive stability”, that is, \(A-kbc^T\) Metzler and Hurwitz, is sufficient to ensure that the zero equilibrium pair of the closed-loop quantised feedback system is ISS with bias and the state remains positive. Consequently, the problem is one of static output feedback control design for linear positive systems via the choice of *k*, which has been considered in, for example [21]. Observe that the constant \(\varGamma \) depends only on *A*, *b*, *c*, *k* and \(\rho \in (0,1)\). Clearly, \(\beta \) is a decreasing function of \(\tau \), which captures the coarseness of the quantisation, and, as such, the ISS with bias bound for \(\Vert x(t) \Vert \) decreases (viz. improves) with decreasing \(\tau \), or finer quantisation, as we would expect, and as seen in Fig. 3. \(\square \)

### Example 4.2

*i*-th stage-class at time-step

*t*. The \(s_i\) and \(h_i\) are probabilities denoting survival (or stasis) within a stage-class, and growth into the next stage-class, respectively. As such \(s_i, h_j \in (0,1]\) and \(s_j + h_j \le 1\) for all \(i \in \{1,2,\ldots ,n\}\) and \(j \in \{2,\ldots ,n-1\}\). The \(c_i\) terms are non-negative constants which capture the fecundity of the

*i*-th stage-class. The set-valued functions \(F_1\) and \(F_2\) capture per-capita reproduction and per-capital survival rates, respectively, and are modelled thus to accommodate for uncertainty in accurately describing these processes. If \(F_1\) and \(F_2\) are constant singleton-valued functions, meaning that there is no density-dependence, then (4.7) may be expressed as matrix population projection model, see [70]. We are assuming, as is often the case, that recruitment is into the first stage-class, which typically denotes eggs, juveniles or seeds, in an insect, animal or plant model, respectively.

Finally, \(u_1, u_2\) and \(u_3\) are forcing (controls or disturbances) affecting the reproductive rates, the number of offspring of reproductive individuals and the number of individuals which transition from the first to second stage-classes, respectively. They are modelled to act multiplicatively so that, in particular, \(u_1 = u_2 = u_3 \equiv 1\) corresponds to the absence of forcing. In an animal model, to reduce a population, the term \(u_1(t) <1 \) could reflect applying contraceptives to (a proportion of) the population, for instance, and \(u_2(t) <1\) could model the removal of new juvenile members of the population, whilst \(u_3(t) <1\) corresponds to the removal of members of the population which are transitioning from first stage class into the second, perhaps by culling, harvesting or moving individuals.

*A*,

*B*and

*C*satisfy (

**B1)**and as \(\sigma (A) = \{s_1,\ldots , s_n\}\), it follows that

*A*satisfies (

**B2)**. An elementary calculation shows thatwhere \(\alpha _1,\alpha _2, \alpha _3 \ge 0\). To apply Theorem 3.1, we require \(v^T = \begin{pmatrix}v_1&v_2 \end{pmatrix} \gg 0\) and \(\rho \in [0, 1]\) such that

*F*, we see that (4.8) holds if

*F*. Indeed, the “worst case” scenario from the point of view of destabilising the zero equilibrium corresponds to the upper bounds \(f_1^+\) and \(f_2^+\) for

*F*, which both appear in (4.9). Furthermore, if \(f_1^+\) and \(f_2^+\) are assumed bounded, then (4.9) can be satisfied for \(\rho <1\), if \(u_2\) and \(u_3\) are sufficiently small. Recall that in the present context control acts multiplicatively, and so “small” \(u_2\) and \(u_3\) correspond to “large” control efforts. For fixed \(\alpha _i\), \(f_i^+\) and \(\rho \), the first inequality in (4.9) may be satisfied by making \(u_2\) sufficiently small. There is then some freedom in how small \(u_3\) has to be to satisfy the second inequality, through the choice of \(v_2\), which itself may be large. However, we note that the functions \(z \mapsto f_1^+(z)\) and \(z \mapsto f_1^+(u_1z)\) have the same image for each fixed positive \(u_1\), and hence \(u_1\) alone cannot necessarily guarantee that the first inequality in (4.9) is satisfied—and small \(u_2\) may be required as well. Arguably these observations would have not been as apparent from the difference equations (4.7) alone. \(\square \)

### Example 4.3

**A1**), (

**A2**). It is readily seen that (

**A3**) holds as well. We seek conditions on \(\alpha , \beta ,\gamma , \delta >0\) which ensure that there exists a strictly positive \(v^T = \begin{pmatrix}v_1&v_2 \end{pmatrix} \in \mathbb R^2_+\) such that

## 5 Discussion

Global stability of the zero equilibrium of systems of positive Lur’e differential and difference inclusions has been considered. We have also considered input-to-state stability of the zero equilibrium pair of these inclusions when subject to exogenous forcing. Under rather mild assumptions on the set-valued term *F*, results available in the literature germane to the existence of local solutions of differential inclusions are readily applicable in the present context. Therefore, in the continuous-time case, we have focussed on stability of global solutions, and have presented conditions in terms of weighted one-norms of the “product” \(\mathbf{G}(0)F(y)\) of the steady-state gain \(\mathbf{G}(0)\) and set-valued nonlinearity *F*, reminiscent of classical small-gain conditions, which are sufficient for global stability properties, including ISS for forced inclusions. As we commented in Remark 2.6, sufficient conditions for the small-gain estimates are the existence of suitable nonnegative matrices *M* such that linear constraints of the form \(\mathbf{G}(0)w \le My\) for all \(w \in F(y)\) and \(y \in \mathbb R^p_+\) hold. The stability conditions presented guarantee existence of global solutions provided standard assumptions for existence of local solutions are satisfied. Our main results are Theorems 2.5 and 2.11 in the continuous-time case, and Theorem 3.1 in discrete-time. As corollaries we obtained a so-called ISS with bias result, Corollary 2.12, for forced inclusions where the small-gain condition fails on some bounded set, and suites of stability results for systems of positive Lur’e differential and difference equations and inequalities, Corollaries 2.13 and 3.3. Two key differences to our earlier work, notably [60], is that the linear components of the Lur’e inclusions are multivariable and, second, here we consider differential inclusions, rather than differential equations. Comparisons to other relevant results available in the literature appear in Remarks 2.14, 3.4 and 3.5 and worked Examples were presented in Sect. 4.

## Footnotes

- 1.
Although this assumption is not needed, it eases exposition.

## Notes

### Acknowledgements

B. Rüffer has been supported by ARC grant DP160102138.

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