text:
Small-gain stability theorems for positive Lur'e inclusions
- 387 Downloads
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.
KeywordsDifferential inclusion Exponential stability Input-to-state stability Lur'e systems Population biology Positive systems
Mathematics Subject Classification34A60 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].
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.
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.
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.
Lemma 2.1
\((A,B,C) \in \mathbb R^{n \times n} \times \mathbb R^{n\times m}_+ \times \mathbb R^{p \times n}_+\), A is Metzler, \(D : \mathbb R_+ \rightarrow P_0(\mathbb R^n_+)\) and \(G : \mathbb R_+ \times \mathbb R^p_+ \rightarrow P_0(\mathbb R^m_+)\);
\(\alpha (A)<0\);
- there exists \(R>0\) such that$$\begin{aligned} \Vert w \Vert \le R \Vert y\Vert \quad \forall \; w \in G(t,y) \; \; \forall \, t \in \mathbb R_+ \; \; \forall \, y \in \mathbb R^p_+. \end{aligned}$$
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 \(t \in [0, \omega )\).
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.
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
- (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 x of (1.1) satisfies$$\begin{aligned} \vert \mathbf{G}(0) w \vert _{v} \le \vert y \vert _{v} \quad \forall \; w \in F(y) \; \; \forall \, y \in \mathbb R^p_+, \end{aligned}$$(2.9)$$\begin{aligned} \Vert x(t) \Vert \le \varGamma \Vert x^0\Vert \quad \forall \, t \in \mathbb R_+. \end{aligned}$$
- (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 x of (1.1) satisfies \(x(t) \rightarrow 0\) as \(t \rightarrow \infty \).$$\begin{aligned} e(y) \; >0 \quad \text {and} \quad \vert \mathbf{G}(0) w \vert _{v} +e(y) \; \le \vert y \vert _{v} \quad \forall \; w \in F(y) \; \; \forall \, y \in \mathbb R^p_+{\setminus }\{0\}, \end{aligned}$$(2.10)
- (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 x of (1.1) satisfies$$\begin{aligned} \vert \mathbf{G}(0) w \vert _{v} \le \rho \vert y \vert _{v} \quad \forall \; w \in F(y) \; \; \forall \, y \in \mathbb R^p_+, \end{aligned}$$(2.11)$$\begin{aligned} \Vert x(t) \Vert \le \varGamma \mathrm{e}^{-\gamma t}\Vert x^0\Vert \quad \forall \, t \in \mathbb R_+. \end{aligned}$$
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
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 \(x^0 \in \mathbb R^n_+\).
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)
\(m =p\) 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;
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
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
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.
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
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
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
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
2.3 Positive Lur'e differential equations and inequalities
Corollary 2.13
- (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 x of (2.53) satisfies$$\begin{aligned} \vert \mathbf{G}(0) g(t,y) \vert _{v} \le \vert y \vert _{v} \quad \forall \, t \in \mathbb R_+, \; \; \forall \, y \in \mathbb R^p_+, \end{aligned}$$(2.55)$$\begin{aligned} \Vert x(t) \Vert \le \varGamma \Vert x^0\Vert \quad \forall \, t \in \mathbb R_+. \end{aligned}$$
- (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 x of (2.53) satisfies \(x(t) \rightarrow 0\) as \(t \rightarrow \infty \).$$\begin{aligned} e(y) \; >0 \quad \text {and} \quad \vert \mathbf{G}(0) g(t,y) \vert _{v} + e(y) \; \le \vert y \vert _{v} \quad \forall \, t \in \mathbb R_+, \; \; \forall \, y \in \mathbb R^p_+\backslash \{0\}\, \end{aligned}$$(2.56)
- (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 x of (2.53) satisfies$$\begin{aligned} \vert \mathbf{G}(0) g(t,y) \vert _{v} \le \rho \vert y \vert _{v} \quad \forall \, t \in \mathbb R_+, \; \; \forall \, y \in \mathbb R^p_+ \; \Vert y \Vert \ge \varTheta , \end{aligned}$$(2.57)Here$$\begin{aligned} \Vert x(t) \Vert \le \varGamma \Big ( \mathrm {e}^{-\gamma t} \Vert x^0\Vert + \Vert d(\tau )\Vert _{L^\infty (0,t)} + \beta \Big )\quad \forall \, t \in \mathbb R_+. \end{aligned}$$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 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].$$\begin{aligned} \Vert \mathbf{G}(0) \Vert _v := \sup _{\begin{array}{c} \xi \in \mathbb R^m_+ \\ \xi \ne 0 \end{array}} \frac{\vert \mathbf{G}(0)\xi \vert _{v}}{\vert \xi \vert _{v}} \quad \text {and} \quad \Vert g \Vert _v := \sup _{\begin{array}{c} \xi \in \mathbb R^m_+ \\ \xi \ne 0 \end{array}} \frac{\vert g(\xi ) \vert _{v}}{\vert \xi \vert _{v}}, \end{aligned}$$
- (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
\((A,B,C) \in \mathbb R^{n \times n}_+ \times \mathbb R^{n\times m}_+ \times \mathbb R^{p \times n}_+\), \(D : \mathbb N_0 \rightarrow P_0(\mathbb R^n_+)\) and \(F : \mathbb R^p_+ \rightarrow P_0(\mathbb R^m_+)\);
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
- (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 x of (1.1) satisfies$$\begin{aligned} \vert \mathbf{G}(1) w \vert _{v} \le \vert y \vert _{v} \quad \forall \; w \in F(y) \; \; \forall \, y \in \mathbb R^p_+, \end{aligned}$$(3.1)$$\begin{aligned} \Vert x(t) \Vert \le \varGamma \Vert x^0\Vert \quad \forall \, t \in \mathbb N_0. \end{aligned}$$
- (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 x of (1.1) satisfies \(x(t) \rightarrow 0\) as \(t \rightarrow \infty \).$$\begin{aligned} e(y) \; >0 \quad \text {and} \quad \vert \mathbf{G}(1) w \vert _{v} +e(y) \; < \vert y \vert _{v} \quad \forall \; w \in F(y) \; \; \forall \, y \in \mathbb R^p_+{\setminus }\{0\}, \end{aligned}$$
- (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 x of (1.1) satisfies$$\begin{aligned} \vert \mathbf{G}(1) w \vert _{v} \le \rho \vert y \vert _{v} \quad \forall \; w \in F(y) \; \; \forall \, y \in \mathbb R^p_+, \end{aligned}$$(3.2)$$\begin{aligned} \Vert x(t) \Vert \le \varGamma \big ( \gamma ^t \Vert x^0\Vert + \max _{\tau \in \underline{t-1}} {\left| \left| \left| D(\tau ) \right| \right| \right| }\big ) \quad \forall \, t \in \mathbb N. \end{aligned}$$(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 \(x^0 \in \mathbb R^n_+\).
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
Corollary 3.3
- (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 x of (2.53) satisfies$$\begin{aligned} \vert \mathbf{G}(1) g(t,y) \vert _{v} \le \vert y \vert _{v} \quad \forall \, y \in \mathbb R^p_+, \; \forall \, t \in \mathbb N_0, \end{aligned}$$$$\begin{aligned} \Vert x(t) \Vert \le \varGamma \Vert x^0\Vert \quad \forall \, t \in \mathbb N_0. \end{aligned}$$
- (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 x of (3.19) satisfies \(x(t) \rightarrow 0\) as \(t \rightarrow \infty \).$$\begin{aligned} e(y) \; >0 \quad \text {and} \quad \vert \mathbf{G}(1) g(t,y) \vert _{v} + e(y) \; \le \vert y \vert _{v} \quad \forall \, y \in \mathbb R^p_+{\setminus }\{0\}, \; t \in \mathbb N_0, \end{aligned}$$
- (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 x of (3.19) satisfies$$\begin{aligned} \vert \mathbf{G}(1) g(t,y) \vert _{v} \le \rho \vert y \vert _{v} \quad \forall \, y \in \mathbb R^p_+, \; \; \Vert y \Vert \ge \varTheta , \; \forall \, t \in \mathbb N_0, \end{aligned}$$Here$$\begin{aligned} \Vert x(t) \Vert \le \varGamma \Big ( \gamma ^{t} \Vert x^0\Vert + \max _{\tau \in \underline{t-1}}\Vert d(\tau ) \Vert + \beta \Big )\quad \forall \, t \in \mathbb N. \end{aligned}$$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 A, B, C and g which ensure that \(y^* >0\) in (3.21) is unique. \(\square \)$$\begin{aligned} \mathbf{G}(1) g(y^*) = y^*, \end{aligned}$$(3.21)
Remark 3.5
4 Examples
We consider three worked examples.
Example 4.1
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 \(t \in \mathbb R_+\).
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
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.
Example 4.3
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.
- 1.
Although this assumption is not needed, it eases exposition.
B. Rüffer has been supported by ARC grant DP160102138.
- 1.Haddad, W.M., Chellaboina, V., Hui, Q.: Nonnegative and Compartmental Dynamical Systems. Princeton University Press, Princeton (2010)CrossRefzbMATHGoogle Scholar
- 2.Krause, U.: Positive Dynamical Systems in Discrete Time. De Gruyter, Berlin (2015)CrossRefzbMATHGoogle Scholar
- 3.Berman, A., Neumann, M., Stern, R.J.: Nonnegative Matrices in Dynamic Systems. Wiley, New York (1989)zbMATHGoogle Scholar
- 4.Krasnosel'skij, M.A., Lifshits, J.A., Sobolev, A.V.: Positive Linear Systems: The Method of Positive Operators. Heldermann Verlag, Berlin (1989)Google Scholar
- 5.Berman, A., Plemmons, R.J.: Nonnegative Matrices in the Mathematical Sciences. SIAM, Philadelphia (1994)CrossRefzbMATHGoogle Scholar
- 6.Luenberger, D.G.: Introduction to Dynamic Systems: Theory, Models, and Applications. Wiley, New York (1979)zbMATHGoogle Scholar
- 7.Gaubert, S., Gunawardena, J.: The Perron–Frobenius theorem for homogeneous, monotone functions. Trans. Am. Math. Soc. 356, 4931–4950 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
- 8.Sine, R.: A nonlinear Perron–Frobenius theorem. Proc. Am. Math. Soc. 109, 331–336 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
- 9.Lemmens, B., Nussbaum, R.: Nonlinear Perron–Frobenius Theory. Cambridge University Press, Cambridge (2012)CrossRefzbMATHGoogle Scholar
- 10.Smith, H.L.: Monotone Dynamical Systems. American Mathematical Society, Providence (1995)Google Scholar
- 11.Hirsch, M.W., Smith, H.: Monotone dynamical systems. In: Cañada, A., Drábek, P., Fonda, A. (eds.) Handbook of Differential Equations: Ordinary Differential Equations, vol. II, pp. 239–357. Elsevier B.V., Amsterdam (2005)Google Scholar
- 12.Hirsch, M.W., Smith, H.L.: Monotone maps: a review. J. Difference Equ. Appl. 11, 379–398 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
- 13.Farina, L., Rinaldi, S.: Positive Linear Systems: Theory and Applications. Wiley, New York (2000)CrossRefzbMATHGoogle Scholar
- 14.Rantzer, A.: Scalable control of positive systems. Eur. J. Control 24, 72–80 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
- 15.Coxson, P.G., Shapiro, H.: Positive input reachability and controllability of positive systems. Linear Algebra Appl. 94, 35–53 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
- 16.Caccetta, L., Rumchev, V.G.: A survey of reachability and controllability for positive linear systems. Ann. Oper. Res. 98, 101–122 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
- 17.Härdin, H.M., van Schuppen, J.H.: Observers for linear positive systems. Linear Algebra Appl. 425, 571–607 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
- 18.Ohta, Y., Maeda, H., Kodama, S.: Reachability, observability, and realizability of continuous-time positive systems. SIAM J. Control Optim. 22, 171–180 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
- 19.Benvenuti, L., Farina, L.: A tutorial on the positive realization problem. IEEE Trans. Automat. Control 49, 651–664 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
- 20.De Leenheer, P., Aeyels, D.: Stabilization of positive linear systems. Syst. Control Lett. 44, 259–271 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
- 21.Roszak, B., Davison, E.J.: Necessary and sufficient conditions for stabilizability of positive LTI systems. Syst. Control Lett. 58, 474–481 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
- 22.Hinrichsen, D., Pritchard, A.J.: Stability radii of linear systems. Syst. Control Lett. 7, 1–10 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
- 23.Hinrichsen, D., Pritchard, A.J.: Stability radius for structured perturbations and the algebraic Riccati equation. Syst. Control Lett. 8, 105–113 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
- 24.Son, N.K., Hinrichsen, D.: Robust stability of positive continuous time systems. Numer. Funct. Anal. Optim. 17, 649–659 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
- 25.Aubin, J.-P., Cellina, A.: Differential Inclusions. Springer, Berlin (1984)CrossRefzbMATHGoogle Scholar
- 26.Deimling, K.: Multivalued Differential Equations. Walter de Gruyter & Co., Berlin (1992)CrossRefzbMATHGoogle Scholar
- 27.Vrabie, I.I.: Differential Equations. World Scientific Publishing Co., Inc., River Edge (2004)CrossRefzbMATHGoogle Scholar
- 28.Liberzon, D.: Calculus of Variations and Optimal Control Theory. Princeton University Press, Princeton (2012)zbMATHGoogle Scholar
- 29.Vinter, R.: Optimal Control. Birkhäuser, Boston (2000)zbMATHGoogle Scholar
- 30.Clarke, F.H.: Optimization and Nonsmooth Analysis, 2nd edn. SIAM, Philadelphia (1990)CrossRefzbMATHGoogle Scholar
- 31.Aubin, J.-P., Frankowska, H.: Set-Valued Analysis. Birkhäuser, Boston (1990)zbMATHGoogle Scholar
- 32.Goebel, R., Sanfelice, R.G., Teel, A.R.: Hybrid dynamical systems: robust stability and control for systems that combine continuous-time and discrete-time dynamics. IEEE Control Syst. Mag. 29, 28–93 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
- 33.Ryan, E.P.: Discontinuous feedback and universal adaptive stabilization. In: Hinrichsen, D., Mårtensson, B. (eds.) Control of Uncertain Systems, pp. 245–258. Birkhäuser, Boston (1990)Google Scholar
- 34.Ryan, E.P.: On Brockett's condition for smooth stabilizability and its necessity in a context of nonsmooth feedback. SIAM J. Control Optim. 32, 1597–1604 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
- 35.Cortés, J.: Discontinuous dynamical systems: a tutorial on solutions, nonsmooth analysis, and stability. IEEE Control Syst. Mag. 28, 36–73 (2008)MathSciNetzbMATHGoogle Scholar
- 36.Jayawardhana, B., Logemann, H., Ryan, E.: The circle criterion and input-to-state stability. IEEE Control Syst. Mag. 31, 32–67 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
- 37.De Glas, M.: Exponential stability revisited. Int. J. Control 46, 1505–1510 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
- 38.Liberzon, M.: Essays on the absolute stability theory. Autom. Remote Control 67, 1610–1644 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
- 39.Vidyasagar, M.: Nonlinear Systems Analysis. SIAM, Philadelphia (2002)CrossRefzbMATHGoogle Scholar
- 40.Khalil, H.K.: Nonlinear Systems. Macmillan Publishing Company, New York (1992)zbMATHGoogle Scholar
- 41.Haddad, W.M., Chellaboina, V.: Nonlinear Dynamical Systems and Control. Princeton University Press, Princeton (2008)zbMATHGoogle Scholar
- 42.Bill, A., Guiver, C., Logemann, H., Townley, S.: The converging-input converging-state property for Lur'e systems. Math. Control Signals Syst. 29, 1–50 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
- 43.Briat, C.: Robust stability and stabilization of uncertain linear positive systems via integral linear constraints: \(L_1\)-gain and \(L_\infty \)-gain characterization. Int. J. Robust Nonlinear Control 23, 1932–1954 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
- 44.Rantzer, A., Rüffer, B., Dirr, G.: Separable Lyapunov functions for monotone systems. In: 52nd IEEE Conference on Decision and Control, Florence (2013)Google Scholar
- 45.Sontag, E.D.: Smooth stabilization implies coprime factorization. IEEE Trans. Automat. Control 34, 435–443 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
- 46.Sontag, E.D.: Input to State Stability: Basic Concepts and Results, in Nonlinear and Optimal Control Theory, pp. 163–220. Springer, Berlin (2008)zbMATHGoogle Scholar
- 47.Sarkans, E., Logemann, H.: Input-to-state stability of Lur'e systems. Math. Control Signals Syst. 27, 439–465 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
- 48.Jayawardhana, B., Logemann, H., Ryan, E.P.: Input-to-state stability of differential inclusions with applications to hysteretic and quantized feedback systems. SIAM J. Control Optim. 48, 1031–1054 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
- 49.Townley, S., Rebarber, R., Tenhumberg, B.: Feedback control systems analysis of density dependent population dynamics. Syst. Control Lett. 61, 309–315 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
- 50.Rebarber, R., Tenhumberg, B., Townley, S.: Global asymptotic stability of density dependent integral population projection models. Theor. Popul. Biol. 81, 81–87 (2012)CrossRefzbMATHGoogle Scholar
- 51.Smith, H.L., Thieme, H.R.: Persistence and global stability for a class of discrete time structured population models. Discrete Contin. Dyn. Syst. 33, 4627–4646 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
- 52.Franco, D., Logemann, H., Perán, J.: Global stability of an age-structured population model. Syst. Control Lett. 65, 30–36 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
- 53.Eager, E.A., Rebarber, R., Tenhumberg, B.: Global asymptotic stability of plant-seed bank models. J. Math. Biol. 69, 1–37 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
- 54.Taylor, C.M., Hastings, A.: Allee effects in biological invasions. Ecol. Lett. 8, 895–908 (2005)CrossRefGoogle Scholar
- 55.Courchamp, F., Berec, L., Gascoigne, J.: Allee Effects in Ecology and Conservation. Oxford University Press, Oxford (2008)CrossRefGoogle Scholar
- 56.Schreiber, S.J.: Allee effects, extinctions, and chaotic transients in simple population models. Theor. Popul. Biol. 64, 201–209 (2003)CrossRefzbMATHGoogle Scholar
- 57.Eager, E., Rebarber, R., Tenhumberg, B.: Choice of density-dependent seedling recruitment function affects predicted transient dynamics: a case study with platte thistle. Theor. Ecol. 5, 1–15 (2011)Google Scholar
- 58.Haddad, W.M., Chellaboina, V.: Stability and dissipativity theory for nonnegative dynamical systems: a unified analysis framework for biological and physiological systems. Nonlinear Anal. Real World Appl. 6, 35–65 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
- 59.Haddad, W.M., Chellaboina, V.S., August, E.: Stability and dissipativity theory for discrete-time non-negative and compartmental dynamical systems. Int. J. Control 76, 1845–1861 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
- 60.Bill, A., Guiver, C., Logemann, H., Townley, S.: Stability of nonnegative Lur'e systems. SIAM J. Control Optim. 54, 1176–1211 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
- 61.Krause, U., Ranft, P.: A limit set trichotomy for monotone nonlinear dynamical systems. Nonlinear Anal. 19, 375–392 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
- 62.Dashkovskiy, S., Rüffer, B.S., Wirth, F.R.: Discrete time monotone systems: criteria for global asymptotic stability and applications. In: Proceeding of 17th International Symposium on Mathematical Theory of Networks and Systems (Kyoto, Japan), pp. 89–97 (2006)Google Scholar
- 63.Rüffer, B.S., Sailer, R.: Input-to-state stability for discrete-time monotone systems. In: Proceeding of 21st International Symposium Mathematical Theory of Networks and Systems (MTNS), pp. 96–102 (2014)Google Scholar
- 64.Franco, D., Guiver, C., Logemann, H., Peràn, J.: Semi-global persistence and stability for a class of forced discrete-time population models. Physica D 360, 46–61 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
- 65.Rüffer, B.S.: Small-gain conditions and the comparison principle. IEEE Trans. Automat. Control 55, 1732–1736 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
- 66.Plemmons, R.J.: \(M\)-matrix characterizations. I. Nonsingular \(M\)-matrices. Linear Algebra Appl. 18, 175–188 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
- 67.Yakubovich, V.A., Leonov, G.A., Gelig, A.K.: Stability of Stationary Sets in Control Systems with Discontinuous Nonlinearities. World Scientific Publishing Co., Inc., River Edge (2004)CrossRefzbMATHGoogle Scholar
- 68.Jiang, Z.P., Teel, A.R., Praly, L.: Small-gain theorem for ISS systems and applications. Math. Control Signals Syst. 7, 95–120 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
- 69.Sontag, E.D., Wang, Y.: New characterizations of input-to-state stability. IEEE Trans. Automat. Control 41, 1283–1294 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
- 70.Caswell, H.: Matrix Population Models: Construction, Analysis, and Interpretation. Sinauer Associates, Sunderland (2001)Google Scholar
Copyright information
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.


text:
|"Love is in the air"
Love is in the air
St.Valentine Networking party organized by TOdOOR
P.S. for one different party
As the Saint Valentine passed, so is the first networking party organized by TOdOOR.
The ambiance was spontaneous and natural, with lots of nice and valuable friends of ours, combined in a very good expected result. Networking party doesn't ever mean boring event with bunch of business cards…the networking could be inspiring, amusing and also beneficial – and we proved it. It may sound not very humble, but the party exceeded the expectations of all of us. We gave a lot of prizes and smiles, and we received even more… On 13.02 we set the beginning of an awesome new tradition and we are thankful to all of you, for being part of it…to be continued…

subset: commoncrawl


text:
My clients have high expectations of clean, and so do I. I am dedicated to being effecient and thorough. I have the experience and skills to tackle any type of situation that comes my way. I am committed to getting the job done while maintaining an environmentally conscious attitude.
I offer deep cleans and wipe downs of vacant houses to get them ready to show or for move in. I clean floors, counters, tubs/showers, toilets, refridgerators, in and under the stove, baseboards, fans, door frames, walls, and windows.
Deep Clean Pricing:
Studio - $100
1bd, 1ba - $125
2bd, 1ba - $150
2bd, 2ba - $175
3bd, 2ba - $200
Email for pricing on larger units.
Wipe Down Pricing:
Studio - 2bd, 1 ba: $50
2bd, 2ba - 3bd, 2ba: $100
Email for pricing on larger units.


text:
Choose your city
Delhi NCR
₹200 / 500 gm
Approximately 0.85 Kg of whole fish is required to get 0.5 Kg of Steak.
Whole Fish Price:
₹ 235.00/ Kg
Whole Fish Quantity:
0.85 Kg
0.35 Kg
Final Weight:
0.5 / Kg
Final Price:
₹ 200.00
Should you need us to deliver the trimmings, pls leave us a comment on checkout
Approximately 0.85 Kg of whole fish is required to get 0.5 Kg of Bengali Cut.
Rohu fish texture fish is white, non-oily and tasty.
It is an excellent source of high quality protein and also rich in omega 3 fatty acids, beneficial to the heart. Rohu (Indian fish) contains Vitamin C.
They are also good sources of vitamin D and niacin.
Rohu is also known as :
*Percent Daily Values are based on a 2000 calorie diet. Your daily values may be higher or lower depending on your calorie needs.
Rohu, Rohitham,rohu, Kannadi Kendai,Bocha Gnadu meenu,Roe, rohiti , row, Rahu, Ruee, Rui, Tapra, Dumra, Tambada masa,Tambada Masa,Tapra,Ruee Fish,Labio
Perch like fish with white flaky flesh and delicate flavor, rich source of protein, recommended for the calorie conscious individuals.
Catla fish high in nutrition and very rich in sulphur and zinc, helps to maintain the skin and boost the immune system as well. It makes for a nice side or starter dish.
We Serve the Finest Seafood, Responsibly!


text:
Salman is a licensed attorney in the State of Texas and focuses his legal practice on corporate and business transactions, and real estate law. Before launching his own law firm (Bhojani Law, PLLC), Salman practiced law at the Dallas office of Haynes and Boone, LLP, an international corporate law firm.
In addition to being a lawyer, Salman is an astute entrepreneur and over the past 15 years, has owned several business ventures in industries such as residential and commercial real estate, convenience and fuel, and software development.
After finishing his high school from Montreal, Canada, Salman immigrated to the United States and received his Bachelor of Science in Business Administration from the University of Texas at Dallas. He received his law degree from Southern Methodist University's Dedman School of Law and a Comparative Law diploma from the University of Oxford.
Salman has lived on three continents but calls Texas his home. He lives in Euless with his wife Nima, his children Aarish (13) and Raisha (10), and his parents. He enjoys traveling, playing ping pong, and spending time with his family!
Patrick is an accomplished attorney with a broad range of legal experience. Patrick currently represents international and domestic business clients in the areas of litigation, corporate governance and transactions, contract review, nonprofit and employment law, real estate and intellectual property. Patrick has been named in Ft. Worth Magazine as a top Civil Law - Litigation and Transactions and Corporate Counsel attorney from 2013 through 2018.
Patrick currently serves on the International Law Section of the Dallas Bar Association as a council member and as section Secretary. Patrick is also an active mentor with the International Law Section JD Mentorship Program through the SMU Dedman School of Law.
After graduating from Holland Hall School in Tulsa, Oklahoma, Patrick attended Pepperdine University in Malibu, California. Patrick earned his bachelor's degree in Psychology from Pepperdine and is a member of the Psi Chi National Honor Society. Patrick returned home to Oklahoma for law school, earning his law degree from The University of Oklahoma College of Law. While in law school, Patrick studied at Oxford University through OU's summer study abroad program.
While not in the office, Patrick enjoys spending time with his family, working on classic muscle cars, and international travel.
David is a partner of the firm focuses his practice on real estate and business transactions. David has significant experience drafting and negotiating commercial leases and purchase and sale agreements. He also has significant experience in working with clients and title companies in all phases of the title review, due diligence and closing process.
David also represents small to medium-size business owners with all aspects of their legal needs, including entity formation, buying and selling of assets or ownership, and business disputes. David has also worked with copyright owners to protect their rights in federal courts throughout the United States.
David also enjoys preparing wills and other estate planning documents for his clients.
David received a B.A. in geography from the University of Texas – Austin, with a concentration in environmental resource management. He received his law degree, with honors, from the University of Texas School of Law.
David is an avid outdoorsman and sports fan. In his free time, he enjoys hiking, running, and playing in a rec basketball league, as well as spending time with his family.
Axel's practice focuses on estate planning and tax planning, with special regard for international clients. Axel has significant experience in estate planning, probate, trust administration, as well as business transactional law. He also serves as counsel for businesses of all sizes, including Fortune 500 companies.
Axel often teaches seminars on topics including estate planning, probate, taxes, and business planning. He currently serves as treasurer and board member of the Houston Chapter of the Society of Financial Service Professionals, and is also on the board of several non-profit organizations.
Axel received a B.S. in political science and criminal justice from Sam Houston State University in Huntsville, Texas. He received his law degree from the University of Texas School of Law in Austin, Texas.
Axel was originally born in Helsinki, Finland. His family moved to South Florida when he was a child, where he lived until moving to Texas in 2005. He is married with two children and enjoys cooking, traveling, and spending time with his family.
Our stellar legal staff brings together 65 years of experience in many fields of expertise. Just as we work together to help one another with our range of knowledge, we would love to bring our skills together to help you through your own legal endeavors.
Copyright © 2017 Bhojani Law, PLLC - All Rights Reserved.


text:
PURPLE 990: Dark Grape outside unrolls to reveal Violet.
132 COLORS AVAILABLE! To order all colors, see "EVERYTHING I GOT" package.
USES: Hand-stitching decorative threads for Quilting, Needlepoint, Cross-stitch, Scrap-booking, & Friendship Bracelets. Not recommended for projects that need to be washed.
Hand-dyed hombre 100% Perle Cotton #5 embroidery thread.
Balls begin as a solid color with a slight sheen. As you begin to peel away the thread, the color changes to a lighter hombre version of the original color. Some colors have a more dramatic color change. It's like unwrapping a beautiful delicious piece of candy!
Because of the unique one-of-a kind uneven dye, every loop/ball can vary per dye lot. Colors are even more beautiful in person!
If you have a preference for loops that are either more solid or variegated, please let me know when you place your order via email at email@example.com. I will do my best to honor your request.
These threads are meant for decorative use. I am still in the testing stages for staining when used for visible mending, so I do not suggest using them for any projects that require washing at this time.
International Shipping available. Shipping rates will be determined after order has been placed.
.These threads are meant for decorative use. Threads are hand-dyed with Procion fiber reactive dyes. Threads are thoroughly rinsed (2-4+ times), though some threads, especially reds and pinks, may still have residual color. Please test your threads before using if bleeding would be an issue.
International Shipping available. Shipping rates will be determined after order has been placed.


text:
Estate 14k Yellow Gold Diamond and Black Enamel Bee Pendant
sold out
Estate 14k Yellow Gold Diamond and Black Enamel Bee Pendant
14K yellow gold, onyx, enamel, and diamond pendant. It just doesn't get more extraordinary than a piece like this! An adorable bee, brought to life by black and yellow enamel, sits atop a gleaming bezel set tablet of black onyx, almost as if the insect laded there with it's sparkling diamond wings, comprised of 0.10 carats total diamond weight (VS1, G/H). The piece is approximately 2" in diamoeter and articulates from a simple yellow gold bale. Chain sold separately.
Item #: 160-300
Add To Cart
Prior to purchase, please view our purchase, shipping, return and repair policies: SHOPPING WITH US ONLINE


text:
11:30am-12:30pm • 5-9pm
$90/30 min, $175/60 min, , Call (650) 988-9800 to reserve a session
The Akashic Record is a treasury of your personal knowledge of the past, present, and future. As soon as your soul begins to experience life, a field of energy is created that records your every thought, word, action, desire and emotion. This field of energy is known as the Akashic Records, your Book of Life. By accessing this personal library of knowledge, you can identify, open or release anything that you have created or blocked through your life's experiences. A reading from Maureen can be powerful, enlightening, cleansing, and inspiring. Maureen St. Germain is an author and founder of Akashic Records International. www.maureenstgermain.com


text:
Project Overview:
Elegant Pride, a residential project located in Hebbal, constructed by one of the well-known developers in the city by Elegant Builders & Developers. This project offers Ready to move 2BHK and 3BHK apartments for sale in Jakkur locality. The size of 2BHK apartments varies from 1055 Sq ft to 1210 Sq ft and the size of 3BHK apartments varies from 1415 Sq ft to 1515 Sq ft.
The project is well equipped with the trending amenities to facilitate the needs of the occupants, the project is in close proximity to various civic utilities which makes your life hassle-free. Elegant Pride's location makes it more admirable and tends to mark its impression in the modern era. Well-known schools, hospitals, colleges, shopping malls, and entertainment zones are there in the proximity. As this project is located in one of the prime localities of Bangalore city you can enjoy the connectivity to well-known localities like Sindhi Colony, Lingarajapuram, Kasturi Nagara, Dooravani Nagar, Dasappa Layout and more.
Along with all these advantages and apartments at Jakkur, Elegant Pride offers lifestyle amenities, some of them are Table Tennis, Vaastu Compliant, 24-hrs Water Supply, Gym, Jogging Track, Indoor Games, Swimming Pool, Multi-purpose Hall, CCTV, Intercom, Parking, and Security.
The project offers:
Location advantages
Suitable/Ideal For:
Young couples
Married couples
Working couples
Couples and senior
Young upwardly families
Higher Secondary School (1.4 km)
Methodist Mission High School (3.1 km)
RBANM's Middle School (3.6 km)
Dr. Sruthi's Homeocure (0.9 km)
Health and Hijama Clinic (3.5 km)
Rashmi Hospital (5.4 km)
Public Transport:
Bus Stations:
Jai Bharat Nagar (1 km)
Doddi (2.1 km)
Coles Park (2.5 km)
Train and Metro Station:
Bangalore East (2.8 km)
Banaswadi (1.1 km)
Halsaruru (4.5 km)
Nearest Landmark:
Near Shree Muneshwara Swamy Temple
Building Structure:RCC Framed Earthquake resistant structure.
Walls:External walls of 6" Solid Blocks and Internal walls with 4" Solid Blocks.
Windows:3 Track alluminium windows with 5mm thick plain glass and MS Safety grills
Doors:Main door : Teak wood door frame with elegantly finished flush shutters. Bedrooms and kitchen doors : Sal wood door frames with flush shutters.
External and Internal Paint:External walls : Ace paints of exterior walls. Internal walls : Emulsion paint for interior walls and ceiling.
Flooring:Living, bedrooms and dining area & kitchen vitrified files. Toilets - Anti skid tiles flooring with ceramic tiles dado up to door height in master bedroom toilet, Anti skid ceramic tiles flooring with ceramic tiles dado upto door height M other toilets. Balconies - Anti skid ceramic tiles with water proofing treatment.
Toilets:Wash basins in all bathrooms. CP fittings of ISI make. Concealed piping system for cold & hot water with provision for geysers. Porcelain sanitary wares of reputed brand. WC of reputed brand.
Water:24 hours water supply from Borewell.
Electrical:Wiring concealed with fire resistant wires. Provision for split AC point in Master bedrooms. Adequate points for lights, fans & other plugs in bedrooms, toilets and kitchen with modular switches of reputed brand. Exhaust fans & geysers points in toilets & kitchen plug points for chimney, water purifier, refrigerator and other kitchen appliances in kitchen.
Kitchen:Granite cooking platform with stainless steel sink. 2' height glazed tiles daddoing above the platform. Reputed ISI make CP fittings.
Lobby & Lift:Entrance lobby finished with marble / granite flooring. Five automated lifts of reputed make with 6 passenger capacity.
Security Features:Round the clock security. Intercom facility with entrance gate from each apartment.
TV & Telephone:TV point in living & Master bedrooms. Telephone points in living & Master bedrooms.
Power Backup:Full backup power for all lighting points in the flats. Additional backup power for Common area, Lifts & Water pumps.
Elegant Builders & Developers are Professional Builders and Promoters in Bangalore. Our aim is to provide reliable, cost-effective, hassle-free, expedient and trusted Proprty services to our esteemed customers. We belive in "Your Comfort Is Our Priority" and keeping this in mind we have set high standards for our selves to achive the goal.


text:
18 March 2016
On March 14th, 2016, the President of the Financial Supervisory Authority (ASF), Mr. Misu Negritoiu, together with a group of students from the pre-university education that are included in ASF's financial education program, have participated in the opening of the trading session of the Bucharest Stock Exchange to mark the start of Global Money Week in Romania.
ASF joins this initiative, which runs internationally between 14 and 22 March 2016, in order to support and promote the importance of education and financial inclusion of young people.
"We are very anxious to find solutions for an optimal functioning of non-banking financial markets, and education in this area represents an important pillar. Therefore, I believe that it is never too early to learn how to take correct financial decisions. A higher level of consumer financial education means a stimulus to increase quality of non-banking financial products and services, while contributing to the maturation and consolidation of a whole system", as ASF's President, Mr. Misu Negritoiu, stated.
In 2015, as part of Global Money Week, the initiative mobilized nearly 1,000 organizations in 125 countries and involved a number of 5.6 million children and teenagers in more than 3,000 activities aimed at financial education.
On this occasion ASF launched www.asfromania.ro/edu website dedicated to financial literacy, which contains resources intended mainly for school-age children.
The website shows the basics of the instruments and mechanisms from the three markets regulated by ASF – insurance, capital market and private pensions. Visitors can access a dictionary of specific terms, a section where they can test their financial knowledge, helpful advice structured by age groups, or references to other external related links. Also, the website provides all teaching materials used in the financial education program developed by ASF.
For this project, 22 trainers from the Authority have given over 330 hours of classes for nearly 2,500 students, coming from 17 schools and high schools, both in Bucharest and in the country.
Further information of Global Money Week may be found here: http://www.globalmoneyweek.org/
Planificarea examenelor de certificare pentru distribuitori
Noua platformă pentru examinări online
Specialiștii constatare daune își desfășoară activitatea profesională în domeniul asigurărilor, în conformitate cu prevederile legale în vigoare și cu respectarea cerințelor privind pregătirea profesională....Click aici pentru detalii >
Vă rugăm să accesați Platforma on-line aici și să urmați pașii menționați în Ghidul ...see all
The Institute of Financial Studies and specialists in the insurance market have developed and published the Insurance Di...see all
ISF organizeaza vineri 29 martie 2019 a doua editie a actiunii de ecologizare a padurilor, urmand astfel sa plante...see all
Bucureşti 25 martie 2019 Digitalizarea, automatizarea şi inovaţia în serviciile financiare, dar ...see all
Institutul de Studii Financiare și specialiști din piața asigurărilor, au elaborat Manualul Distribuitorului de Asig...see all


text:
Prodigy Burger & Bar - Carmel
Birthday celebration. Great experience and food.
Delicious and creative burgers. Our server, Cody, was super! He was friendly, knowledgeable of the menu, attentive, really made the experience a great one. The live music made for a fun evening out.
My wife and I came in with our two young children last night. We had an enjoyable time, but we weren't blown away. The burgers were decent, but they seemed to be missing something. We both got the "Sheriff" which is a cheeseburger with a Jack Daniels sauce. It was decent, but there was no garnish on the burger, no bacon, etc...it was just kind of boring and needed something more. The fries were really good, and it is a comfortable atmosphere. Maybe I would enjoy a different burger more if I return.
It was a great night to sit on the patio. Service was excellent. Carson our waiter was very good and worked hard to make our dining experience enjoyable. This will become our new regular.
Steak tacos were way overcooked.
Yes I would recommend but my food did come out cold, we had a larger party that may have been the issue. They offered to remake but took it off the bill instead. My waitress was attentive, I was only bothered that manager didn't inquire about the issue. I just feel if I have to pay extra for an item like sweet potato fries they should be hot at least warm when they arrive with my meal. They weren't too busy for he/she to check with me. I usually don't write reviews but thought I would share my experience.
Good burgers, good mix of food. Little pricey, but not too bad. Only complaint was live music was waaaay too loud for size of space, couldn't hear anyone at my table.
Service was very good and cheerful! The Ted Turner with a friend egg was very good! We also had a burger with blue cheese and onion straws on it. Again very tasty!!
We had a reservation Saturday night at 7:15. We got there to hear the manager say to not sit anyone as the kitchen was behind. They told us that we would get a table as soon as one opened. After 30 mins of waiting we asked about our table as plenty were open and were told it would be another 30 mins. We left without being seated as we had three kids needing to eat. When I reinforced we had a reservation, I was told it's an automated system and they don't guarantee a table which has never been the case at any other restaurant using open table. Very disappointed at the lack of transparency when we arrived as we would have left right away.
Very good seasoning on the French fries, we enjoyed the live music and great selection of craft beers
Bar-b-que pulled pork sandwich was excellent!
There were four of us and we ordered a variety of food, from salads to burgers. The restaurant is located in Clay Terrace. The server forgot the bacon, lettuce and tomato on one of the burgers and after she was told, it took a long time to get it, by which time the burger was no longer hot (and was somewhat overdone, but at this point it was too late to send it back). One of the managers brought the forgotten items and he really tried to make up for the mistake. Everything else came out as ordered and was tasty. We liked the industrial décor (although it made it a bit loud) and the menu was varied enough to satisfy most tastes. We'll probably give them another chance.


text:
Contact us WhatsApp:+86 17862811103 Email : [email protected] Chat Online
Explore Now
A wet belt permanent high gradient magnetic separator (referred here as WBHGMS) was developed on the principle of HGMS in an inclined slurry flow of several
Returning this item:This is non hematite; they look like tumbled hematite and they are strongly magnetic. These are man made and they are not even an iron oxide.
Smokey Quartz Crystals are some of the premier grounding and anchoring stones. These stones are strongly protective and are excellent stones for all of us to keep
Quartz crystal properties and definitions of metaphysical crystals. Quartz crystals all posses certain properties but some crystals have extra extraordinary properties.
Domain Theory. A remarkable property of ferrimagnetic materials is not so much that they have a spontaneous magnetization, but rather that their magnetization can be
Research interests and outcomes: Magnetism of graphene Room temperature organic magnets (Nat Commun 2017), effect of doping on magnetic properties (Adv Mater 2016
Detailed description, properties, and locality information guide about the iron mineral hematite.
Orgone Pyramids by Peter Stathis. Fifth Element Orgone products are powerful Spiritual tools that can be used to enhance all areas of your lifephysically, mentally
Wholesale Gemstone Beads supplies with cheap price. nbeads provide a great selection of quality Gemstone Beads at wholesale price.
Photomicrograph (100 X) of soil from a crop circle showing traces of melted magnetic material adhering to soil grains.
(ī'ərn) Symbol Fe A silvery white, hard metallic element that occurs abundantly in minerals such as hematite, magnetite, pyrite, and ilmenite.
A ferrofluid (portmanteau of ferromagnetic and fluid) is a liquid that becomes strongly magnetized in the presence of a magnetic field. A grinding process for
Minerals With Magnetic Properties & Types Magnetism Ferromagnetism Paramagnetism Diamagnetism When Heated Bismuth Repelled Field Magnet Weak Strong
What is attracted to magnets? Take a wand magnet and go around the house to see what will stick to it or feel like it is attracted to it. Keep a list of the items you
Detailed description, properties, and locality information guide about the magnetic mineral magnetite.
Hematite crystals give pain relief, grounding & balancing energy, stimulate the mind, enhance memory, evoke deep thought, self confidence, will power & self esteem.
The modification of a hematite nanoarray photoanode with a TiO 2 interlayer by atomic layer deposition is described by J. Gong and co workers in their Communication
Maghemite (Fe 2 O 3, γ Fe 2 O 3) is a member of the family of iron oxides. It has the same spinel ferrite structure as magnetite and is also ferrimagnetic.
Li ion batteries are the powerhouse for the digital electronic revolution in this modern mobile society, exclusively used in mobile phones and laptop computers. The
Feldspar Feldspar refers to a group of minerals containing aluminum and silica. They all show good cleavage in two directions at about 90 degrees.
The Gallery of Minerals has examples of common minerals that are categorized by type. Each has a thumbnail picture of the specimen and a brief description of it.
Iron: Iron (Fe), chemical element and one of the transition elements, the most used and cheapest metal.
Hematite as a widespread natural pigment has been used by man since the earliest times. Red color in cave paintings comes from fine
Fabrication and characterization of novel iodine doped hollow and mesoporous hematite (Fe 2 O 3) particles derived from sol gel method and their photocatalytic
Is Hematite Magnetic? One of the most confusing questions about hematite is whether or not it is magnetic. When you are at a gem store, you may see certain magnetized
Grinding Equipment
Flotation Equipment
Gold Extraction Equipment
My Personal info
Screening Equipment
Tel / Whatsapp: +86 17862811103
Wechat : 17862811103
Email : [email protected]
Copyright © 2017 Shandong Xinhai Mining Technology & Equipment Inc. All rights reserved.|sitemap|


text:
Dmanisi - located in Georgia, Southwest fo the capital of Tbilisi, contains some of the oldest hominid remains ever found in the area.
Happisburgh Footprints with Martin Bates - Episode 180
On today's episode Kim Biddulph of the Prehis/Stories podcast interviews Martin Bates. Martin is a Pleistocene geoarchaeologist. He talks about the oldest hominid footprints found outside of Africa in Happisburgh, UK.


text:
various primary impact crusher pictures
different types of crushers mining
all different pictures of crushers test rig. different types of primary crushers, jaw crusher, cone crusher, impact crusher,Get Price
primary and secondary crusher
primary and secondary crusher 2.impact crusher is an ideal choice for benefits increase the lifetime of many different types of crushers used in theGet Price
crushers are machines used to crush various the jaw crusher is one of the most widely used primary crushers and is suited the impact crusher isGet Price
primary jaw crushers plant units pictures
primary jaw crushers plant units pictures primary jaw crushers impact crusher, from atlastypes and pictures of jaw crushers norway,different types ofGet Price
for crusher blueprint pictures
for crusher blueprint pictures home products solutions project about contact get area, primary impact crusher pdk,Get Price
the crushing machine image crusher, grinding mill
crushing machine pictures, you can buy various high quality crushing machine pictures products from global crushing machine mobile primary jaw crusher photosGet Price
quarry photos with primary crushers
quarry photos with primary crushers. almeda the impact crusher is used for primary stone crushing plant buy various high quality stone crushing photos ofGet Price
types and pictures of jaw crushers
parts for kue ken jaw crushers pictures of kue ken primary crushing plants for wholesale various high quality pf impact crusher for concrete productGet Price
types and pictures of jaw crushers
different types of cone crusher pictures etc. cone crushers can be used in a primary two types of cone crushers impact crushers used; pictures ofGet Price
pictures of primary secondary and tertiary crushers in
granite various primary impact crusher pictures they are popular secondary or tertiary crushers . get information; pictures of primary secondary and tertiary crushersGet Price
a gyratory crusher is one of the main types of primary crushers in a hydraulic cone crusher and gyratory crusher. according to different impact crusher impactGet Price
types and pictures of jaw crushers
mobile jaw crushers primary jaw crusher jaw crushers size reduction in impact crushers occurs types and pictures of jaw crushers. different types ofGet Price
technical notes 5 crushers
jaw and gyratory crushers are used mostly for primary crushing. crushers of various size as a function of the open side set. 5.1.2 impact crushersGet Price
cone crusher of various types
crusher; sand primary crushers different types different types of cone crusher pictures of various types of crusher and mill. how impactGet Price
top quality impact crusher pictures for stone quarry
micronizer how does impact crusher work; pictures for sand picture and photos of used stone crusher. buy various high from large primary jaw crusher andGet Price
impact crusher from inside
· from large primary jaw crusher and impact crusher to cone crusher, pictures show used impact crushers crusher) is applicable for variousGet Price
primary jaw crusher for sale canada
various primary impact crusher pictures; primary crusher jaw used primary jaw crusher wholesale various high quality jaw crusher for sale in canadaGet Price
different types of crushers machine of primary
different types of primary crushers quality crusher machine manufacturer provide good price primary impact crusher for sa we produce various typesGet Price
mobile crusher pictures
and of course various types of stone impact mobile crusher manufacturers has uploaded 3188 impact mobile crusher pictures for mobile primary jaw crusherGet Price
primary secondary and tertiary crusher in granite industry
complete crushing circuit.dove ® jaw crushers are supplied in various primary and secondary impact crushers grinding primary crusher pictures of primaryGet Price
crusher type primary
primary jaw crusher, wholesale various high quality primary jaw crusher products from secondary cone crusher + scre. hot pictures bp primary impact crusher.Get Price
what is the advantages of different types of rock crusher?
what is the advantages of different types of rock crusher? primary rock crusher, jaw crusher and impact crusher.Get Price
various pictures of crusher plant
various type of crusher plant. request for quotation. different types of crushers mining mine grinder plant . various primary impact crusher pictures;Get Price
small jaw crusher for concrete waste recycling
· like us on facebook to stay in touch our crusher equipment, pictures, impact crusher, the pe jaw crusher is suitable for primary crushing and theGet Price
- gold ore washing equipment company
- quary stone crushing plant machinery
- high efficiency stone box crusher from manufacturer
- coarse jaw crusher specification
- 1618 reversible impact crusher
- quarries looking for loaders
- vietnam stone crushing machine for sale
- rock crusher for sand gravel making
- agma kilns and mills open gearing
- crushing plants rock used
- dredging machine manufacturers in china
- ul manufacturers cassava grinding machines


text:
Thoroughly Thursday - the "Bucketsful of Rain" Edition
Welcome back to Thoroughly Thursday, the "Bucketsful of Rain" edition. Last week, while visiting the East Coast to see a client, I stepped outside to get a folder from my rental car. I had already walked 15ft into the parking lot before realizing that it wasn't raining as much as being pummeled by water balloons from the sky. There was a moment's pause, and then I looked up and thought, "oh, right. It was cloudy, wasn't it?" I have been in this exact position half-a-dozen times when traveling during the last few years, and strangely, I'm ALWAYS SURPRISED. If I may say, there is an astounding amount of weather in places that are not the Southwest (notwithstanding hurricanes). As I've lived for over two decades in a place where for 98.3% of the year it is sunny, I've acclimated to the idea that the only surprising weather patterns are earthquakes.
What I'm Reading (Article) - Smile, darn ya, smile!
This week I read the article "How to Be Happy" by Adam Sternbergh. At Yale University, there is now a class on how to be happy, which has become the most popular course in the schools' history. We seem to be at an interesting place in history, with the U.S. being one of the richest countries in the world, but with record numbers of people of all ages who feel anxious, depressed or generally miserable. As it turns out, we're terrible at determining what actually makes us happy. Additionally, being happy is a skill that takes practice. If you are interested in learning more, you can take the Authentic Happiness Inventory questionnaire developed by University of Pennsylvania professor Martin Seligman.
What I'm Listening To - What could be more than Fun?
This last week, I've been listening to a lot to Strange Desire by Bleachers which is fronted by music impresario Jack Antonoff, who is best-known as lead guitarist of Fun. It's a great pop album with a lot of expansive anthems and strong hooks. You've heard "I Wanna Get Better" (trust me), but I suggest "Wild Heart" or this version of "Rollercoaster" with Charli XCX. If you are interested in something a little more loungey, check out "Born Free'ky" by Jeff Goldblum and the Mildred Snitzer Orchestra.
What I'm Doing for Work - Vegas, Baby!
As I mentioned last week, I will be attending the Global Security Conference (GSX) next Wednesday. During the show, I'm planning on making a series of videos around the exhibition hall for clients and industry leaders, showcasing their new products (see a sample here). I've set up a couple meetings, but if you just want to meet up, let's do this!
What I'm Thinking About -
"Happiness Equals Reality Minus Expectations." - Tom Magliozzi (recently mentioned by Elon Musk during that infamous podcast episode).
All the best, and remember, tomorrow is Friday. :)
Scott Moe
P.S. If you like this email, send it to a friend. They can sign up here.
P.P.S. I'm always on the lookout for new items to write about. If you notice something that you think would be worth sharing, send it my way.


text: LinkedIn Launches Volunteer Marketplace, Adds Unpaid Work To Its Recruitment Platform, Posting For A Fee Jane Ullah Programming January 15, 2014 0 Minutes Advertisements Share this:TwitterFacebookLike this:Like Loading... Related Published by Jane Ullah I am a wife, student, consumer, and a tech. enthusiast! @janetalkstech. View all posts by Jane Ullah Published January 15, 2014


text:
The Key to Earning the Intercession of Rasulullah (sallallahu 'alaihi wasallam)Written by user2
قال القطب الحلبي: رأيت أبا إسحاق إبراهيم بن علي بن عطية التليدمي وقال لي: رأيت النبي صلى الله عليه وسلم في المنام فقلت: يا رسول الله أسألك شفاعتك فقال: أكثر من الصلاة علي صلى الله عليه وسلم (القول البديع ص267)
Qutb Al-Halabi (rahimahullah) mentions, "I once met Abu Ishaaq, Ebrahim bin 'Ali bin 'Atiyyah Al-Taleedami (rahimahullah). He said to me, 'I was blessed with the mubaarak vision of Rasulullah (sallallahu 'alaihi wasallam) in a dream. On seeing him, I said to him, 'O Rasul of Allah (sallallahu 'alaihi wasallam)! I request you to intercede for me on the Day of Qiyaamah!' Rasulullah (sallallahu 'alaihi wasallam) replied, 'Recite abundant durood upon me.''"
يَا رَبِّ صَلِّ وَسَلِّمْ دَائِمًا اَبَدًا عَلَى حَبِيْبِكَ خَيْرِ الْخَلْقِ كُلِّهِمِ
Caution in Fatwa – Part TwoWritten by user2
'Amr bin Yazeed was a friend of Imaam Maalik (rahimahullah) who lived in Egypt. He narrates the following:
I once said to Imaam Maalik (rahimahullah), "O Abu 'Abdillah! People travel to you from different lands. They wear out their animals and spend their money in coming to you so that they can ask you their masaa'il and benefit from the knowledge of Deen which Allah Ta'ala has blessed you with. Thereafter, when they ask you their question, you often tell them, 'I do not know the answer.'"
The Desire of Hazrat 'Ali (radhiyallahu 'anhu) to Meet Allah Ta'ala with the Actions of Hazrat 'Umar (radhiyallahu 'anhu)Written by user2
Hazrat 'Ali's (radhiyallahu 'anhu) deep love for Hazrat 'Umar (radhiyallahu 'anhu) could be gauged from the following incident. In this incident, besides Hazrat 'Ali (radhiyallahu 'anhu) informing the ummah of the high and lofty position of Hazrat 'Umar (radhiyallahu 'anhu), he also expressed the desire to meet Allah Ta'ala with the actions of Hazrat 'Umar (radhiyallahu 'anhu). Hence, we find that Hazrat 'Ali (radhiyallahu 'anhu) greatly loved Hazrat 'Umar (radhiyallahu 'anhu) during his lifetime and even after his demise. He held him in the highest esteem and regarded him to be the greatest leader of this ummah after Hazrat Abu Bakr (radhiyallahu 'anhu).
Sunnats of Sitting in a Deeni Discourse – Part TwoWritten by user2
3. The primary purpose for one participating in a Deeni discourse is for one to correct and reform himself and to acquire the knowledge of the sunnah (i.e. the mubaarak lifestyle of Rasulullah (sallallahu 'alaihi wasallam)).
4. One should consider the Deeni discourse as a means of spiritual medication and consider it to be directed to oneself.
The Key to JannahWritten by user2
عن أبي هريرة أن رسول الله صلى الله عليه وسلم قال: كل أمتي يدخلون الجنة إلا من أبى قالوا: يا رسول الله ومن يأبى قال: من أطاعني دخل الجنة ومن عصاني فقد أبى (صحيح البخاري رقم 7280)
Hazrat Abu Hurairah (radhiyallahu 'anhu) reports that Rasulullah (sallallahu 'alaihi wasallam) said, "Every person in my ummah will enter Jannah, except those who refuse." The Sahaabah (radhiyallahu 'anhum) asked, "O Rasul of Allah (sallallahu 'alaihi wasallam)! Who are those who refuse?" Rasulullah (sallallahu 'alaihi wasallam) replied, "Those who obey me (believe in me and follow my way) will enter Jannah (forever), and those who disobey me (they do not believe in me) have refused (hence they will remain in Jahannum forever)."
Ten Durood in the Morning and EveningWritten by user2
عن أبي الدرداء قال: قال رسول الله صلى الله عليه وسلم: من صلى علي حين يصبح عشرا وحين يمسي عشرا أدركته شفاعتي يوم القيامة رواه الطبراني بإسنادين وإسناد أحدهما جيد ورجاله وثقوا (مجمع الزوائد رقم 17022)
Hazrat Abu Dardaa (radhiyallahu 'anhu) reports that Rasulullah (sallallahu 'alaihi wasallam) said, "Whoever recites ten durood upon me in the morning and ten durood upon me in the evening, he will receive my intercession on the Day of Qiyaamah."
يَا رَبِّ صَلِّ وَسَلِّمْ دَائِمًا اَبَدًا عَلَى حَبِيْبِكَ خَيْرِ الْخَلْقِ كُلِّهِمِ
Hazrat 'Uthmaan (radhiyallahu 'anhu) – Part Thirty SixWritten by user2
Appointment as the Khalifah (continued)
Before passing away, Hazrat 'Umar (radhiyallahu 'anhu) formed a Shura (council) consisting of the following six Sahaabah (radhiyallahu 'anhum):
Hazrat 'Ali (radhiyallahu 'anhu), Hazrat 'Uthmaan (radhiyallahu 'anhu), Hazrat Zubair (radhiyallahu 'anhu), Hazrat Talhah (radhiyallahu 'anhu), Hazrat Sa'd (radhiyallahu 'anhu) and Hazrat 'Abdur Rahmaan bin 'Auf (radhiyallahu 'anhu).
The Islam of Hazrat 'Ammaar bin Yaasir (radhiyallahu 'anhu) and Hazrat Suhaib bin Sinaan (radhiyallahu 'anhu)Written by user2
Hazrat 'Ammaar bin Yaasir (radhiyallahu 'anhu) says:
Hazrat Suhaib bin Sinaan (radhiyallahu 'anhu) and I had come to Daar-e-Arqam at the same time. We met at the door of Daar-e-Arqam, while Rasulullah (sallallahu 'alaihi wasallam) was inside, and asked each other what our intention for coming to see Rasulullah (sallallahu 'alaihi wasallam) was.
I replied, "I intend to go to Rasulullah (sallallahu 'alaihi wasallam) to listen to his message." Hazrat Suhaib (radhiyallahu 'anhu) expressed that he had the same intention. Both of us then entered Daar-e-Arqam where Rasulullah (sallallahu 'alaihi wasallam) presented the teachings of Islam to us and invited us to bring Imaan. Without any delay, we both accepted his invitation to Islam and were blessed with Imaan.
(Extracted from Seeratul Mustafa 1/168-169)


text:
Copper conductor - Wikipedia
· A copper cable consists of two or more copper wires running side by side and bonded, twisted or braided together to form a single assembly. Electrical cables may be made more flexible by stranding the wires.Get Price >>
Electric cable multicore stranded power cable and wire .
· Looking for PVC insulated copper wire, PVC cables, Electrical power cable, PVC Wires & Cables. PVC Stranded Copper Wire & Cable – Philatron Wire & Stranded Copper Wires & Cables by Philatron are suitable to Electric Vehicle (EV) Cable; Flexible Multi Strand Copper Electrical Cables, Multi Multi Strand Copper Electrical .Get Price >>
Electrical wiring - Wikipedia
· Lead-cased electrical cable from a circa 1912 house in Southern England. . The first rubber-insulated cables for USA building wiring were introduced in 1922 with US patent . wire", issued , assigned to Boston Insulated Wire And Cable . [citation needed] These were two or more solid copper electrical wires with rubber .Get Price >>
Stranded Copper Wire Electric Cable, Stranded
Alibaba offers 68,594 stranded copper wire electric cable products. About 92% of these are electrical wires, 6% are power cables, and 1% are communication cables. A wide variety of stranded copper wire electric cable options are available to you, such as high voltage, low voltage, and medium voltage.Get Price >>
10mm electric wire and cable bare copper ground
Optical Ground Wire. How do I set Bare Aluminum /ACSR/AAC 100mm2 overhead conductor AAAC 7/4.65mm · Bare Aluminum 1.5mm 2.5mm 4mm 6mm 10mm house wiring electrical cable. 1.5mm 2.5mm .. armoured cable Xlpe swa pvc power supply cable copper conductor 5 core copper power cable · armoured stranded copper single core pvc insulated .Get Price >>
cable wire electrical copper - alibaba
Alibaba offers 129,454 cable wire electrical copper products. About 50% of these are electrical wires, 26% are power cables, and 3% are terminals. A wide variety of cable wire electrical copper options are available to you, such as insulated, low voltage, and medium voltage.Get Price >>
Stranded vs Solid Wire - CustomCable
Beyond choosing between aluminum or copper, the wiring expert will understand the difference between stranded and solid wire and will choose the appropriate wire core to use for their chosen project. While solid wire consists of a single metal core, a stranded wire is composed of numerous thinner wires twisted together into a cohesive bunch.Get Price >>
Copper In Your Home: Home Planning Series - House
· House Wiring "Bundles Could Be a Fire Hazard. March 2005. New Research Leads to Electrical Code Change. Almost every home has electrical cables bundled together in various locations, and those bundled wires typically run through holes in the wood framing.Get Price >>
Difference between Wire and Cable - Copper wire
· HOME >> Difference between Wire and Cable. . There are two main types of wires: solid or stranded. A solid wire is a single conductor that is either bare or insulated by a protective colored sheath. It offers low resistance and are perfect for use in higher frequencies. . Types of Electrical Cables There are various types of cables .Get Price >>
Multi Strand Copper House Wiring China . - copper
Jun 30, 2015 round copper enameled wire singapore · transformer enamelled wire machine Electrical Cables & Wires – Multistrand Single Core Copper Panel . wires, teflon cables, RF coaxial cable, copper wire, wire harness, house wiring, safety cables at. china enameled copper clad aluminum for earphone.Get Price >>


text:
- 5 years ago
- Wedding: December 2014 - The Villa
So I was going to post some of the ABSOLUTELY GORGEOUS teaser photos from my bargain-priced photographer for you girls on the Bee, but I decided to wait. It's killing me slowly b/c I reeeeaaalllyyyy wanna recap before I forget all the details and get lost in a wedding haze but no one cares about the words anyway… MORE PICTURES!! Lol. So since I've opted against, I thought I would share that my wedding is currently being run (online) & will run in print in this Sunday's Washington Post. Whooooop!
Check the link: http://www.washingtonpost.com/lifestyle/style/on-love-candace-butler-and-anthony-williams/2015/01/22/f816302a-a182-11e4-b146-577832eafcb4_story.html


text:
19355 Seaton Ave. Perris CA 92570 United States
19355 Seaton Ave. Perris California 92570 US
(951) 657-1004(951) 657-1004
I rescue dogs purely for the love of dogs and save one dog's life at a time. I have had dogs for many decades and for 25 years, when my kennel was built, I started rescuing dogs, more now, because the need is greater. I like to specialize in small dogs, but sometimes people bring me larger dogs and I have accepted them, but prefer to stick with smaller dogs.


text:
Modernize legacy systems, applications and mainframes by abstracting the business functionality to make more efficient consumption patterns from front-end channels.
Decouple front-end channels from back-end systems through automation, governance and streamlined deployment via microservices, APIs and service-mesh architectures.
MERCURY is an accelerator that automates the development, testing and deployment of microservices and APIs, to help enable legacy systems and modernize front-end applications.
ARRAY accelerates the externalization of back-end services locked in legacy systems and the subsequent transition to a service-oriented architecture.
SHIFT automates the conversion and translation of legacy code when moving to Big Data environments or the Cloud.
Copyright © 2019 Next Pathway Inc.
All rights reserved. Cornerstone® is a registered trademark of Next Pathway Inc. under Canadian trademark reg. no. TMA964,114.
SHIFT™ is an existing, applied for or registered trademark of Next Pathway Inc.


text:
Login Form
or login
Established in 1986, G&G (Guay & Guay) Armament has been designing and manufacturing tactical equipment in Taiwan for military and law enforcement use. It is with this experience that they are also able to design and manufacture such an excellent range of high quality airsoft guns, with something for every player, whatever level they are at.
G&G use an automated computer programme to cut all of the parts required for their guns, meaning they are made in pretty much the same way as their live-firing counterparts. Take a look at our range of G&G AEG for sale below:
Valken V12 Optima Block-II 6mm Airsoft Assault Rifle RIF HPA
Special Order Only, Please Call For Details...6mm BB HPA Rifle with adjustable power and a 300rd high capacity magaz...


text:
12 Best Ideas Living Room Wall Mirror – From the thousands of pictures on the web in relation to Living Room Wall Mirror
, selects the top selections having best quality exclusively for you, and now this photographs is usually one among photos series within our best photos gallery with regards to 12 Best Ideas Living Room Wall Mirror. I am hoping you'll as it.
This graphic (Living Room Wall Mirror Lovely Bloombety Wall Mirrors for Living Room with Decorative Lighting Wall Mirrors for Living Room) above is branded with:
published through admin-moscowbiennale on 2017-11-13 15:42:49. To view all photos throughout 12 Best Ideas Living Room Wall Mirror pictures gallery you should abide by this specific website link.
The Incredible and Attractive Living Room Wall Mirror
pertaining to Inspire Your own home Present Property|Warm WishProperty


text:
Amsterdam, Netherlands – Well I couldn't spend all that money to come to Europe to race and not take a peak at what Amsterdam's all about! I had never been here before, and likely will not race in the Netherlands again, so I took a day to wander the infamous city.
After yesterday's race I drove back to the airport (about 2 and a half hour drive), dropped off the car and stored my luggage, and took the train into the center of Amsterdam. I wasn't quite sure what to expect outside of all the usual reasons why people go to Amsterdam, so I was a bit more nervous traveling solo here than any of the other solo travels I've taken.
After a day of touring the Red Light District, Amsterdam as a whole, fishing villages, cheese and clog makers, windmills and the canals, I was exhausted. I was also completely amazed at the beauty of the city. It was nothing like the preconceived notions that I had about the place. Sure the Red Light District was true to its reputation (and also very safe), but the architecture of the buildings, bridges and city itself was breathtaking. Amsterdam seems to be a cultural mecca for the arts, and given a few more days I might have been able to partake in them.
It was a great end to my trip. Going from the misery of being jet-lagged and sick, to the disappointment of the first race, to the growth of a good lesson of the second race, and finally to the appreciation and awe of the city of Amsterdam was quite an emotional journey over the course of five days. But this will be yet another trip that I'll remember for my lifetime.