We study a nonlinear stationary system of transport equations with specific boundary conditions describing the transport of solutes dissolved in a fluid
circulating in a countercurrent tubular architecture, which constitutes a simplified model of a kidney nephron.
We prove that for every Lipschitz and monotonic nonlinearity (which stems from active transport across the ascending limb),
the dynamic system, a PDE which we study through contraction properties, relaxes toward the unique stationary state.
A study of the linearized stationary operator enables us, using eigenelements, to further show that under certain conditions regarding the
nonlinearity, the relaxation is exponential. We also describe a finite volume scheme which allows us to
efficiently approach the numerical solution to the stationary system.
Finally, we apply this numerical method to illustrate how the countercurrent arrangement of tubes enhances the axial concentration
gradient, thereby favoring the production of highly concentrated urine.
1.
Introduction
A time delay widely exists in many fields such as chemistry, biology, industry, and so on. Since a time delay arising in a system may cause instability, the stability analysis of time-delay systems has been wildly studied in the past few decades [1,2]. The main purpose is paid to determine the admissible delay, for which the systems remain stable.
It is well known that the LKF method has been widely used to obtain stability conditions for time-delay systems [3,4]. The main purpose of the LKF method is to estimate the integral term arising in the time derivative of double integral term in the LKF. Therefore, to get less conservative stability criteria, many integral inequality methods are derived. Those inequality methods include Jensen inequality [5,6], Wirtinger inequality [7,8,9], double integral inequlity [10,11,12], various improved integral inequalities [13,14,15,16,17,18,19,20,21,22,23,24,25,26]. The Jensen inequality expressed as $ V_{ab}(\dot y) = \int_a^b\dot y^T(t)R\dot y(t)dt\geq \frac{1}{b-a}\Omega_0^TR\Omega_0 = V_{Jensen} $, where $ a < b, R = R^T > 0, \Omega_0 = y(b)-y(a) $. The Wirtinger-based inequality expressed as $ V_{ab}(\dot y)\geq V_{Jensen}+\frac{3}{b-a}\Omega_1^TR\Omega_1 = V_{Seuret} $, where $ \Omega_1 = y(b)+y(a)-\frac{2}{b-a}\int_a^by(t)dt $. The further improved inequality expressed as$ V_{ab}(\dot y)\geq V_{Seuret}+ \frac{5}{b-a}\Omega_2^TR\Omega_2 $, $ V_{ab}(\dot y)\geq V_{Seuret}+ \frac{5}{b-a}\Omega_2^TR\Omega_2 +\frac{7}{b-a}\Omega_3^TR\Omega_3 $, $ V_{ab}(\dot y)\geq V_{Seuret}+ \frac{5}{b-a}\Omega_2^TR\Omega_2 +\frac{7}{b-a}\Omega_3^TR\Omega_3 +\frac{9}{b-a}\Omega_4^TR\Omega_4 $ in [13,14,15], respectively, where $ \Omega_2, \Omega_3, \Omega_4 $ are defined in Lemma 4 [15]. However, these results only estimate the integral term arising in the time derivative of double integral term in the LKF. This paper presents a generalized double integral inequality which includes those in [10,11,12] as special cases. A new stability criterion is proposed by choosing a new LKF and using the generalized double integral inequality. Both the generalized integral inequality and the new LKF include fourth integrals, which may yield less conservative results. Two examples are introduced to show the effectiveness of the proposed criterion. The contributions of our paper are as follows:
$ \bullet $ The integral $ -\int_a^b\int_u^b\dot x^T(s)P\dot x(s)dsdu $ is estimated as$ -\int_a^b\int_u^b\dot x^T(s)P\dot x(s)dsdu\le \zeta^T\omega\zeta $, where $ \omega $ and $ \zeta $ are defined in Lemma 3. The above inequality includes those in [10,11,12] as special cases.
$ \bullet $ Both the new double integral inequality and the new LKF include fourth integrals, which may obtain more general results.
$ \bf{Notation} $: See Table 1.
2.
Preliminary
Consider the time delay systems as
where $ y(t)\in R^{n} $ is the state vector, $ h > 0 $ is constant time-delay and the initial condition $ \phi(t) $ is a continuous function.
Lemma 1. [15] For a matrix $ P\in S^n_+ $, and any continuously differentiable function $ x:[a, b]\longrightarrow R^n $, then we can obtain
where
Lemma 2. [27] For a positive define matrix $ P\in S_+^{n} $, a integrable function $ \left\{ {x(s)\left| {s \in \left[ {a, \; b} \right]} \right.} \right\} $, and any auxiliary functions $ \left\{ {f_i(s)\left| {i \in \left[0, n\right]}, {s \in \left[ {a, \; b} \right]} \right.}, f_0(s) = 1 \right\} $ satisfying $ \int_a^b\int_u^bf_i(s)f_j(s)dsdu = 0 $, $ (0\le i, j\le n, i\ne j) $ with $ f_i(s)\ne 0, i = 1, 2, \cdots, n $. Let $ \lambda_i\in R^{n\times k}, i = 0, 1, \cdots, n $ and a vector $ \zeta\in R^k $, such that$ \int_a^b\int_u^bf_i(s)x(s)dsdu = \lambda_i\zeta $. Then for any matrices $ M_i\in R^{k\times n}(i = 0, 1, \cdots, n) $, the following inequality holds
Proof. Define $ M = \left[ {MT0MT1⋯MTn
} \right]^T $, $ \xi(s) = \left[ {
f0(s)ζTf1(s)ζT⋯fn(s)ζT
} \right]^T $.
It is easy to obtain that
Integrating the inequality (2.5) from $ \left[ {a, b} \right]\times \left[ {u, b} \right] $ yields
This completes the proof.
Lemma 3. For a differential function $ x: \left[a, b\right]\to R^n $, a matrix $ P\in R_+^n $, a vector $ \zeta\in R^k $, and any matrices $ M_i\in R^{k\times n}(i = 1, 2, 3, 4) $, then the following inequality holds:
where
Proof. The result can be easily obtained by choosing $ n = 3 $, $ f_1(s) = \frac{3}{b-a}\left(s-\frac{2b+a}{3}\right) $, $ f_2(s) = \frac{10}{(b-a)^2}\left[\left(s-\frac{3b+2a}{5}\right)^2-\frac{3(b-a)^2}{50}\right] $, $ f_3(s) = -4+30\frac{s-a}{b-a}-60\left(\frac{s-a}{b-a}\right)^2+35\left(\frac{s-a}{b-a}\right)^3 $in (2.4). So the details of proof is omitted.
Remark 1. The inequality (25) of Lemma 5.1 in [10] is a special case of Lemma 3 by setting $ M_1 = -\frac{2}{b-a}\lambda_1^TR $, $ M_2 = -\frac{4}{b-a}\lambda_2^TR $, $ M_3 = 0 $, and $ M_4 = 0 $. The inequality (4) of Lemma 2.3 in [11] is a special case of Lemma 3 by setting $ M_1 = -\frac{2}{b-a}\lambda_1^TR $, $ M_2 = -\frac{4}{b-a}\lambda_2^TR $, $ M_3 = -\frac{6}{b-a}\lambda_3^TR $, and $ M_4 = 0 $. In addition, the inequality (12) of Lemma 5 in [12] is a special case of Lemma 3 by set ting $ M_1 = -\frac{2}{b-a}\lambda_1^TR $, $ M_2 = -\frac{4}{b-a}\lambda_2^TR $, $ M_3 = -\frac{6}{b-a}\lambda_3^TR $, and $ M_4 = -\frac{8}{b-a}\lambda_4^TR $.
3.
Results
Based on Lemma 1 and Lemma 3, a new stability condition can be obtained.
Theorem 1. System (1) is asymptotically stable if there exist matrices $ P\in S_+^{5n} $, $ R_1, R_2, R_3\in S_+^{n} $, and any matrices $ M_1, M_2, M_3, M_4\in R^{6n\times n} $ such that
where
$ \Pi_1 = \left[ {δT1δT3δT4δT5δT6
} \right]^T $,
$ \Pi_2 = \left[ {δT0δT1−δT2hδT1−δT3h22δT1−δT4h36δT1−δT5
} \right] $,
$ \Pi_3 = \delta_1-\delta_2 $,
$ \Pi_4 = \delta_1+\delta_2-\frac{2}{h}\delta_3 $,
$ \Pi_5 = \delta_1-\delta_2+\frac{6}{h}\delta_3-\frac{12}{h^2}\delta_4 $,
$ \Pi_6 = \delta_1+\delta_2-\frac{12}{h}\delta_3+\frac{60}{h^2}\delta_4-\frac{120}{h^3}\delta_5 $,
$ \Pi_7 = \delta_1-\delta_2+\frac{20}{h}\delta_3-\frac{180}{h^2}\delta_4+\frac{840}{h^3}\delta_5-\frac{1680}{h^4}\delta_6 $,
$ \Pi_8 = \delta_1-\frac{1}{h}\delta_3 $,
$ \Pi_9 = \delta_1+\frac{2}{h}\delta_3-\frac{6}{h^2}\delta_4 $,
$ \Pi_{10} = \delta_1-\frac{3}{h}\delta_3+\frac{24}{h^2}\delta_4-\frac{60}{h^3}\delta_5 $,
$ \Pi_{11} = \delta_1+\frac{4}{h}\delta_3-\frac{60}{h^2}\delta_4+\frac{360}{h^3}\delta_5-\frac{840}{h^4}\delta_6 $,
$ \delta_0 = A\delta_1+B\delta_2+C\delta_3 $,
$ \delta_i = \left[ {0n×(i−1)nIn0n×(7−i)n
} \right] $, $ i = 1, 2, \cdots, 6. $
Proof. Introduce a LKF as
where
$ ζ(t)=[yT(t)∫tt−hyT(s)dsvT1(t)vT2(t)vT3(t)]T
$
$ v_1^T(t) = \int_{t-h}^t\int_{u_1}^ty^T(s)dsdu_1 $
$ v_2^T(t) = \int_{t-h}^t\int_{u_1}^t\int_{u_2}^t y^T(s)dsdu_2 du_1 $
$ v_3^T(t) = \int_{t-h}^t\int_{u_1}^t\int_{u_2}^t\int_{u_3}^t y^T(s)dsdu_3 du_2du_1 $
Then, the time derivative of $ V(y_t) $ along the trajectories of system (1) as follows
where
$ η(t)=[yT(t)yT(t−h)∫tt−hyT(s)dsvT1(t)vT2(t)vT3(t)]T
$
By Lemma 1, we have
By Lemma 3, we have
Thus, according to (3.2)–(3.5), we have $ \dot V(y_t)\le \eta^T(t)\Psi \eta(t) $. Thus, if (3.1)holds, then, for a sufficient small scalar $ \varepsilon > 0 $, $ \dot V(y_t)\le -\varepsilon\left\| {y(t)} \right\|^2 $ holds, which ensures system (1) is asymptotically stable. The proof is completed.
Remark 2. Both the double integral inequality and the new LKF include fourth integrals, which may yield novel stability results. Furthermore, in order to fully consider relevant information of the double integral inequality in Lemma 3, the $ \int_{t-h}^t\int_{u_1}^t\int_{u_2}^t\int_{u_3}^t y^T(s)dsdu_3 du_2du_1 $ is added as a state vector.
4.
Numerical examples
In this section, we demonstrate the advantages of our proposed criterion by two numerical examples.
Example 1. Consider system(1) with:
$ A = \left[ {0.200.20.1
} \right] $, $ B = \left[ {
0000
} \right] $, $ C = \left[ {
−10−1−1
} \right] $.
Table 2 lists the allowable upper bounds of $ h $ by different methods. Table 2 shows that the maximum delay bounds of $ h $ obtained by our method are much larger than those in [4,6,7,9,11].
Example 2. Consider system(1) with:
$ A = \left[ {01−100−1
} \right] $, $ B = \left[ {
0.00.10.10.2
} \right] $, $ C = \left[ {
0000
} \right] $
Table 3 lists the allowable upper bounds of $ h $ by different methods. Table 3 shows that the maximum delay bounds of $ h $ obtained by our method are much larger than those in [4,7,9,10,11,14]. For $ h = 0.750 $, $ y(0) = (0.001, -0.001)^T $, the state trajectories of the system(1) is given in Figure 1.
Remark 3. According to Example 1 and Example 2, although our method can reduce the conservatism of the system effectively, it increases the computational burden.
5.
Conclusion
This paper focus on a new stability condition for a class of time delay systems. By using two generalized integral inequalities and a new augmented LKF, a new stability criterion is obtained. Both the double integral inequality and the new LKF include fourth integrals, which may yield more general results. Two numerical examples are proposed to show the effectiveness of the proposed criterion.
Acknowledgments
This work was supported by New Academic Talents and Innovation Exploration Project of Department of Science and Technology of Guizhou Province of China under Grant (Qian ke he pingtai rencai [2017] 5727-19); Innovative Groups of Education Department of Guizhou Province (Qian jiao he KY [2016] 046).
Conflict of interest
The authors declare that there are no conflicts of interest.