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Journal articles on the topic 'Asymptotically Stable'

Author: Grafiati

Published: 3 June 2025

Last updated: 6 July 2025

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1

Nadzieja, Tadek. "Construction of a smooth Lyapunov function for an asymptotically stable set." Czechoslovak Mathematical Journal 40, no. 2 (1990): 195–99. http://dx.doi.org/10.21136/cmj.1990.102373.

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2

Choi, Sung Kyu, Yoon Hoe Goo, and Namjip Koo. "Variationally Asymptotically Stable Difference Systems." Advances in Difference Equations 2007 (2007): 1–22. http://dx.doi.org/10.1155/2007/35378.

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3

Elaydi, Saber, and Hani R. Farran. "Exponentially asymptotically stable dynamical systems." Applicable Analysis 25, no. 4 (1987): 243–52. http://dx.doi.org/10.1080/00036818708839688.

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4

Yoneyama, T., and J. Sugie. "Exponentially asymptotically stable dynamical systems." Applicable Analysis 27, no. 1-3 (1988): 235–42. http://dx.doi.org/10.1080/00036818808839736.

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5

Driesse, R., and A. J. Homburg. "Essentially asymptotically stable homoclinic networks." Dynamical Systems 24, no. 4 (2009): 459–71. http://dx.doi.org/10.1080/14689360903039664.

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6

Ding, Changming, and J. M. Soriano. "Uniformly asymptotically Zhukovskij stable orbits." Computers & Mathematics with Applications 49, no. 1 (2005): 81–84. http://dx.doi.org/10.1016/j.camwa.2005.01.007.

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7

Hasina, A. T. R., R. Sedra, and R. Raft. "ASYMPTOTICALLY STABLE PROCESS AND APPLICATIONS." Advances in Mathematics: Scientific Journal 12, no. 1 (2023): 153–74. http://dx.doi.org/10.37418/amsj.12.1.10.

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We remark that some stationary processes do not verify $x_\infty|x_\infty$ is equal to its value. To do this, we propose a new definitions to differentiate it in which a process is asymptotically stable if it verifies this property. We also remark that all processes in all financial models have missed this property. Which leads us to reexamine the models and look the impact and importance of this property.

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8

WANG, XIA, and XINYU SONG. "GLOBAL PROPERTIES OF A MODEL OF IMMUNE EFFECTOR RESPONSES TO VIRAL INFECTIONS." Advances in Complex Systems 10, no. 04 (2007): 495–503. http://dx.doi.org/10.1142/s0219525907001252.

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This article proposes a mathematical model which has been used to investigate the importance of lytic and non-lytic immune responses for the control of viral infections. By means of Lyapunov functions, the global properties of the model are obtained. The virus is cleared if the basic reproduction number R0 ≤ 1 and the virus persists in the host if R0 > 1. Furthermore, if R0 > 1 and other conditions hold, the immune-free equilibrium E0 is globally asymptotically stable. The equilibrium E1 exists and is globally asymptotically stale if the CTL immune response reproductive number R1 < 1 and the antibody immune response reproductive number R2 > 1. The equilibrium E2 exists and is globally asymptotically stable if R1 > 1 and R2 < 1. Finally, the endemic equilibrium E3 exists and is globally asymptotically stable if R1 > 1 and R2 > 1.

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9

Kaczorek, Tadeusz. "Approximation of fractional positive stable continuous-time linear systems by fractional positive stable discrete-time systems." International Journal of Applied Mathematics and Computer Science 23, no. 3 (2013): 501–6. http://dx.doi.org/10.2478/amcs-2013-0038.

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Abstract Fractional positive asymptotically stable continuous-time linear systems are approximated by fractional positive asymptotically stable discrete-time systems using a linear Padé-type approximation. It is shown that the approximation preserves the positivity and asymptotic stability of the systems. An optional system approximation is also discussed.

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10

W. Hirsch, M., and Hal L. Smith. "Asymptotically stable equilibria for monotone semiflows." Discrete & Continuous Dynamical Systems - A 14, no. 3 (2006): 385–98. http://dx.doi.org/10.3934/dcds.2006.14.385.

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11

Dimopoulos, Nikitas J., John T. Dorocicz, Chris Jubien, and Stephen Neville. "Training Asymptotically Stable Recurrent Neural Networks." Intelligent Automation & Soft Computing 2, no. 4 (1996): 375–88. http://dx.doi.org/10.1080/10798587.1996.10750681.

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12

de la Calle Ysern, Bernardo. "Asymptotically Stable Equilibria of Gradient Systems." American Mathematical Monthly 126, no. 10 (2019): 936–39. http://dx.doi.org/10.1080/00029890.2019.1684152.

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13

MIMNA, R., and T. STEELE. "Asymptotically stable sets for semi-homeomorphisms." Nonlinear Analysis 59, no. 6 (2004): 849–55. http://dx.doi.org/10.1016/s0362-546x(04)00293-7.

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14

Mimna, R. A., and T. H. Steele. "Asymptotically stable sets for semi-homeomorphisms." Nonlinear Analysis: Theory, Methods & Applications 59, no. 6 (2004): 849–55. http://dx.doi.org/10.1016/j.na.2004.07.041.

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15

Moulay, Emmanuel, and Sanjay P. Bhat. "Topological properties of asymptotically stable sets." Nonlinear Analysis: Theory, Methods & Applications 73, no. 4 (2010): 1093–97. http://dx.doi.org/10.1016/j.na.2010.04.043.

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16

Vanualailai, Jito, Bibhya Sharma, and Shin-ichi Nakagiri. "An asymptotically stable collision-avoidance system." International Journal of Non-Linear Mechanics 43, no. 9 (2008): 925–32. http://dx.doi.org/10.1016/j.ijnonlinmec.2008.06.012.

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17

Zaharopol, Radu. "Strongly asymptotically stable Frobenius-Perron operators." Proceedings of the American Mathematical Society 128, no. 12 (2000): 3547–52. http://dx.doi.org/10.1090/s0002-9939-00-05473-3.

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18

Kanev, S., and M. Verhaegen. "Robustly asymptotically stable finite-horizon MPC." Automatica 42, no. 12 (2006): 2189–94. http://dx.doi.org/10.1016/j.automatica.2006.07.011.

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19

Beck, Nikolai. "Stable parabolic Higgs bundles as asymptotically stable decorated swamps." Journal of Geometry and Physics 104 (June 2016): 229–41. http://dx.doi.org/10.1016/j.geomphys.2016.02.014.

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20

XIUXIANG, YANG, and XUE CHUNRONG. "AN SIQS INFECTION MODEL WITH NONLINEAR AND ISOLATION." International Journal of Biomathematics 01, no. 02 (2008): 239–45. http://dx.doi.org/10.1142/s1793524508000199.

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By means of asymptotically stable theory and infection model theory of ordinary differential equation, we do research on SIQS model with nonlinear and isolation. Firstly, we obtain the existence of threshold value R0 of disease-free equilibration point and local disease equilibration point. Secondly, we prove disease-free equilibration point is locally asymptotically stable when R0 < 1, and local disease equilibration point is locally asymptotically stable when R0 > 1. Furthermore, we have disease-free equilibration point and local disease equilibration point are globally asymptotically stable with the help of Liapunov function. Lastly, we explain at the point of biology.

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21

Kaczorek, Tadeusz. "Comparison of approximation methods of positive stable continuous-time linear systems by positive stable discrete-time systems." Archives of Electrical Engineering 62, no. 2 (2013): 345–55. http://dx.doi.org/10.2478/aee-2013-0027.

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Abstract The positive asymptotically stable continuous-time linear systems are approximated by corresponding asymptotically stable discrete-time linear systems. Two methods of the approximation are presented and the comparison of the methods is addressed. The considerations are illustrated by three numerical examples and an example of positive electrical circuit.

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22

Wang, Juan, Xue-Zhi Li, and Souvik Bhattacharya. "The backward bifurcation of a model for malaria infection." International Journal of Biomathematics 11, no. 02 (2018): 1850018. http://dx.doi.org/10.1142/s1793524518500183.

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In this paper, an epidemic model of a vector-borne disease, namely, malaria, is considered. The explicit expression of the basic reproduction number is obtained, the local and global asymptotical stability of the disease-free equilibrium is proved under certain conditions. It is shown that the model exhibits the phenomenon of backward bifurcation where the stable disease-free equilibrium coexists with a stable endemic equilibrium. Further, it is proved that the unique endemic equilibrium is globally asymptotically stable under certain conditions.

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23

Lee, Kwang Sung, and Abid Ali Lashari. "Global Stability of a Host-Vector Model for Pine Wilt Disease with Nonlinear Incidence Rate." Abstract and Applied Analysis 2014 (2014): 1–11. http://dx.doi.org/10.1155/2014/219173.

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Based on classical epidemic models, this paper considers a deterministic epidemic model for the spread of the pine wilt disease which has vector mediated transmission. The analysis of the model shows that its dynamics are completely determined by the basic reproduction numberR0. Using a Lyapunov function and a LaSalle's invariant set theorem, we proved the global asymptotical stability of the disease-free equilibrium. We find that ifR0≤1, the disease free equilibrium is globally asymptotically stable, and the disease will be eliminated. IfR0>1, a unique endemic equilibrium exists and is shown to be globally asymptotically stable, under certain restrictions on the parameter values, using the geometric approach method for global stability, due to Li and Muldowney and the disease persists at the endemic equilibrium state if it initially exists.

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24

Khan, Muhammad Altaf, Yasir Khan, Sehra Khan, and Saeed Islam. "Global stability and vaccination of an SEIVR epidemic model with saturated incidence rate." International Journal of Biomathematics 09, no. 05 (2016): 1650068. http://dx.doi.org/10.1142/s1793524516500686.

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This study considers SEIVR epidemic model with generalized nonlinear saturated incidence rate in the host population horizontally to estimate local and global equilibriums. By using the Routh–Hurwitz criteria, it is shown that if the basic reproduction number [Formula: see text], the disease-free equilibrium is locally asymptotically stable. When the basic reproduction number exceeds the unity, then the endemic equilibrium exists and is stable locally asymptotically. The system is globally asymptotically stable about the disease-free equilibrium if [Formula: see text]. The geometric approach is used to present the global stability of the endemic equilibrium. For [Formula: see text], the endemic equilibrium is stable globally asymptotically. Finally, the numerical results are presented to justify the mathematical results.

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25

Cui, Jing’an, and Zhanmin Wu. "AnSIRSModel for Assessing Impact of Media Coverage." Abstract and Applied Analysis 2014 (2014): 1–6. http://dx.doi.org/10.1155/2014/424610.

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AnSIRSmodel incorporating a general nonlinear contact function is formulated and analyzed. When the basic reproduction numberℛ0<1, the disease-free equilibrium is locally asymptotically stable. There is a unique endemic equilibrium that is locally asymptotically stable ifℛ0>1. Under some conditions, the endemic equilibrium is globally asymptotically stable. At last, we conduct numerical simulations to illustrate some results which shed light on the media report that may be the very effective method for infectious disease control.

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26

Yang, Yu. "Global Analysis of a Virus Dynamics Model with General Incidence Function and Cure Rate." Abstract and Applied Analysis 2014 (2014): 1–7. http://dx.doi.org/10.1155/2014/726349.

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A virus dynamics model with logistic function, general incidence function, and cure rate is considered. By carrying out mathematical analysis, we show that the infection-free equilibrium is globally asymptotically stable if the basic reproduction numberℛ0≤1. Ifℛ0>1, then the infection equilibrium is globally asymptotically stable under some assumptions. Furthermore, we also obtain the conditions for which the model exists an orbitally asymptotically stable periodic solution. Examples are provided to support our analytical conclusions.

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27

Guerman, Anna, Ana Seabra, and Georgi Smirnov. "Optimization of Parameters of Asymptotically Stable Systems." Mathematical Problems in Engineering 2011 (2011): 1–19. http://dx.doi.org/10.1155/2011/526167.

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This work deals with numerical methods of parameter optimization for asymptotically stable systems. We formulate a special mathematical programming problem that allows us to determine optimal parameters of a stabilizer. This problem involves solutions to a differential equation. We show how to chose the mesh in order to obtain discrete problem guaranteeing the necessary accuracy. The developed methodology is illustrated by an example concerning optimization of parameters for a satellite stabilization system.

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28

Xu, Lan, Beimei Chen, Yun Zhao, and Yongluo Cao. "Normal Lyapunov exponents and asymptotically stable attractors." Dynamical Systems 23, no. 2 (2008): 207–18. http://dx.doi.org/10.1080/14689360802058765.

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29

Brodbeck, Othmar, Simonetta Frittelli, Peter Hübner, and Oscar A. Reula. "Einstein’s equations with asymptotically stable constraint propagation." Journal of Mathematical Physics 40, no. 2 (1999): 909–23. http://dx.doi.org/10.1063/1.532694.

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30

Slyusarchuk, V. E. "Nonlinear differential equations with asymptotically stable solutions." Ukrainian Mathematical Journal 50, no. 2 (1998): 302–13. http://dx.doi.org/10.1007/bf02513453.

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31

Slyusarchuk, V. E. "Nonlinear difference equations with asymptotically stable solutions." Ukrainian Mathematical Journal 49, no. 7 (1997): 1089–101. http://dx.doi.org/10.1007/bf02528754.

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32

Shevlyakov, Georgy, Vladimir Shin, Seokhyoung Lee, and Kiseon Kim. "Asymptotically stable detection of a weak signal." International Journal of Adaptive Control and Signal Processing 28, no. 9 (2013): 848–58. http://dx.doi.org/10.1002/acs.2405.

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33

Liu, Guang-Jun, and Andrew A. Goldenberg. "Asymptotically stable robust control of robot manipulators." Mechanism and Machine Theory 31, no. 5 (1996): 607–18. http://dx.doi.org/10.1016/0094-114x(95)00098-j.

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34

Caraballo, Tomás, and Peter E. Kloeden. "The persistence of synchronization under environmental noise." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 461, no. 2059 (2005): 2257–67. http://dx.doi.org/10.1098/rspa.2005.1484.

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It is shown that the synchronization of dissipative systems persists when they are disturbed by additive noise, no matter how large the intensity of the noise, provided asymptotically stable stationary-stochastic solutions are used instead of asymptotically stable equilibria.

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35

Dikansky, Arnold. "Periodic solutions and Galerkin approximations to the autonomous reaction-diffusion equations." Bulletin of the Australian Mathematical Society 48, no. 1 (1993): 127–40. http://dx.doi.org/10.1017/s0004972700015537.

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The assumption that a Galerkin equation of the reaction-diffusion system of high order has an asymptotically orbitally stable time-periodic solution implies that the full reaction-diffusion system has a nearby asymptotically orbitally stable time-periodic solution with asymptotic phase.

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36

Kaczorek, T. "Positive and asymptotically stable realizations for descriptor discrete-time linear systems." Bulletin of the Polish Academy of Sciences: Technical Sciences 61, no. 1 (2013): 229–37. http://dx.doi.org/10.2478/bpasts-2013-0022.

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Abstract Conditions for the existence of positive and asymptotically stable realizations for descriptor discrete-time linear systems are established. Procedures for computation of positive and asymptotically stable realizations for improper transfer matrices are proposed. The effectiveness of the methods is demonstrated on numerical examples

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37

Davis, Richard A., and Sidney I. Resnick. "Crossings of max-stable processes." Journal of Applied Probability 31, no. 1 (1994): 130–38. http://dx.doi.org/10.2307/3215240.

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38

Davis, Richard A., and Sidney I. Resnick. "Crossings of max-stable processes." Journal of Applied Probability 31, no. 01 (1994): 130–38. http://dx.doi.org/10.1017/s0021900200107387.

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39

Keyoumu, Tuersunjiang, Wanbiao Ma, and Ke Guo. "Global Stability of a MERS-CoV Infection Model with CTL Immune Response and Intracellular Delay." Mathematics 11, no. 4 (2023): 1066. http://dx.doi.org/10.3390/math11041066.

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In this paper, we propose and study a Middle East respiratory syndrome coronavirus (MERS-CoV) infection model with cytotoxic T lymphocyte (CTL) immune response and intracellular delay. This model includes five compartments: uninfected cells, infected cells, viruses, dipeptidyl peptidase 4 (DPP4), and CTL immune cells. We obtained an immunity-inactivated reproduction number R0 and an immunity-activated reproduction number R1. By analyzing the distributions of roots of the corresponding characteristic equations, the local stability results of the infection-free equilibrium, the immunity-inactivated equilibrium, and the immunity-activated equilibrium were obtained. Moreover, by constructing suitable Lyapunov functionals and combining LaSalle’s invariance principle and Barbalat’s lemma, some sufficient conditions for the global stability of the three types of equilibria were obtained. It was found that the infection-free equilibrium is globally asymptotically stable if R0≤1 and unstable if R0>1; the immunity-inactivated equilibrium is locally asymptotically stable if R0>1>R1 and globally asymptotically stable if R0>1>R1 and condition (H1) holds, but unstable if R1>1; and the immunity-activated equilibrium is locally asymptotically stable if R1>1 and is globally asymptotically stable if R1>1 and condition (H1) holds.

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40

Li, Yunfei, Rui Xu, Zhe Li, and Shuxue Mao. "Global Dynamics of a Delayed HIV-1 Infection Model with CTL Immune Response." Discrete Dynamics in Nature and Society 2011 (2011): 1–13. http://dx.doi.org/10.1155/2011/673843.

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A delayed HIV-1 infection model with CTL immune response is investigated. By using suitable Lyapunov functionals, it is proved that the infection-free equilibrium is globally asymptotically stable if the basic reproduction ratio for viral infection is less than or equal to unity; if the basic reproduction ratio for CTL immune response is less than or equal to unity and the basic reproduction ratio for viral infection is greater than unity, the CTL-inactivated infection equilibrium is globally asymptotically stable; if the basic reproduction ratio for CTL immune response is greater than unity, the CTL-activated infection equilibrium is globally asymptotically stable.

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41

Simpson, David J. W. "Grazing-Sliding Bifurcations Creating Infinitely Many Attractors." International Journal of Bifurcation and Chaos 27, no. 12 (2017): 1730042. http://dx.doi.org/10.1142/s0218127417300427.

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As the parameters of a piecewise-smooth system of ODEs are varied, a periodic orbit undergoes a bifurcation when it collides with a surface where the system is discontinuous. Under certain conditions this is a grazing-sliding bifurcation. Near grazing-sliding bifurcations, structurally stable dynamics are captured by piecewise-linear continuous maps. Recently it was shown that maps of this class can have infinitely many asymptotically stable periodic solutions of a simple type. Here this result is used to show that at a grazing-sliding bifurcation an asymptotically stable periodic orbit can bifurcate into infinitely many asymptotically stable periodic orbits. For an abstract ODE system the periodic orbits are continued numerically revealing subsequent bifurcations at which they are destroyed.

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42

Yuan, Haiyan, and Cheng Song. "Nonlinear Stability and Convergence of Two-Step Runge-Kutta Methods for Volterra Delay Integro-Differential Equations." Abstract and Applied Analysis 2013 (2013): 1–13. http://dx.doi.org/10.1155/2013/679075.

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This paper introduces the stability and convergence of two-step Runge-Kutta methods with compound quadrature formula for solving nonlinear Volterra delay integro-differential equations. First, the definitions of(k,l)-algebraically stable and asymptotically stable are introduced; then the asymptotical stability of a(k,l)-algebraically stable two-step Runge-Kutta method with0<k<1is proved. For the convergence, the concepts ofD-convergence, diagonally stable, and generalized stage order are firstly introduced; then it is proved by some theorems that if a two-step Runge-Kutta method is algebraically stable and diagonally stable and its generalized stage order isp, then the method with compound quadrature formula isD-convergent of order at leastmin{p,ν}, whereνdepends on the compound quadrature formula.

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43

Wang, Lei, Zhidong Teng, and Long Zhang. "Global Behaviors of a Class of Discrete SIRS Epidemic Models with Nonlinear Incidence Rate." Abstract and Applied Analysis 2014 (2014): 1–18. http://dx.doi.org/10.1155/2014/249623.

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We study a class of discrete SIRS epidemic models with nonlinear incidence rateF(S)G(I)and disease-induced mortality. By using analytic techniques and constructing discrete Lyapunov functions, the global stability of disease-free equilibrium and endemic equilibrium is obtained. That is, if basic reproduction numberℛ0<1, then the disease-free equilibrium is globally asymptotically stable, and ifℛ0>1, then the model has a unique endemic equilibrium and when some additional conditions hold the endemic equilibrium also is globally asymptotically stable. By using the theory of persistence in dynamical systems, we further obtain that only whenℛ0>1, the disease in the model is permanent. Some special cases ofF(S)G(I)are discussed. Particularly, whenF(S)G(I)=βSI/(1+λI), it is obtained that the endemic equilibrium is globally asymptotically stable if and only ifℛ0>1. Furthermore, the numerical simulations show that for general incidence rateF(S)G(I)the endemic equilibrium may be globally asymptotically stable only asℛ0>1.

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44

Kaczorek, Tadeusz. "Stability of Interval Positive Fractional Discrete–Time Linear Systems." International Journal of Applied Mathematics and Computer Science 28, no. 3 (2018): 451–56. http://dx.doi.org/10.2478/amcs-2018-0034.

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Abstract The aim of this work is to show that interval positive fractional discrete-time linear systems are asymptotically stable if and only if the respective lower and upper bound systems are asymptotically stable. The classical Kharitonov theorem is extended to interval positive fractional linear systems.

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45

Kaczorek, Tadeusz. "Positive stable realizations for fractional descriptor continuous-time linear systems." Archives of Control Sciences 22, no. 3 (2012): 303–13. http://dx.doi.org/10.2478/v10170-011-0026-y.

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Abstract A method for computation of positive asymptotically stable realizations of fractional descriptor continuous-time linear systems with regular pencil is proposed. The method is based on the decomposition of the improper transfer matrix into strictly proper matrix and a polynomial matrix. A procedure for decomposition of a positive asymptotically stable realization is given and illustrated by a numerical example.

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46

Kaczorek, Tadeusz. "Inverse systems of linear systems." Archives of Electrical Engineering 59, no. 3-4 (2010): 203–16. http://dx.doi.org/10.2478/s10171-010-0016-x.

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Inverse systems of linear systemsThe concept of inverse systems for standard and positive linear systems is introduced. Necessary and sufficient conditions for the existence of the positive inverse system for continuous-time and discrete-time linear systems are established. It is shown that: 1) The inverse system of continuous-time linear system is asymptotically stable if and only if the standard system is asymptotically stable. 2) The inverse system of discrete-time linear system is asymptotically stable if and only if the standard system is unstable. 3) The inverse system of continuous-time and discrete-time linear systems are reachable if and only if the standard systems are reachable. The considerations are illustrated by numerical examples.

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47

Karakostas, George L. "On a conjecture for the difference equation $ x_{n+1} = 1+p\frac{x_{n-m}}{x_n^2} $." AIMS Mathematics 8, no. 10 (2023): 22714–29. http://dx.doi.org/10.3934/math.20231156.

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<abstract><p>In <sup>[<xref ref-type="bibr" rid="b24">24</xref>]</sup>, E. Tasdemir, et al. proved that the positive equilibrium of the nonlinear discrete equation $ x_{n+1} = 1+p\frac{x_{n-m}}{x_n^2} $ is globally asymptotically stable for $ p\in(0, \frac{1}{2}) $, {locally} asymptotically stable for $ p\in(\frac{1}{2}, \frac{3}{4}) $ and it was { conjectured} that for any $ p $ in the open interval $ (\frac{1}{2}, \frac{3}{4}) $ the equilibrium is { globally} asymptotically stable. In this paper, we prove that this conjecture is true for the closed interval $ [\frac{1}{2}, \frac{3}{4}]. $ In addition, it is shown that for $ p\in(\frac{3}{4}, 1) $ the behaviour of the solutions depend on the delay $ m. $ Indeed, here we show that in case $ m = 1 $, there is an unstable equilibrium and an asymptotically stable 2-periodic solution. But, in case $ m = 2 $, there is an asymptotically stable equilibrium. These results are obtained by using linearisation, a method lying on the well known Perron's stability theorem (<sup>[<xref ref-type="bibr" rid="b17">17</xref>]</sup>, p. 18). Finally, a conjecture is posed about the behaviour of the solutions for $ m &gt; 2 $ and $ p\in(\frac{3}{4}, 1) $.</p></abstract>

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48

Edelman, Mark. "Cycles in asymptotically stable and chaotic fractional maps." Nonlinear Dynamics 104, no. 3 (2021): 2829–41. http://dx.doi.org/10.1007/s11071-021-06379-2.

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49

Özdemi̇r, İsmet. "Asymptotically stable solutions of a nonlinear integral equation." Methods of Functional Analysis and Topology 27, no. 1 (2021): 57–73. http://dx.doi.org/10.31392/mfat-npu26_1.2021.08.

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50

PADEN, BRAD, and RAVI PANJA. "Globally asymptotically stable ‘PD+’ controller for robot manipulators." International Journal of Control 47, no. 6 (1988): 1697–712. http://dx.doi.org/10.1080/00207178808906130.

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