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July  2011, 16(1): 283-317. doi: 10.3934/dcdsb.2011.16.283

Feedback stabilization methods for the numerical solution of ordinary differential equations

Iasson Karafyllis 1, and  Lars Grüne 2,

1. 

Department of Environmental Engineering, Technical University of Crete, 73100 Chania, Greece
2. 

Mathematisches Institute, Universität Bayreuth, 95440 Bayreuth

Received  May 2010 Revised  August 2010 Published  April 2011

In this work we study the problem of step size selection for numerical schemes, which guarantees that the numerical solution presents the same qualitative behavior as the original system of ordinary differential equations. We apply tools from nonlinear control theory, specifically Lyapunov function and small-gain based feedback stabilization methods for systems with a globally asymptotically stable equilibrium point. Proceeding this way, we derive conditions under which the step size selection problem is solvable (including a nonlinear generalization of the well-known A-stability property for the implicit Euler scheme) as well as step size selection strategies for several applications.
Keywords: feedback stabilization, Asymptotic stability of numerical approximations, nonlinear differential equations..
Mathematics Subject Classification: Primary: 65L07, 34D20; Secondary: 65L06, 93D1.
Citation: Iasson Karafyllis, Lars Grüne. Feedback stabilization methods for the numerical solution of ordinary differential equations. Discrete & Continuous Dynamical Systems - B, 2011, 16 (1) : 283-317. doi: 10.3934/dcdsb.2011.16.283
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show all references

References:
[1]

Nonlinear Anal., 7 (1983), 1163-1173. doi: 10.1016/0362-546X(83)90049-4.  Google Scholar

[2]

Progress in Nonlinear Differential Equations and their Applications, 58, Birkhäuser Boston Inc., Boston, MA, 2004.  Google Scholar

[3]

Math. Control Signals Systems, 19 (2007), 93-122. doi: 10.1007/s00498-007-0014-8.  Google Scholar

[4]

Birkhäuser, Boston, MA, 1996.  Google Scholar

[5]

Discrete Contin. Dyn. Syst., 13 (2005), 827-841. doi: 10.3934/dcds.2005.13.827.  Google Scholar

[6]

SIAM J. Sci. Comput., 24 (2003), 1091-1106. doi: 10.1137/S1064827501388157.  Google Scholar

[7]

J. Comput. Phys., 187 (2003), 95-109. doi: 10.1016/S0021-9991(03)00082-2.  Google Scholar

[8]

Springer, Berlin, 2007.  Google Scholar

[9]

J. Optim. Theory Appl., 92 (1997), 581-604. doi: 10.1023/A:1022607507153.  Google Scholar

[10]

BIT, 45 (2005), 709-723. doi: 10.1007/s10543-005-0034-z.  Google Scholar

[11]

Springer, Berlin, 2002.  Google Scholar

[12]

SIAM J. Numer. Anal., 41 (2003), 2096-2113. doi: 10.1137/S003614290139411X.  Google Scholar

[13]

Syst. Control Lett., 38 (1999), 127-134.  Google Scholar

[14]

ACM Trans. Math. Software, 17 (1991), 533-554. doi: 10.1145/210232.210242.  Google Scholar

[15]

ACM Trans. Math. Software, 20 (1994), 496-517. doi: 10.1145/198429.198437.  Google Scholar

[16]

BIT, 28 (1988), 270-287.  Google Scholar

[17]

Springer, Berlin, second edition, 2006.  Google Scholar

[18]

Springer, Berlin, second edition, 1993.  Google Scholar

[19]

Springer, Berlin, second edition, 1996.  Google Scholar

[20]

Math. Control Signals Systems, 7 (1994), 95-120. doi: 10.1007/BF01211469.  Google Scholar

[21]

IMA J. Math. Control Inform., 23 (2006), 11-41. doi: 10.1093/imamci/dni037.  Google Scholar

[22]

J. Math. Anal. Appl., 328 (2007), 876-899. doi: 10.1016/j.jmaa.2006.05.059.  Google Scholar

[23]

SIAM J. Control Optim., 46 (2007), 1483-1517. doi: 10.1137/060669310.  Google Scholar

[24]

In "Proceedings of the 48th IEEE Conference on Decision and Control,'' pages 7996-8001, Shanghai, China, 2009. Google Scholar

[25]

Prentice Hall, Upper Saddle River, third edition, 2002.  Google Scholar

[26]

SIAM J. Numer. Anal., 23 (1986), 986-995. doi: 10.1137/0723066.  Google Scholar

[27]

Discrete Contin. Dynam. Systems, 2 (1996), 163-172. doi: 10.3934/dcds.1996.2.163.  Google Scholar

[28]

Marcel Dekker, New York, second edition, 2002.  Google Scholar

[29]

BIT, 40 (2000), 314-335. doi: 10.1023/A:1022395124683.  Google Scholar

[30]

SIAM J. Control Optim., 34 (1996), 124-160. doi: 10.1137/S0363012993259981.  Google Scholar

[31]

IEEE Trans. Circuits Syst. I Regul. Pap., 52 (2005), 804-814. doi: 10.1109/TCSI.2005.844366.  Google Scholar

[32]

IEEE Trans. Automat. Control, 34 (1989), 435-443. doi: 10.1109/9.28018.  Google Scholar

[33]

Systems Control Lett., 13 (1989), 117-123.  Google Scholar

[34]

Springer, New York, second edition, 1998.  Google Scholar

[35]

Cambridge University Press, Cambridge, 1996.  Google Scholar

[36]

Preprint, 2005. Google Scholar

[37]

IEEE Trans. Circuits Syst., 51 (2004), 1385-1394. doi: 10.1109/TCSI.2004.830694.  Google Scholar

[38]

Math. Programming, 18 (1980), 155-168. doi: 10.1007/BF01588311.  Google Scholar

[39]

Appl. Math. Comput., 184 (2007), 789-797. doi: 10.1016/j.amc.2006.05.190.  Google Scholar

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