American Institute of Mathematical Sciences

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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
References:
[1]

Z. Artstein, Stabilization with relaxed controls, Nonlinear Anal., 7 (1983), 1163-1173. doi: 10.1016/0362-546X(83)90049-4.  Google Scholar

[2]

P. Cannarsa and C. Sinestrari, "Semiconcave Functions, Hamilton-Jacobi Equations, and Optimal Control,'' Progress in Nonlinear Differential Equations and their Applications, 58, Birkhäuser Boston Inc., Boston, MA, 2004.  Google Scholar

[3]

S. Dashkovskiy, B. S. Rüffer and F. R. Wirth, An ISS small gain theorem for general networks, Math. Control Signals Systems, 19 (2007), 93-122. doi: 10.1007/s00498-007-0014-8.  Google Scholar

[4]

R. A. Freeman and P. V. Kokotović, "Robust Nonlinear Control Design - State-Space and Lyapunov Techniques,'' Birkhäuser, Boston, MA, 1996.  Google Scholar

[5]

B. M. Garay and K. Lee, Attractors under discretization with variable stepsize, Discrete Contin. Dyn. Syst., 13 (2005), 827-841. doi: 10.3934/dcds.2005.13.827.  Google Scholar

[6]

C. W. Gear and I. G. Kevrekidis, Projective methods for stiff differential equations: Problems with gaps in their eigenvalue spectrum, SIAM J. Sci. Comput., 24 (2003), 1091-1106. doi: 10.1137/S1064827501388157.  Google Scholar

[7]

C. W. Gear and I. G. Kevrekidis, Telescopic projective methods for parabolic differential equations, J. Comput. Phys., 187 (2003), 95-109. doi: 10.1016/S0021-9991(03)00082-2.  Google Scholar

[8]

P. Giesl, "Construction of Global Lyapunov Functions Using Radial Basis Functions,'' volume 1904 of "Lecture Notes in Mathematics," Springer, Berlin, 2007.  Google Scholar

[9]

B. S. Goh, Algorithms for unconstrained optimization problems via control theory, J. Optim. Theory Appl., 92 (1997), 581-604. doi: 10.1023/A:1022607507153.  Google Scholar

[10]

V. Grimm and G. R. W. Quispel, Geometric integration methods that preserve Lyapunov functions, BIT, 45 (2005), 709-723. doi: 10.1007/s10543-005-0034-z.  Google Scholar

[11]

L. Grüne, "Asymptotic Behavior of Dynamical and Control Systems under Perturbation and Discretization,'' volume 1783 of "Lecture Notes in Mathematics," Springer, Berlin, 2002.  Google Scholar

[12]

L. Grüne, Attraction rates, robustness, and discretization of attractors, SIAM J. Numer. Anal., 41 (2003), 2096-2113. doi: 10.1137/S003614290139411X.  Google Scholar

[13]

L. Grüne, E. D. Sontag and F. R. Wirth, Asymptotic stability equals exponential stability, and ISS equals finite energy gain-if you twist your eyes, Syst. Control Lett., 38 (1999), 127-134.  Google Scholar

[14]

K. Gustafsson, Control-theoretic techniques for stepsize selection in explicit Runge-Kutta methods, ACM Trans. Math. Software, 17 (1991), 533-554. doi: 10.1145/210232.210242.  Google Scholar

[15]

K. Gustafsson, Control-theoretic techniques for stepsize selection in implicit Runge-Kutta methods, ACM Trans. Math. Software, 20 (1994), 496-517. doi: 10.1145/198429.198437.  Google Scholar

[16]

K. Gustafsson, M. Lundh and G. Söderlind, A {PI stepsize control for the numerical solution of ordinary differential equations}, BIT, 28 (1988), 270-287.  Google Scholar

[17]

E. Hairer, C. Lubich and G. Wanner, "Geometric Numerical Integration. Structure-Preserving Algorithms for Ordinary Differential Equations,'' Springer, Berlin, second edition, 2006.  Google Scholar

[18]

E. Hairer, S. P. Nørsett and G. Wanner, "Solving Ordinary Differential Equations. I Nonstiff Problems,'' Springer, Berlin, second edition, 1993.  Google Scholar

[19]

E. Hairer and G. Wanner, "Solving Ordinary Differential Equations. {II} Stiff and Differential-Algebraic Problems,'' Springer, Berlin, second edition, 1996.  Google Scholar

[20]

Z.-P. Jiang, A. R. Teel and L. Praly, Small-gain theorem for ISS systems and applications, Math. Control Signals Systems, 7 (1994), 95-120. doi: 10.1007/BF01211469.  Google Scholar

[21]

I. Karafyllis, Non-uniform robust global asymptotic stability for discrete-time systems and applications to numerical analysis, IMA J. Math. Control Inform., 23 (2006), 11-41. doi: 10.1093/imamci/dni037.  Google Scholar

[22]

I. Karafyllis, A system-theoretic framework for a wide class of systems. I, Applications to numerical analysis, J. Math. Anal. Appl., 328 (2007), 876-899. doi: 10.1016/j.jmaa.2006.05.059.  Google Scholar

[23]

I. Karafyllis and Z.-P. Jiang, A small-gain theorem for a wide class of feedback systems with control applications, SIAM J. Control Optim., 46 (2007), 1483-1517. doi: 10.1137/060669310.  Google Scholar

[24]

I. Karafyllis and Z.-P. Jiang, A vector small-gain theorem for general nonlinear control systems, In "Proceedings of the 48th IEEE Conference on Decision and Control,'' pages 7996-8001, Shanghai, China, 2009. Google Scholar

[25]

H. K. Khalil, "Nonlinear Systems,'' Prentice Hall, Upper Saddle River, third edition, 2002.  Google Scholar

[26]

P. E. Kloeden and J. Lorenz, Stable attracting sets in dynamical systems and in their one-step discretizations, SIAM J. Numer. Anal., 23 (1986), 986-995. doi: 10.1137/0723066.  Google Scholar

[27]

P. E. Kloeden and B. Schmalfuss, Lyapunov functions and attractors under variable time-step discretization, Discrete Contin. Dynam. Systems, 2 (1996), 163-172. doi: 10.3934/dcds.1996.2.163.  Google Scholar

[28]

V. Lakshmikantham and D. Trigiante, "Theory of Difference Equations: Numerical Methods and Applications,'' Marcel Dekker, New York, second edition, 2002.  Google Scholar

[29]

H. Lamba, Dynamical systems and adaptive timestepping in ODE solvers, BIT, 40 (2000), 314-335. doi: 10.1023/A:1022395124683.  Google Scholar

[30]

Y. Lin, E. D. Sontag and Y. Wang, A smooth converse Lyapunov theorem for robust stability, SIAM J. Control Optim., 34 (1996), 124-160. doi: 10.1137/S0363012993259981.  Google Scholar

[31]

J. Peng, Z.-B. Xu, H. Qiao and B. Zhang, A critical analysis on global convergence of Hopfield-type neural networks, IEEE Trans. Circuits Syst. I Regul. Pap., 52 (2005), 804-814. doi: 10.1109/TCSI.2005.844366.  Google Scholar

[32]

E. D. Sontag, Smooth stabilization implies coprime factorization, IEEE Trans. Automat. Control, 34 (1989), 435-443. doi: 10.1109/9.28018.  Google Scholar

[33]

E. D. Sontag, A "universal'' construction of Artstein's theorem on nonlinear stabilization, Systems Control Lett., 13 (1989), 117-123.  Google Scholar

[34]

E. D. Sontag, "Mathematical Control Theory,'' Springer, New York, second edition, 1998.  Google Scholar

[35]

A. M. Stuart and A. R. Humphries, "Dynamical Systems And Numerical Analysis,'' Cambridge University Press, Cambridge, 1996.  Google Scholar

[36]

A. R. Teel, Input-to-state stability and the nonlinear small gain theorem, Preprint, 2005. Google Scholar

[37]

Y. Xia and J. Wang, A recurrent neural network for nonlinear convex optimization subject to nonlinear inequality constraints, IEEE Trans. Circuits Syst., 51 (2004), 1385-1394. doi: 10.1109/TCSI.2004.830694.  Google Scholar

[38]

H. Yamashita, A differential equation approach to nonlinear programming, Math. Programming, 18 (1980), 155-168. doi: 10.1007/BF01588311.  Google Scholar

[39]

L. Zhou, Y. Wu, L. Zhang and G. Zhang, Convergence analysis of a differential equation approach for solving nonlinear programming problems, Appl. Math. Comput., 184 (2007), 789-797. doi: 10.1016/j.amc.2006.05.190.  Google Scholar

show all references

References:
[1]

Z. Artstein, Stabilization with relaxed controls, Nonlinear Anal., 7 (1983), 1163-1173. doi: 10.1016/0362-546X(83)90049-4.  Google Scholar

[2]

P. Cannarsa and C. Sinestrari, "Semiconcave Functions, Hamilton-Jacobi Equations, and Optimal Control,'' Progress in Nonlinear Differential Equations and their Applications, 58, Birkhäuser Boston Inc., Boston, MA, 2004.  Google Scholar

[3]

S. Dashkovskiy, B. S. Rüffer and F. R. Wirth, An ISS small gain theorem for general networks, Math. Control Signals Systems, 19 (2007), 93-122. doi: 10.1007/s00498-007-0014-8.  Google Scholar

[4]

R. A. Freeman and P. V. Kokotović, "Robust Nonlinear Control Design - State-Space and Lyapunov Techniques,'' Birkhäuser, Boston, MA, 1996.  Google Scholar

[5]

B. M. Garay and K. Lee, Attractors under discretization with variable stepsize, Discrete Contin. Dyn. Syst., 13 (2005), 827-841. doi: 10.3934/dcds.2005.13.827.  Google Scholar

[6]

C. W. Gear and I. G. Kevrekidis, Projective methods for stiff differential equations: Problems with gaps in their eigenvalue spectrum, SIAM J. Sci. Comput., 24 (2003), 1091-1106. doi: 10.1137/S1064827501388157.  Google Scholar

[7]

C. W. Gear and I. G. Kevrekidis, Telescopic projective methods for parabolic differential equations, J. Comput. Phys., 187 (2003), 95-109. doi: 10.1016/S0021-9991(03)00082-2.  Google Scholar

[8]

P. Giesl, "Construction of Global Lyapunov Functions Using Radial Basis Functions,'' volume 1904 of "Lecture Notes in Mathematics," Springer, Berlin, 2007.  Google Scholar

[9]

B. S. Goh, Algorithms for unconstrained optimization problems via control theory, J. Optim. Theory Appl., 92 (1997), 581-604. doi: 10.1023/A:1022607507153.  Google Scholar

[10]

V. Grimm and G. R. W. Quispel, Geometric integration methods that preserve Lyapunov functions, BIT, 45 (2005), 709-723. doi: 10.1007/s10543-005-0034-z.  Google Scholar

[11]

L. Grüne, "Asymptotic Behavior of Dynamical and Control Systems under Perturbation and Discretization,'' volume 1783 of "Lecture Notes in Mathematics," Springer, Berlin, 2002.  Google Scholar

[12]

L. Grüne, Attraction rates, robustness, and discretization of attractors, SIAM J. Numer. Anal., 41 (2003), 2096-2113. doi: 10.1137/S003614290139411X.  Google Scholar

[13]

L. Grüne, E. D. Sontag and F. R. Wirth, Asymptotic stability equals exponential stability, and ISS equals finite energy gain-if you twist your eyes, Syst. Control Lett., 38 (1999), 127-134.  Google Scholar

[14]

K. Gustafsson, Control-theoretic techniques for stepsize selection in explicit Runge-Kutta methods, ACM Trans. Math. Software, 17 (1991), 533-554. doi: 10.1145/210232.210242.  Google Scholar

[15]

K. Gustafsson, Control-theoretic techniques for stepsize selection in implicit Runge-Kutta methods, ACM Trans. Math. Software, 20 (1994), 496-517. doi: 10.1145/198429.198437.  Google Scholar

[16]

K. Gustafsson, M. Lundh and G. Söderlind, A {PI stepsize control for the numerical solution of ordinary differential equations}, BIT, 28 (1988), 270-287.  Google Scholar

[17]

E. Hairer, C. Lubich and G. Wanner, "Geometric Numerical Integration. Structure-Preserving Algorithms for Ordinary Differential Equations,'' Springer, Berlin, second edition, 2006.  Google Scholar

[18]

E. Hairer, S. P. Nørsett and G. Wanner, "Solving Ordinary Differential Equations. I Nonstiff Problems,'' Springer, Berlin, second edition, 1993.  Google Scholar

[19]

E. Hairer and G. Wanner, "Solving Ordinary Differential Equations. {II} Stiff and Differential-Algebraic Problems,'' Springer, Berlin, second edition, 1996.  Google Scholar

[20]

Z.-P. Jiang, A. R. Teel and L. Praly, Small-gain theorem for ISS systems and applications, Math. Control Signals Systems, 7 (1994), 95-120. doi: 10.1007/BF01211469.  Google Scholar

[21]

I. Karafyllis, Non-uniform robust global asymptotic stability for discrete-time systems and applications to numerical analysis, IMA J. Math. Control Inform., 23 (2006), 11-41. doi: 10.1093/imamci/dni037.  Google Scholar

[22]

I. Karafyllis, A system-theoretic framework for a wide class of systems. I, Applications to numerical analysis, J. Math. Anal. Appl., 328 (2007), 876-899. doi: 10.1016/j.jmaa.2006.05.059.  Google Scholar

[23]

I. Karafyllis and Z.-P. Jiang, A small-gain theorem for a wide class of feedback systems with control applications, SIAM J. Control Optim., 46 (2007), 1483-1517. doi: 10.1137/060669310.  Google Scholar

[24]

I. Karafyllis and Z.-P. Jiang, A vector small-gain theorem for general nonlinear control systems, In "Proceedings of the 48th IEEE Conference on Decision and Control,'' pages 7996-8001, Shanghai, China, 2009. Google Scholar

[25]

H. K. Khalil, "Nonlinear Systems,'' Prentice Hall, Upper Saddle River, third edition, 2002.  Google Scholar

[26]

P. E. Kloeden and J. Lorenz, Stable attracting sets in dynamical systems and in their one-step discretizations, SIAM J. Numer. Anal., 23 (1986), 986-995. doi: 10.1137/0723066.  Google Scholar

[27]

P. E. Kloeden and B. Schmalfuss, Lyapunov functions and attractors under variable time-step discretization, Discrete Contin. Dynam. Systems, 2 (1996), 163-172. doi: 10.3934/dcds.1996.2.163.  Google Scholar

[28]

V. Lakshmikantham and D. Trigiante, "Theory of Difference Equations: Numerical Methods and Applications,'' Marcel Dekker, New York, second edition, 2002.  Google Scholar

[29]

H. Lamba, Dynamical systems and adaptive timestepping in ODE solvers, BIT, 40 (2000), 314-335. doi: 10.1023/A:1022395124683.  Google Scholar

[30]

Y. Lin, E. D. Sontag and Y. Wang, A smooth converse Lyapunov theorem for robust stability, SIAM J. Control Optim., 34 (1996), 124-160. doi: 10.1137/S0363012993259981.  Google Scholar

[31]

J. Peng, Z.-B. Xu, H. Qiao and B. Zhang, A critical analysis on global convergence of Hopfield-type neural networks, IEEE Trans. Circuits Syst. I Regul. Pap., 52 (2005), 804-814. doi: 10.1109/TCSI.2005.844366.  Google Scholar

[32]

E. D. Sontag, Smooth stabilization implies coprime factorization, IEEE Trans. Automat. Control, 34 (1989), 435-443. doi: 10.1109/9.28018.  Google Scholar

[33]

E. D. Sontag, A "universal'' construction of Artstein's theorem on nonlinear stabilization, Systems Control Lett., 13 (1989), 117-123.  Google Scholar

[34]

E. D. Sontag, "Mathematical Control Theory,'' Springer, New York, second edition, 1998.  Google Scholar

[35]

A. M. Stuart and A. R. Humphries, "Dynamical Systems And Numerical Analysis,'' Cambridge University Press, Cambridge, 1996.  Google Scholar

[36]

A. R. Teel, Input-to-state stability and the nonlinear small gain theorem, Preprint, 2005. Google Scholar

[37]

Y. Xia and J. Wang, A recurrent neural network for nonlinear convex optimization subject to nonlinear inequality constraints, IEEE Trans. Circuits Syst., 51 (2004), 1385-1394. doi: 10.1109/TCSI.2004.830694.  Google Scholar

[38]

H. Yamashita, A differential equation approach to nonlinear programming, Math. Programming, 18 (1980), 155-168. doi: 10.1007/BF01588311.  Google Scholar

[39]

L. Zhou, Y. Wu, L. Zhang and G. Zhang, Convergence analysis of a differential equation approach for solving nonlinear programming problems, Appl. Math. Comput., 184 (2007), 789-797. doi: 10.1016/j.amc.2006.05.190.  Google Scholar

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