# American Institute of Mathematical Sciences

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

## Feedback stabilization methods for the numerical solution of ordinary differential equations

 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.
Citation: Iasson Karafyllis, Lars Grüne. Feedback stabilization methods for the numerical solution of ordinary differential equations. Discrete and 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. [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. [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. [4] R. A. Freeman and P. V. Kokotović, "Robust Nonlinear Control Design - State-Space and Lyapunov Techniques,'' Birkhäuser, Boston, MA, 1996. [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. [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. [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. [8] P. Giesl, "Construction of Global Lyapunov Functions Using Radial Basis Functions,'' volume 1904 of "Lecture Notes in Mathematics," Springer, Berlin, 2007. [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. [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. [11] L. Grüne, "Asymptotic Behavior of Dynamical and Control Systems under Perturbation and Discretization,'' volume 1783 of "Lecture Notes in Mathematics," Springer, Berlin, 2002. [12] L. Grüne, Attraction rates, robustness, and discretization of attractors, SIAM J. Numer. Anal., 41 (2003), 2096-2113. doi: 10.1137/S003614290139411X. [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. [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. [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. [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. [17] E. Hairer, C. Lubich and G. Wanner, "Geometric Numerical Integration. Structure-Preserving Algorithms for Ordinary Differential Equations,'' Springer, Berlin, second edition, 2006. [18] E. Hairer, S. P. Nørsett and G. Wanner, "Solving Ordinary Differential Equations. I Nonstiff Problems,'' Springer, Berlin, second edition, 1993. [19] E. Hairer and G. Wanner, "Solving Ordinary Differential Equations. {II} Stiff and Differential-Algebraic Problems,'' Springer, Berlin, second edition, 1996. [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. [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. [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. [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. [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. [25] H. K. Khalil, "Nonlinear Systems,'' Prentice Hall, Upper Saddle River, third edition, 2002. [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. [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. [28] V. Lakshmikantham and D. Trigiante, "Theory of Difference Equations: Numerical Methods and Applications,'' Marcel Dekker, New York, second edition, 2002. [29] H. Lamba, Dynamical systems and adaptive timestepping in ODE solvers, BIT, 40 (2000), 314-335. doi: 10.1023/A:1022395124683. [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. [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. [32] E. D. Sontag, Smooth stabilization implies coprime factorization, IEEE Trans. Automat. Control, 34 (1989), 435-443. doi: 10.1109/9.28018. [33] E. D. Sontag, A "universal'' construction of Artstein's theorem on nonlinear stabilization, Systems Control Lett., 13 (1989), 117-123. [34] E. D. Sontag, "Mathematical Control Theory,'' Springer, New York, second edition, 1998. [35] A. M. Stuart and A. R. Humphries, "Dynamical Systems And Numerical Analysis,'' Cambridge University Press, Cambridge, 1996. [36] A. R. Teel, Input-to-state stability and the nonlinear small gain theorem, Preprint, 2005. [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. [38] H. Yamashita, A differential equation approach to nonlinear programming, Math. Programming, 18 (1980), 155-168. doi: 10.1007/BF01588311. [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.

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##### References:
 [1] Z. Artstein, Stabilization with relaxed controls, Nonlinear Anal., 7 (1983), 1163-1173. doi: 10.1016/0362-546X(83)90049-4. [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. [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. [4] R. A. Freeman and P. V. Kokotović, "Robust Nonlinear Control Design - State-Space and Lyapunov Techniques,'' Birkhäuser, Boston, MA, 1996. [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. [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. [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. [8] P. Giesl, "Construction of Global Lyapunov Functions Using Radial Basis Functions,'' volume 1904 of "Lecture Notes in Mathematics," Springer, Berlin, 2007. [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. [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. [11] L. Grüne, "Asymptotic Behavior of Dynamical and Control Systems under Perturbation and Discretization,'' volume 1783 of "Lecture Notes in Mathematics," Springer, Berlin, 2002. [12] L. Grüne, Attraction rates, robustness, and discretization of attractors, SIAM J. Numer. Anal., 41 (2003), 2096-2113. doi: 10.1137/S003614290139411X. [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. [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. [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. [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. [17] E. Hairer, C. Lubich and G. Wanner, "Geometric Numerical Integration. Structure-Preserving Algorithms for Ordinary Differential Equations,'' Springer, Berlin, second edition, 2006. [18] E. Hairer, S. P. Nørsett and G. Wanner, "Solving Ordinary Differential Equations. I Nonstiff Problems,'' Springer, Berlin, second edition, 1993. [19] E. Hairer and G. Wanner, "Solving Ordinary Differential Equations. {II} Stiff and Differential-Algebraic Problems,'' Springer, Berlin, second edition, 1996. [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. [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. [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. [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. [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. [25] H. K. Khalil, "Nonlinear Systems,'' Prentice Hall, Upper Saddle River, third edition, 2002. [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. [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. [28] V. Lakshmikantham and D. Trigiante, "Theory of Difference Equations: Numerical Methods and Applications,'' Marcel Dekker, New York, second edition, 2002. [29] H. Lamba, Dynamical systems and adaptive timestepping in ODE solvers, BIT, 40 (2000), 314-335. doi: 10.1023/A:1022395124683. [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. [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. [32] E. D. Sontag, Smooth stabilization implies coprime factorization, IEEE Trans. Automat. Control, 34 (1989), 435-443. doi: 10.1109/9.28018. [33] E. D. Sontag, A "universal'' construction of Artstein's theorem on nonlinear stabilization, Systems Control Lett., 13 (1989), 117-123. [34] E. D. Sontag, "Mathematical Control Theory,'' Springer, New York, second edition, 1998. [35] A. M. Stuart and A. R. Humphries, "Dynamical Systems And Numerical Analysis,'' Cambridge University Press, Cambridge, 1996. [36] A. R. Teel, Input-to-state stability and the nonlinear small gain theorem, Preprint, 2005. [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. [38] H. Yamashita, A differential equation approach to nonlinear programming, Math. Programming, 18 (1980), 155-168. doi: 10.1007/BF01588311. [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.
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