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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 |
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. |
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. |
[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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