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The Magneto–Hydrodynamic equations: Local theory and blow-up of solutions
Construction of a contraction metric by meshless collocation
1. | Department of Mathematics, University of Sussex, Falmer, BN1 9QH, UK |
2. | Department of Mathematics, University of Bayreuth, 95440 Bayreuth, Germany |
A contraction metric for an autonomous ordinary differential equation is a Riemannian metric such that the distance between adjacent solutions contracts over time. A contraction metric can be used to determine the basin of attraction of an equilibrium and it is robust to small perturbations of the system, including those varying the position of the equilibrium.
The contraction metric is described by a matrix-valued function $ M(x) $ such that $ M(x) $ is positive definite and $ F(M)(x) $ is negative definite, where $ F $ denotes a certain first-order differential operator. In this paper, we show existence, uniqueness and continuous dependence on the right-hand side of the matrix-valued partial differential equation $ F(M)(x) = -C(x) $. We then use a construction method based on meshless collocation, developed in the companion paper [
References:
[1] |
J. Anderson and A. Papachristodoulou,
Advances in computational Lyapunov analysis using sum-of-squares programming, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 2361-2381.
doi: 10.3934/dcdsb.2015.20.2361. |
[2] |
E. Aylward, P. Parrilo and J.-J. Slotine,
Stability and robustness analysis of nonlinear systems via contraction metrics and SOS programming, Automatica, 44 (2008), 2163-2170.
doi: 10.1016/j.automatica.2007.12.012. |
[3] |
M. Buhmann, Radial Basis Functions: Theory and Implementations, volume 12 of Cambridge Monographs on Applied and Computational Mathematics, Cambridge University Press, Cambridge, 2003.
doi: 10.1017/CBO9780511543241.![]() ![]() ![]() |
[4] |
C. Chicone, Ordinary Differential Equations with Applications, Springer, Texts in Applied Mathematics 34, 1999. |
[5] |
F. Forni and R. Sepulchre,
A differential Lyapunov framework for Contraction Analysis, IEEE Trans. Automat. Control, 59 (2014), 614-628.
doi: 10.1109/TAC.2013.2285771. |
[6] |
P. Giesl, Construction of Global Lyapunov Functions Using Radial Basis Functions, Lecture Notes in Math. 1904, Springer, 2007. |
[7] |
P. Giesl,
On the determination of the basin of attraction of periodic orbits in three- and higher-dimensional systems, J. Math. Anal. Appl., 354 (2009), 606-618.
doi: 10.1016/j.jmaa.2009.01.027. |
[8] |
P. Giesl,
Converse theorems on contraction metrics for an equilibrium, J. Math. Anal. Appl., 424 (2015), 1380-1403.
doi: 10.1016/j.jmaa.2014.12.010. |
[9] |
P. Giesl and S. Hafstein,
Construction of a CPA contraction metric for periodic orbits using semidefinite optimization, Nonlinear Anal., 86 (2013), 114-134.
doi: 10.1016/j.na.2013.03.012. |
[10] |
P. Giesl and S. Hafstein,
Revised CPA method to compute Lyapunov functions for nonlinear systems, J. Math. Anal. Appl., 410 (2014), 292-306.
doi: 10.1016/j.jmaa.2013.08.014. |
[11] |
P. Giesl and S. Hafstein,
Review on computational methods for Lyapunov functions, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 2291-2331.
doi: 10.3934/dcdsb.2015.20.2291. |
[12] |
P. Giesl and H. Wendland,
Kernel-based discretisation for solving matrix-valued PDEs, SIAM J. Numer. Anal., 56 (2018), 3386-3406.
|
[13] |
S. Hafstein, An Algorithm for Constructing Lyapunov Functions, volume 8 of Electronic Journal of Differential Equations, Texas State University - San Marcos, Department of Mathematics, San Marcos, TX, 2007. available from: http://ejde.math.txstate.edu/. |
[14] |
W. Hahn, Theory and Application of Liapunov's Direct Method, English edition prepared by Siegfried H. Lehnigk; translation by Hans H. Losenthien and Siegfried H. Lehnigk. Prentice-Hall, Inc., Englewood Cliffs, N.J., 1963. |
[15] | |
[16] |
Ch. M. Kellett,
Classical converse theorems in Lyapunov's second method, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 2333-2360.
doi: 10.3934/dcdsb.2015.20.2333. |
[17] |
N. Krasovski$\breve{{\rm{i}}}$, Problems of the Theory of Stability of Motion, Mir, Moskow, 1959. English translation by Stanford University Press, 1963.
![]() |
[18] |
G. Leonov, I. Burkin and A. Shepelyavyi, Frequency Methods in Oscillation Theory, Ser. Math. and its Appl.: Vol. 357, Kluwer, 1996.
doi: 10.1007/978-94-009-0193-3. |
[19] |
D. Lewis,
Metric properties of differential equations, Amer. J. Math., 71 (1949), 294-312.
doi: 10.2307/2372245. |
[20] |
W. Lohmiller and J.-J. Slotine,
On contraction analysis for non-linear systems, Automatica, 34 (1998), 683-696.
doi: 10.1016/S0005-1098(98)00019-3. |
[21] |
A. M. Lyapunov,
The general problem of the stability of motion, Internat. J. Control, 55 (1992), 521-790.
doi: 10.1080/00207179208934253. |
[22] |
I. Mezić,
Spectral properties of dynamical systems, model reduction and decompositions, Nonlinear Dynam., 41 (2005), 309-325.
doi: 10.1007/s11071-005-2824-x. |
[23] |
P. Parrilo, Structured Semidefinite Programs and Semialgebraic Geometry Methods in Robustness and Optimiziation, PhD thesis, California Institute of Technology Pasadena, 2000. |
[24] |
M. J. D. Powell, The theory of radial basis function approximation in 1990, In Advances
in numerical analysis, Vol. Ⅱ (Lancaster, 1990), Oxford Sci. Publ., pages 105–210. Oxford
Univ. Press, New York, 1992. |
[25] |
A. Rantzer,
A dual to Lyapunov's stability theorem, Systems Control Lett., 42 (2001), 161-168.
doi: 10.1016/S0167-6911(00)00087-6. |
[26] |
R. Schaback and H. Wendland,
Kernel techniques: From machine learning to meshless methods, Acta Numer., 15 (2006), 543-639.
doi: 10.1017/S0962492906270016. |
[27] |
H. Wendland,
Error estimates for interpolation by compactly supported radial basis functions of minimal degree, J. Approx. Theory, 93 (1998), 258-272.
doi: 10.1006/jath.1997.3137. |
[28] |
H. Wendland, Scattered Data Approximation, volume 17 of Cambridge Monographs on Applied and Computational Mathematics, Cambridge University Press, Cambridge, 2005.
![]() ![]() |
[29] |
V. I. Zubov, The Methods of A. M. Lyapunov and Their Applications, Izdat. Leningrad. Univ., Moscow, 1957. |
show all references
References:
[1] |
J. Anderson and A. Papachristodoulou,
Advances in computational Lyapunov analysis using sum-of-squares programming, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 2361-2381.
doi: 10.3934/dcdsb.2015.20.2361. |
[2] |
E. Aylward, P. Parrilo and J.-J. Slotine,
Stability and robustness analysis of nonlinear systems via contraction metrics and SOS programming, Automatica, 44 (2008), 2163-2170.
doi: 10.1016/j.automatica.2007.12.012. |
[3] |
M. Buhmann, Radial Basis Functions: Theory and Implementations, volume 12 of Cambridge Monographs on Applied and Computational Mathematics, Cambridge University Press, Cambridge, 2003.
doi: 10.1017/CBO9780511543241.![]() ![]() ![]() |
[4] |
C. Chicone, Ordinary Differential Equations with Applications, Springer, Texts in Applied Mathematics 34, 1999. |
[5] |
F. Forni and R. Sepulchre,
A differential Lyapunov framework for Contraction Analysis, IEEE Trans. Automat. Control, 59 (2014), 614-628.
doi: 10.1109/TAC.2013.2285771. |
[6] |
P. Giesl, Construction of Global Lyapunov Functions Using Radial Basis Functions, Lecture Notes in Math. 1904, Springer, 2007. |
[7] |
P. Giesl,
On the determination of the basin of attraction of periodic orbits in three- and higher-dimensional systems, J. Math. Anal. Appl., 354 (2009), 606-618.
doi: 10.1016/j.jmaa.2009.01.027. |
[8] |
P. Giesl,
Converse theorems on contraction metrics for an equilibrium, J. Math. Anal. Appl., 424 (2015), 1380-1403.
doi: 10.1016/j.jmaa.2014.12.010. |
[9] |
P. Giesl and S. Hafstein,
Construction of a CPA contraction metric for periodic orbits using semidefinite optimization, Nonlinear Anal., 86 (2013), 114-134.
doi: 10.1016/j.na.2013.03.012. |
[10] |
P. Giesl and S. Hafstein,
Revised CPA method to compute Lyapunov functions for nonlinear systems, J. Math. Anal. Appl., 410 (2014), 292-306.
doi: 10.1016/j.jmaa.2013.08.014. |
[11] |
P. Giesl and S. Hafstein,
Review on computational methods for Lyapunov functions, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 2291-2331.
doi: 10.3934/dcdsb.2015.20.2291. |
[12] |
P. Giesl and H. Wendland,
Kernel-based discretisation for solving matrix-valued PDEs, SIAM J. Numer. Anal., 56 (2018), 3386-3406.
|
[13] |
S. Hafstein, An Algorithm for Constructing Lyapunov Functions, volume 8 of Electronic Journal of Differential Equations, Texas State University - San Marcos, Department of Mathematics, San Marcos, TX, 2007. available from: http://ejde.math.txstate.edu/. |
[14] |
W. Hahn, Theory and Application of Liapunov's Direct Method, English edition prepared by Siegfried H. Lehnigk; translation by Hans H. Losenthien and Siegfried H. Lehnigk. Prentice-Hall, Inc., Englewood Cliffs, N.J., 1963. |
[15] | |
[16] |
Ch. M. Kellett,
Classical converse theorems in Lyapunov's second method, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 2333-2360.
doi: 10.3934/dcdsb.2015.20.2333. |
[17] |
N. Krasovski$\breve{{\rm{i}}}$, Problems of the Theory of Stability of Motion, Mir, Moskow, 1959. English translation by Stanford University Press, 1963.
![]() |
[18] |
G. Leonov, I. Burkin and A. Shepelyavyi, Frequency Methods in Oscillation Theory, Ser. Math. and its Appl.: Vol. 357, Kluwer, 1996.
doi: 10.1007/978-94-009-0193-3. |
[19] |
D. Lewis,
Metric properties of differential equations, Amer. J. Math., 71 (1949), 294-312.
doi: 10.2307/2372245. |
[20] |
W. Lohmiller and J.-J. Slotine,
On contraction analysis for non-linear systems, Automatica, 34 (1998), 683-696.
doi: 10.1016/S0005-1098(98)00019-3. |
[21] |
A. M. Lyapunov,
The general problem of the stability of motion, Internat. J. Control, 55 (1992), 521-790.
doi: 10.1080/00207179208934253. |
[22] |
I. Mezić,
Spectral properties of dynamical systems, model reduction and decompositions, Nonlinear Dynam., 41 (2005), 309-325.
doi: 10.1007/s11071-005-2824-x. |
[23] |
P. Parrilo, Structured Semidefinite Programs and Semialgebraic Geometry Methods in Robustness and Optimiziation, PhD thesis, California Institute of Technology Pasadena, 2000. |
[24] |
M. J. D. Powell, The theory of radial basis function approximation in 1990, In Advances
in numerical analysis, Vol. Ⅱ (Lancaster, 1990), Oxford Sci. Publ., pages 105–210. Oxford
Univ. Press, New York, 1992. |
[25] |
A. Rantzer,
A dual to Lyapunov's stability theorem, Systems Control Lett., 42 (2001), 161-168.
doi: 10.1016/S0167-6911(00)00087-6. |
[26] |
R. Schaback and H. Wendland,
Kernel techniques: From machine learning to meshless methods, Acta Numer., 15 (2006), 543-639.
doi: 10.1017/S0962492906270016. |
[27] |
H. Wendland,
Error estimates for interpolation by compactly supported radial basis functions of minimal degree, J. Approx. Theory, 93 (1998), 258-272.
doi: 10.1006/jath.1997.3137. |
[28] |
H. Wendland, Scattered Data Approximation, volume 17 of Cambridge Monographs on Applied and Computational Mathematics, Cambridge University Press, Cambridge, 2005.
![]() ![]() |
[29] |
V. I. Zubov, The Methods of A. M. Lyapunov and Their Applications, Izdat. Leningrad. Univ., Moscow, 1957. |




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