June  2020, 7(1): 57-81. doi: 10.3934/jcd.2020003

Approximation of Lyapunov functions from noisy data

1. 

Department of Mathematics, University of Sussex, Falmer, BN1 9QH, UK

2. 

Department of Mathematics, Imperial College London, London SW7 2AZ, UK

Received  May 2017 Revised  October 2019 Published  December 2019

Fund Project: B. Hamzi was supported by Marie Curie Fellowships. M. Rasmussen was supported by an EPSRC Career Acceleration Fellowship EP/I004165/1 and K.N. Webster was supported by the EPSRC Grant EP/L00187X/1 and a Marie Skł odowska-Curie Individual Fellowship Grant Number 660616

Methods have previously been developed for the approximation of Lyapunov functions using radial basis functions. However these methods assume that the evolution equations are known. We consider the problem of approximating a given Lyapunov function using radial basis functions where the evolution equations are not known, but we instead have sampled data which is contaminated with noise. We propose an algorithm in which we first approximate the underlying vector field, and use this approximation to then approximate the Lyapunov function. Our approach combines elements of machine learning/statistical learning theory with the existing theory of Lyapunov function approximation. Error estimates are provided for our algorithm.

Citation: Peter Giesl, Boumediene Hamzi, Martin Rasmussen, Kevin Webster. Approximation of Lyapunov functions from noisy data. Journal of Computational Dynamics, 2020, 7 (1) : 57-81. doi: 10.3934/jcd.2020003
References:
[1] R. A. Adams, Sobolev Spaces, Adademic Press, New York, 1975. 
[2]

N. Bhatia, On asymptotic stability in dynamical systems, Math. Systems Theory, 1 (1967), 113-127.  doi: 10.1007/BF01705521.

[3]

N. Bhatia and G. Szegö, Stability Theory of Dynamical Systems, Grundlehren der mathematischen Wissenschaften, 161, Springer, Berlin, 1970.

[4]

J. Bouvrie and B. Hamzi, Balanced reduction of nonlinear control systems in reproducing kernel hilbert space, in Proc. 48th Annual Allerton Conference on Communication, Control, and Computing, (2010), 294–301, http://arXiv.org/abs/1011.2952.

[5]

J. Bouvrie and B. Hamzi, Empirical estimators for the controllability energy and invariant measure of stochastically forced nonlinear systems, in Proc. of the 2012 American Control Conference, (2012), (long version at http://arXiv.org/abs/1204.0563).

[6]

J. Bouvrie and B. Hamzi, Kernel methods for the approximation of some key quantities of nonlinear systems, in Journal of Computational Dynamics, 4 (2017), http://arXiv.org/abs/1204.0563.

[7]

J. Bouvrie and B. Hamzi, Kernel methods for the approximation of nonlinear systems, in SIAM J. Control & Optimization, 55 (2017), http://arXiv.org/abs/1108.2903.

[8]

F. CamilliL. Grüne and F. Wirth, A generalization of Zubov's method to perturbed systems, SIAM J. Control Optim., 40 (2001), 496-515.  doi: 10.1137/S036301299936316X.

[9]

F. Cucker and S. Smale, On the mathematical foundations of learning, Bull. Amer. Math. Soc., 39 (2001), 1-49.  doi: 10.1090/S0273-0979-01-00923-5.

[10]

F. Cucker and S. Smale, Best choices for regularisation parameters in learning theory, Found. Comput. Math., 2 (2002), 413-428.  doi: 10.1007/s102080010030.

[11]

L. Evans, Partial Differential Equations, vol. 19 of Graduate Studies in Mathematics, AMS, Providence, Rhode Island, 1998.

[12]

T. EvgeniouM. Pontil and T. Poggio, Regularization networks and support vector machines, Adv. Comput. Math., 13 (2000), 1-50.  doi: 10.1023/A:1018946025316.

[13]

P. Giesl, Construction of Global Lyapunov Functions Using Radial Basis Functions, Lecture Notes in Mathematics. Springer Berlin Heidelberg, 2007.

[14]

P. Giesl and S. Hafstein, Computation and verification of Lyapunov functions, SIAM J. Appl. Dyn. Syst., 14 (2015), 1663-1698.  doi: 10.1137/140988802.

[15]

P. Giesl and S. Hafstein, Review on computational methods for Lyapunov functions, Discrete and Continuous Dynamical Systems Series B, 20 (2015), 2291-2331.  doi: 10.3934/dcdsb.2015.20.2291.

[16]

P. Giesl and H. Wendland, Meshless collocation: Error estimates with application to dynamical systems, SIAM J. Num. Anal., 45 (2007), 1723-1741.  doi: 10.1137/060658813.

[17]

L. Grüne, Asymptotic Behavior of Dynamical and Control Systems Under Perturbation and Discretization, Lecture Notes in Mathematics. Springer-Verlag, Berlin, 2002. doi: 10.1007/b83677.

[18]

S. Hafstein, An Algorithm for Constructing Lyapunov Functions, Electronic Journal of Differential Equations. Monograph, 8. Texas State UniversityCSan Marcos, Department of Mathematics, San Marcos, TX, 2007.

[19]

W. Hahn, Theorie und Anwendung der direkten Methode von Ljapunov, Ergebnisse der Mathematik und ihrer Grenzgebiete 22, Springer, Berlin, 1959.

[20]

W. Hahn, Stability of Motion, Springer, New York, 1967.

[21]

A. M. Lyapunov, Problème général de la stabilité du mouvement, Ann. Fac. Sci. Toulouse, 9 (1907), 203–474. Translation of the Russian version, published 1893 in Comm. Soc. math. Kharkow. Newly printed: Ann. of math. Stud. 17, Princeton, 1949.

[22]

Y. LinE. 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.

[23]

C. 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.

[24]

J. L. Massera, On Liapounoff's conditions of stability, Ann. of Math., 50 (1949), 705-721.  doi: 10.2307/1969558.

[25]

F. J. NarcowichJ. D. Ward and H. Wendland, Sobolev bounds on functions with scattered zeros, with applications to radial basis function surface fitting, Mathematics of Computation, 74 (2005), 743-763.  doi: 10.1090/S0025-5718-04-01708-9.

[26]

R. Opfer, Multiscale kernels, Adv. Comput. Math., 25 (2006), 357-380.  doi: 10.1007/s10444-004-7622-3.

[27]

R. Opfer, Tight frame expansions of multiscale reproducing kernels in Sobolev spaces, Appl. Comput. Harmon. Anal., 20 (2006), 357-374.  doi: 10.1016/j.acha.2005.05.003.

[28]

A. Papachristodoulou and S. Prajna, On the Construction of Lyapunov Functions Using the Sum of Squares Decomposition, Proceedings of the 41st IEEE Conference on Decision and Control, 2002. doi: 10.1109/CDC.2002.1184414.

[29]

G. Pagès, A space quantization method for numerical integration, J. Comp. Appl. Math., 89 (1998), 1-38.  doi: 10.1016/S0377-0427(97)00190-8.

[30]

R. Rifkin and R. A. Lippert, ., Notes on Regularized Least-Squares, CBCL Paper 268/AI Technical Report 2007-019, Massachusetts Institute of Technology, Cambridge, MA, May, 2007.

[31]

S. Smale and D.-X. Zhou, Shannon Sampling and Function Reconstruction from Point Values, Bull. Amer. Math. Soc., 41 (2004), 279-305.  doi: 10.1090/S0273-0979-04-01025-0.

[32]

S. Smale and D.-X. Zhou, Shannon sampling II: Connections to learning theory, Appl. Comput. Harmon. Anal., 19 (2005), 285-302.  doi: 10.1016/j.acha.2005.03.001.

[33]

S. Smale and D.-X. Zhou, Learning theory estimates via their integral operators and their approximations, Constr. Approx., 26 (2007), 153-172.  doi: 10.1007/s00365-006-0659-y.

[34]

S. Smale and D.-X. Zhou, Online learning with Markov sampling, Anal. Appl., 7 (2009), 87-113.  doi: 10.1142/S0219530509001293.

[35]

G. Voronoi, Recherches sur les parallelodres primitives, J. Reine Angew. Math., 134 (1908), 198-287.  doi: 10.1515/crll.1908.134.198.

[36]

F. Wesley Wilson and Jr ., Smoothing derivatives of functions and applications, Trans. Amer. Math. Soc., 139 (1969), 413-428.  doi: 10.1090/S0002-9947-1969-0251747-9.

[37]

H. Wendland, Piecewise polynomial, positive definite and compactly supported radial functions of minimal degree, Adv. Comput. Math., 4 (1995), 389-396.  doi: 10.1007/BF02123482.

[38] H. Wendland, Scattered Data Approximation, Cambridge Monogr. Appl. Comput. Math., Cambridge University Press, Cambridge, UK, 2005. 
[39]

H. Wendland and C. Rieger, Approximate interpolation with applications to selecting smoothing parameters, Numer. Math., 101 (2005), 729-748.  doi: 10.1007/s00211-005-0637-y.

show all references

References:
[1] R. A. Adams, Sobolev Spaces, Adademic Press, New York, 1975. 
[2]

N. Bhatia, On asymptotic stability in dynamical systems, Math. Systems Theory, 1 (1967), 113-127.  doi: 10.1007/BF01705521.

[3]

N. Bhatia and G. Szegö, Stability Theory of Dynamical Systems, Grundlehren der mathematischen Wissenschaften, 161, Springer, Berlin, 1970.

[4]

J. Bouvrie and B. Hamzi, Balanced reduction of nonlinear control systems in reproducing kernel hilbert space, in Proc. 48th Annual Allerton Conference on Communication, Control, and Computing, (2010), 294–301, http://arXiv.org/abs/1011.2952.

[5]

J. Bouvrie and B. Hamzi, Empirical estimators for the controllability energy and invariant measure of stochastically forced nonlinear systems, in Proc. of the 2012 American Control Conference, (2012), (long version at http://arXiv.org/abs/1204.0563).

[6]

J. Bouvrie and B. Hamzi, Kernel methods for the approximation of some key quantities of nonlinear systems, in Journal of Computational Dynamics, 4 (2017), http://arXiv.org/abs/1204.0563.

[7]

J. Bouvrie and B. Hamzi, Kernel methods for the approximation of nonlinear systems, in SIAM J. Control & Optimization, 55 (2017), http://arXiv.org/abs/1108.2903.

[8]

F. CamilliL. Grüne and F. Wirth, A generalization of Zubov's method to perturbed systems, SIAM J. Control Optim., 40 (2001), 496-515.  doi: 10.1137/S036301299936316X.

[9]

F. Cucker and S. Smale, On the mathematical foundations of learning, Bull. Amer. Math. Soc., 39 (2001), 1-49.  doi: 10.1090/S0273-0979-01-00923-5.

[10]

F. Cucker and S. Smale, Best choices for regularisation parameters in learning theory, Found. Comput. Math., 2 (2002), 413-428.  doi: 10.1007/s102080010030.

[11]

L. Evans, Partial Differential Equations, vol. 19 of Graduate Studies in Mathematics, AMS, Providence, Rhode Island, 1998.

[12]

T. EvgeniouM. Pontil and T. Poggio, Regularization networks and support vector machines, Adv. Comput. Math., 13 (2000), 1-50.  doi: 10.1023/A:1018946025316.

[13]

P. Giesl, Construction of Global Lyapunov Functions Using Radial Basis Functions, Lecture Notes in Mathematics. Springer Berlin Heidelberg, 2007.

[14]

P. Giesl and S. Hafstein, Computation and verification of Lyapunov functions, SIAM J. Appl. Dyn. Syst., 14 (2015), 1663-1698.  doi: 10.1137/140988802.

[15]

P. Giesl and S. Hafstein, Review on computational methods for Lyapunov functions, Discrete and Continuous Dynamical Systems Series B, 20 (2015), 2291-2331.  doi: 10.3934/dcdsb.2015.20.2291.

[16]

P. Giesl and H. Wendland, Meshless collocation: Error estimates with application to dynamical systems, SIAM J. Num. Anal., 45 (2007), 1723-1741.  doi: 10.1137/060658813.

[17]

L. Grüne, Asymptotic Behavior of Dynamical and Control Systems Under Perturbation and Discretization, Lecture Notes in Mathematics. Springer-Verlag, Berlin, 2002. doi: 10.1007/b83677.

[18]

S. Hafstein, An Algorithm for Constructing Lyapunov Functions, Electronic Journal of Differential Equations. Monograph, 8. Texas State UniversityCSan Marcos, Department of Mathematics, San Marcos, TX, 2007.

[19]

W. Hahn, Theorie und Anwendung der direkten Methode von Ljapunov, Ergebnisse der Mathematik und ihrer Grenzgebiete 22, Springer, Berlin, 1959.

[20]

W. Hahn, Stability of Motion, Springer, New York, 1967.

[21]

A. M. Lyapunov, Problème général de la stabilité du mouvement, Ann. Fac. Sci. Toulouse, 9 (1907), 203–474. Translation of the Russian version, published 1893 in Comm. Soc. math. Kharkow. Newly printed: Ann. of math. Stud. 17, Princeton, 1949.

[22]

Y. LinE. 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.

[23]

C. 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.

[24]

J. L. Massera, On Liapounoff's conditions of stability, Ann. of Math., 50 (1949), 705-721.  doi: 10.2307/1969558.

[25]

F. J. NarcowichJ. D. Ward and H. Wendland, Sobolev bounds on functions with scattered zeros, with applications to radial basis function surface fitting, Mathematics of Computation, 74 (2005), 743-763.  doi: 10.1090/S0025-5718-04-01708-9.

[26]

R. Opfer, Multiscale kernels, Adv. Comput. Math., 25 (2006), 357-380.  doi: 10.1007/s10444-004-7622-3.

[27]

R. Opfer, Tight frame expansions of multiscale reproducing kernels in Sobolev spaces, Appl. Comput. Harmon. Anal., 20 (2006), 357-374.  doi: 10.1016/j.acha.2005.05.003.

[28]

A. Papachristodoulou and S. Prajna, On the Construction of Lyapunov Functions Using the Sum of Squares Decomposition, Proceedings of the 41st IEEE Conference on Decision and Control, 2002. doi: 10.1109/CDC.2002.1184414.

[29]

G. Pagès, A space quantization method for numerical integration, J. Comp. Appl. Math., 89 (1998), 1-38.  doi: 10.1016/S0377-0427(97)00190-8.

[30]

R. Rifkin and R. A. Lippert, ., Notes on Regularized Least-Squares, CBCL Paper 268/AI Technical Report 2007-019, Massachusetts Institute of Technology, Cambridge, MA, May, 2007.

[31]

S. Smale and D.-X. Zhou, Shannon Sampling and Function Reconstruction from Point Values, Bull. Amer. Math. Soc., 41 (2004), 279-305.  doi: 10.1090/S0273-0979-04-01025-0.

[32]

S. Smale and D.-X. Zhou, Shannon sampling II: Connections to learning theory, Appl. Comput. Harmon. Anal., 19 (2005), 285-302.  doi: 10.1016/j.acha.2005.03.001.

[33]

S. Smale and D.-X. Zhou, Learning theory estimates via their integral operators and their approximations, Constr. Approx., 26 (2007), 153-172.  doi: 10.1007/s00365-006-0659-y.

[34]

S. Smale and D.-X. Zhou, Online learning with Markov sampling, Anal. Appl., 7 (2009), 87-113.  doi: 10.1142/S0219530509001293.

[35]

G. Voronoi, Recherches sur les parallelodres primitives, J. Reine Angew. Math., 134 (1908), 198-287.  doi: 10.1515/crll.1908.134.198.

[36]

F. Wesley Wilson and Jr ., Smoothing derivatives of functions and applications, Trans. Amer. Math. Soc., 139 (1969), 413-428.  doi: 10.1090/S0002-9947-1969-0251747-9.

[37]

H. Wendland, Piecewise polynomial, positive definite and compactly supported radial functions of minimal degree, Adv. Comput. Math., 4 (1995), 389-396.  doi: 10.1007/BF02123482.

[38] H. Wendland, Scattered Data Approximation, Cambridge Monogr. Appl. Comput. Math., Cambridge University Press, Cambridge, UK, 2005. 
[39]

H. Wendland and C. Rieger, Approximate interpolation with applications to selecting smoothing parameters, Numer. Math., 101 (2005), 729-748.  doi: 10.1007/s00211-005-0637-y.

Figure 1.  Domains and sets used in the statement and proof of Theorem 2.1. The dotted lines show the boundary of the set $ \mathcal{D} = \Omega\setminus B_{\varepsilon}(\overline{x}) $ where the Lyapunov functions are approximated. We also have $ \Omega_V,\Omega_T\subset A(\overline{x}) $
Figure 2.  Errors between $ f_1 $ and $ {f}^1_{{\bf z},\lambda} $
Figure 3.  Errors between $ f_2 $ and $ {f}^2_{{\bf z},\lambda} $
Figure 4.  Lyapunov function using Algorithm 2 with 360 points
Figure 5.  Lyapunov function using Algorithm 2 with 1520 points
Figure 6.  Orbital derivative of the Lyapunov function with respect to the original system using Algorithm 2 with 360 points
Figure 7.  Orbital derivative of the Lyapunov function with respect to the original system using Algorithm 2 with 1520 points
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