# American Institute of Mathematical Sciences

August  2019, 24(8): 3843-3863. doi: 10.3934/dcdsb.2018333

## 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

* Corresponding author

Received  March 2018 Revised  July 2018 Published  January 2019

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 [12], to approximate the solution of the matrix-valued PDE. In this paper, we justify error estimates showing that the approximate solution itself is a contraction metric. The method is applied to several examples.

Citation: Peter Giesl, Holger Wendland. Construction of a contraction metric by meshless collocation. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 3843-3863. doi: 10.3934/dcdsb.2018333
##### 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [4] C. Chicone, Ordinary Differential Equations with Applications, Springer, Texts in Applied Mathematics 34, 1999.  Google Scholar [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.  Google Scholar [6] P. Giesl, Construction of Global Lyapunov Functions Using Radial Basis Functions, Lecture Notes in Math. 1904, Springer, 2007.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [12] P. Giesl and H. Wendland, Kernel-based discretisation for solving matrix-valued PDEs, SIAM J. Numer. Anal., 56 (2018), 3386-3406.   Google Scholar [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/.  Google Scholar [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.  Google Scholar [15] W. Hahn, Stability of Motion, Springer, Berlin, 1967.  Google Scholar [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.  Google Scholar [17] N. Krasovski$\breve{{\rm{i}}}$, Problems of the Theory of Stability of Motion, Mir, Moskow, 1959. English translation by Stanford University Press, 1963.   Google Scholar [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.  Google Scholar [19] D. Lewis, Metric properties of differential equations, Amer. J. Math., 71 (1949), 294-312.  doi: 10.2307/2372245.  Google Scholar [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.  Google Scholar [21] A. M. Lyapunov, The general problem of the stability of motion, Internat. J. Control, 55 (1992), 521-790.  doi: 10.1080/00207179208934253.  Google Scholar [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.  Google Scholar [23] P. Parrilo, Structured Semidefinite Programs and Semialgebraic Geometry Methods in Robustness and Optimiziation, PhD thesis, California Institute of Technology Pasadena, 2000. Google Scholar [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.  Google Scholar [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.  Google Scholar [26] R. Schaback and H. Wendland, Kernel techniques: From machine learning to meshless methods, Acta Numer., 15 (2006), 543-639.  doi: 10.1017/S0962492906270016.  Google Scholar [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.  Google Scholar [28] H. Wendland, Scattered Data Approximation, volume 17 of Cambridge Monographs on Applied and Computational Mathematics, Cambridge University Press, Cambridge, 2005.   Google Scholar [29] V. I. Zubov, The Methods of A. M. Lyapunov and Their Applications, Izdat. Leningrad. Univ., Moscow, 1957.  Google Scholar

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##### 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [4] C. Chicone, Ordinary Differential Equations with Applications, Springer, Texts in Applied Mathematics 34, 1999.  Google Scholar [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.  Google Scholar [6] P. Giesl, Construction of Global Lyapunov Functions Using Radial Basis Functions, Lecture Notes in Math. 1904, Springer, 2007.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [12] P. Giesl and H. Wendland, Kernel-based discretisation for solving matrix-valued PDEs, SIAM J. Numer. Anal., 56 (2018), 3386-3406.   Google Scholar [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/.  Google Scholar [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.  Google Scholar [15] W. Hahn, Stability of Motion, Springer, Berlin, 1967.  Google Scholar [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.  Google Scholar [17] N. Krasovski$\breve{{\rm{i}}}$, Problems of the Theory of Stability of Motion, Mir, Moskow, 1959. English translation by Stanford University Press, 1963.   Google Scholar [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.  Google Scholar [19] D. Lewis, Metric properties of differential equations, Amer. J. Math., 71 (1949), 294-312.  doi: 10.2307/2372245.  Google Scholar [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.  Google Scholar [21] A. M. Lyapunov, The general problem of the stability of motion, Internat. J. Control, 55 (1992), 521-790.  doi: 10.1080/00207179208934253.  Google Scholar [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.  Google Scholar [23] P. Parrilo, Structured Semidefinite Programs and Semialgebraic Geometry Methods in Robustness and Optimiziation, PhD thesis, California Institute of Technology Pasadena, 2000. Google Scholar [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.  Google Scholar [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.  Google Scholar [26] R. Schaback and H. Wendland, Kernel techniques: From machine learning to meshless methods, Acta Numer., 15 (2006), 543-639.  doi: 10.1017/S0962492906270016.  Google Scholar [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.  Google Scholar [28] H. Wendland, Scattered Data Approximation, volume 17 of Cambridge Monographs on Applied and Computational Mathematics, Cambridge University Press, Cambridge, 2005.   Google Scholar [29] V. I. Zubov, The Methods of A. M. Lyapunov and Their Applications, Izdat. Leningrad. Univ., Moscow, 1957.  Google Scholar
System (45) with $\epsilon = 0$. The collocation points used for the approximation together with the boundaries of the areas where $\mathop{\mathrm{sign}}\limits ( \mathop{\mathrm{tr}}\limits F(S)(x,y))- \mathop{\mathrm{sign}}\limits (\det F(S)(x,y)) = -2$ (red) and $\mathop{\mathrm{sign}}\limits ( \mathop{\mathrm{tr}}\limits S(x,y))+ \mathop{\mathrm{sign}}\limits (\det S(x,y)) = 2$ (blue). Blue and red lines are lines where one of the requirements of a contraction metric is violated. The constructed metric is thus a valid contraction metric where the collocation points are placed, but not beyond the first red or blue line
$\mathop{\mathrm{sign}}\limits ( \mathop{\mathrm{tr}}\limits F_\epsilon(S)(x,y))- \mathop{\mathrm{sign}}\limits (\det F_\epsilon(S)(x,y))$. If this function is $-2$, then $F_\epsilon(S)(x,y)$ is negative definite, which is one of the requirements for $S$ to be a contraction metric for the system with $\epsilon = 0.1$
The collocation points used for the approximation with $f_0$ together with the boundaries of the areas where $\mathop{\mathrm{sign}}\limits ( \mathop{\mathrm{tr}}\limits F_\epsilon(S)(x,y))- \mathop{\mathrm{sign}}\limits (\det F_\epsilon(S)(x,y)) = -2$ (red) and $\mathop{\mathrm{sign}}\limits ( \mathop{\mathrm{tr}}\limits S(x,y))+ \mathop{\mathrm{sign}}\limits (\det S(x,y)) = 2$ (blue). Blue and red lines are lines where one of the requirements of a contraction metric is violated. Hence, there are collocation points, where the constructed metric is not a contraction metric, since it was computed using a different system, namely with $\epsilon = 0$
The collocation points used for the approximation together with the boundary of the area where $F(S)$ is not negative definite (green). Note that $S$ is positive definite in the whole area displayed. Hence, the constructed metric is a contraction metric inside the cube bounded by the green areas
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