Existence of piecewise linear Lyapunov functions in arbitrary dimensions

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October 2012, 32(10): 3539-3565. doi: 10.3934/dcds.2012.32.3539

Existence of piecewise linear Lyapunov functions in arbitrary dimensions

Peter Giesl ^1, and Sigurdur Hafstein ^2,

1.	Department of Mathematics, University of Sussex, Falmer BN1 9QH, United Kingdom
2.	School of Science and Engineering, Reykjavik University, Menntavegi 1, IS-101 Reykjavik, Iceland

Received May 2011 Revised October 2011 Published May 2012

Abstract
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Lyapunov functions are an important tool to determine the basin of attraction of exponentially stable equilibria in dynamical systems. In Marinósson (2002), a method to construct Lyapunov functions was presented, using finite differences on finite elements and thus transforming the construction problem into a linear programming problem. In Hafstein (2004), it was shown that this method always succeeds in constructing a Lyapunov function, except for a small, given neighbourhood of the equilibrium.
For two-dimensional systems, this local problem was overcome by choosing a fan-like triangulation around the equilibrium. In Giesl/Hafstein (2010) the existence of a piecewise linear Lyapunov function was shown, and in Giesl/Hafstein (2012) it was shown that the above method with a fan-like triangulation always succeeds in constructing a Lyapunov function, without any local exception. However, the previous papers only considered two-dimensional systems. This paper generalises the existence of piecewise linear Lyapunov functions to arbitrary dimensions.

Keywords: existence, triangulation., piecewise linear, Lyapunov function, exponentially stable equilibrium.

Mathematics Subject Classification: Primary: 37B25; Secondary: 34D20, 37C7.

Citation: Peter Giesl, Sigurdur Hafstein. Existence of piecewise linear Lyapunov functions in arbitrary dimensions. Discrete and Continuous Dynamical Systems, 2012, 32 (10) : 3539-3565. doi: 10.3934/dcds.2012.32.3539

References:

[1]	R. Baier, L. Grüne and S. Hafstein, Linear programming based Lyapunov function computation for differential inclusions, Discrete Contin. Dyn. Syst. Ser. B, 17 (2012), 33-56. doi: 10.3934/dcdsb.2012.17.33.
[2]	R. Bartels and G. Stewart, Algorithm 432: Solution of the matrix equation $AX + XB = C$, Comm. ACM, 15 (1972), 820-826. doi: 10.1145/361573.361582.
[3]	F. Clarke, "Optimization and Nonsmooth Analysis,'' Second edition, Classics in Applied Mathematics, 5, SIAM, Philadephia, PA, 1990.
[4]	A. Garcia and O. Agamennoni, Attraction and stability of nonlinear ODE's using continuous piecewise linear approximations, submitted.
[5]	P. Giesl, "Construction of Global Lyapunov Functions Using Radial Basis Functions,'' Lecture Notes in Mathematics, 1904, Springer, Berlin, 2007.
[6]	P. Giesl and S. Hafstein, Existence of piecewise affine Lyapunov functions in two dimensions, J. Math. Anal. Appl., 371 (2010), 233-248. doi: 10.1016/j.jmaa.2010.05.009.
[7]	P. Giesl and S. Hafstein, Construction of Lyapunov functions for nonlinear planar systems by linear programming, J. Math. Anal. Appl., 388 (2012), 463-479. doi: 10.1016/j.jmaa.2011.10.047.
[8]	S. Hafstein, A constructive converse Lyapunov theorem on exponential stability, Discrete Contin. Dyn. Syst., 10 (2004), 657-678. doi: 10.3934/dcds.2004.10.657.
[9]	S. Hafstein, A constructive converse Lyapunov theorem on asymptotic stability for nonlinear autonomous ordinary differential equations, Dynamical Systems, 20 (2005), 281-299. doi: 10.1080/14689360500164873.
[10]	S. Hafstein, "An Algorithm for Constructing Lyapunov Functions,'' Electron. J. Differential Equ. Monogr., 8, Texas State Univ.-San Marcos, Dep. of Mathematics, San Marcos, TX, 2007. Available from: http://ejde.math.txstate.edu/.
[11]	T. Johansen, Computation of Lyapunov functions for smooth nonlinear systems using convex optimization, Automatica J. IFAC, 36 (2000), 1617-1626. doi: 10.1016/S0005-1098(00)00088-1.
[12]	M. Johansson and A. Rantzer, On the computation of piecewise quadratic Lyapunov functions, in "Proceedings of the 36^th IEEE Conference on Decision and Control,'' 1997.
[13]	P. Julian, "A High-Level Canonical Piecewise Linear Representation: Theory and Applications,'' Ph.D. thesis, Universidad Nacional del Sur, Bahia Blanca, Argentina, 1999.
[14]	P. Julián, J. Guivant and A. Desages, A parametrization of piecewise linear Lyapunov function via linear programming. Multiple model approaches to modelling and control, Int. Journal of Control, 72 (1999), 702-715.
[15]	H. K. Khalil, "Nonlinear Systems,'' 3^rd edition, Prentice Hall, New Jersey, 2002.
[16]	S. Marinósson, "Stability Analysis of Nonlinear Systems with Linear Programming: A Lyapunov Functions Based Approach,'' Ph.D. thesis, Gerhard-Mercator-University, Duisburg, Germany, 2002.
[17]	S. Marinósson, Lyapunov function construction for ordinary differential equations with linear programming, Dynamical Systems, 17 (2002), 137-150. doi: 10.1080/0268111011011847.
[18]	A. Papachristodoulou and S. Prajna, The construction of Lyapunov functions using the sum of squares decomposition, in "Proceedings of the 41^st IEEE Conference on Decision and Control,'' (2002), 3482-3487.
[19]	P. Parrilo, "Structured Semidefinite Programs and Semialgebraic Geometry Methods in Robustness and Optimization,'' Ph.D. thesis, Caltech, Pasadena, USA, 2000.
[20]	M. Peet, Exponentially stable nonlinear systems have polynomial Lyapunov functions on bounded regions, IEEE Trans. Automatic Control, 54 (2009), 979-987. doi: 10.1109/TAC.2009.2017116.
[21]	V. Zubov, "Methods of A. M. Lyapunov and Their Application,'' Translation prepared under the auspices of the United States Atomic Energy Commission, edited by Leo F. Boron, P. Noordhoff Ltd, Groningen, 1964.

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References:

[1]	R. Baier, L. Grüne and S. Hafstein, Linear programming based Lyapunov function computation for differential inclusions, Discrete Contin. Dyn. Syst. Ser. B, 17 (2012), 33-56. doi: 10.3934/dcdsb.2012.17.33.
[2]	R. Bartels and G. Stewart, Algorithm 432: Solution of the matrix equation $AX + XB = C$, Comm. ACM, 15 (1972), 820-826. doi: 10.1145/361573.361582.
[3]	F. Clarke, "Optimization and Nonsmooth Analysis,'' Second edition, Classics in Applied Mathematics, 5, SIAM, Philadephia, PA, 1990.
[4]	A. Garcia and O. Agamennoni, Attraction and stability of nonlinear ODE's using continuous piecewise linear approximations, submitted.
[5]	P. Giesl, "Construction of Global Lyapunov Functions Using Radial Basis Functions,'' Lecture Notes in Mathematics, 1904, Springer, Berlin, 2007.
[6]	P. Giesl and S. Hafstein, Existence of piecewise affine Lyapunov functions in two dimensions, J. Math. Anal. Appl., 371 (2010), 233-248. doi: 10.1016/j.jmaa.2010.05.009.
[7]	P. Giesl and S. Hafstein, Construction of Lyapunov functions for nonlinear planar systems by linear programming, J. Math. Anal. Appl., 388 (2012), 463-479. doi: 10.1016/j.jmaa.2011.10.047.
[8]	S. Hafstein, A constructive converse Lyapunov theorem on exponential stability, Discrete Contin. Dyn. Syst., 10 (2004), 657-678. doi: 10.3934/dcds.2004.10.657.
[9]	S. Hafstein, A constructive converse Lyapunov theorem on asymptotic stability for nonlinear autonomous ordinary differential equations, Dynamical Systems, 20 (2005), 281-299. doi: 10.1080/14689360500164873.
[10]	S. Hafstein, "An Algorithm for Constructing Lyapunov Functions,'' Electron. J. Differential Equ. Monogr., 8, Texas State Univ.-San Marcos, Dep. of Mathematics, San Marcos, TX, 2007. Available from: http://ejde.math.txstate.edu/.
[11]	T. Johansen, Computation of Lyapunov functions for smooth nonlinear systems using convex optimization, Automatica J. IFAC, 36 (2000), 1617-1626. doi: 10.1016/S0005-1098(00)00088-1.
[12]	M. Johansson and A. Rantzer, On the computation of piecewise quadratic Lyapunov functions, in "Proceedings of the 36^th IEEE Conference on Decision and Control,'' 1997.
[13]	P. Julian, "A High-Level Canonical Piecewise Linear Representation: Theory and Applications,'' Ph.D. thesis, Universidad Nacional del Sur, Bahia Blanca, Argentina, 1999.
[14]	P. Julián, J. Guivant and A. Desages, A parametrization of piecewise linear Lyapunov function via linear programming. Multiple model approaches to modelling and control, Int. Journal of Control, 72 (1999), 702-715.
[15]	H. K. Khalil, "Nonlinear Systems,'' 3^rd edition, Prentice Hall, New Jersey, 2002.
[16]	S. Marinósson, "Stability Analysis of Nonlinear Systems with Linear Programming: A Lyapunov Functions Based Approach,'' Ph.D. thesis, Gerhard-Mercator-University, Duisburg, Germany, 2002.
[17]	S. Marinósson, Lyapunov function construction for ordinary differential equations with linear programming, Dynamical Systems, 17 (2002), 137-150. doi: 10.1080/0268111011011847.
[18]	A. Papachristodoulou and S. Prajna, The construction of Lyapunov functions using the sum of squares decomposition, in "Proceedings of the 41^st IEEE Conference on Decision and Control,'' (2002), 3482-3487.
[19]	P. Parrilo, "Structured Semidefinite Programs and Semialgebraic Geometry Methods in Robustness and Optimization,'' Ph.D. thesis, Caltech, Pasadena, USA, 2000.
[20]	M. Peet, Exponentially stable nonlinear systems have polynomial Lyapunov functions on bounded regions, IEEE Trans. Automatic Control, 54 (2009), 979-987. doi: 10.1109/TAC.2009.2017116.
[21]	V. Zubov, "Methods of A. M. Lyapunov and Their Application,'' Translation prepared under the auspices of the United States Atomic Energy Commission, edited by Leo F. Boron, P. Noordhoff Ltd, Groningen, 1964.

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