# American Institute of Mathematical Sciences

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

## Existence of piecewise linear Lyapunov functions in arbitrary dimensions

 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

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.
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 36th 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,'' 3rd 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 41st 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 36th 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,'' 3rd 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 41st 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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