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Dimension and measure of baker-like skew-products of $\boldsymbol{\beta}$-transformations
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 |
  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.
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. |
show all references
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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