`a`
Discrete and Continuous Dynamical Systems - Series A (DCDS-A)
 

Existence of piecewise linear Lyapunov functions in arbitrary dimensions
Pages: 3539 - 3565, Issue 10, October 2012

doi:10.3934/dcds.2012.32.3539      Abstract        References        Full text (1016.8K)           Related Articles

Peter Giesl - Department of Mathematics, University of Sussex, Falmer BN1 9QH, United Kingdom (email)
Sigurdur Hafstein - School of Science and Engineering, Reykjavik University, Menntavegi 1, IS-101 Reykjavik, Iceland (email)

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.       
2 R. Bartels and G. Stewart, Algorithm 432: Solution of the matrix equation $AX + XB = C$, Comm. ACM, 15 (1972), 820-826.
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.       
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.       
8 S. Hafstein, A constructive converse Lyapunov theorem on exponential stability, Discrete Contin. Dyn. Syst., 10 (2004), 657-678.       
9 S. Hafstein, A constructive converse Lyapunov theorem on asymptotic stability for nonlinear autonomous ordinary differential equations, Dynamical Systems, 20 (2005), 281-299.       
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.       
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.       
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.       
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.       

Go to top