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Linear programming based Lyapunov function computation for differential inclusions
1. | Chair of Applied Mathematics, University of Bayreuth, 95440 Bayreuth, Germany, Germany |
2. | School of Science and Engineering, Reykjavík University, Menntavegur 1, 101 Reykjavík, Iceland |
References:
[1] |
A. Bacciotti and F. Ceragioli, Stability and stabilization of discontinuous systems and nonsmooth Lyapunov functions, ESAIM Control Optim. Calc. Var., 4 (1999), 361-376 (electronic).
doi: 10.1051/cocv:1999113. |
[2] |
F. Camilli, L. Grüne and F. Wirth, A regularization of Zubov's equation for robust domains of attraction, in "Nonlinear Control in the Year 2000, Vol. 1," Lecture Notes in Control and Inform. Sci., 258, Springer, London, (2001), 277-289. |
[3] |
G. Chesi, Estimating the domain of attraction for uncertain polynomial systems, Automatica J. IFAC, 40 (2004), 1981-1986.
doi: 10.1016/j.automatica.2004.06.014. |
[4] |
F. H. Clarke, "Optimization and Nonsmooth Analysis," Second edition, Classics in Applied Mathematics, 5, SIAM, Philadelphia, PA, 1990, First edition published in John Wiley & Sons, Inc., New York, 1983. |
[5] |
F. H. Clarke, Yu. S. Ledyaev and R. J. Stern, Asymptotic stability and smooth Lyapunov functions, J. Differential Equations, 149 (1998), 69-114. |
[6] |
T. Donchev, V. Ríos and P. Wolenski, Strong invariance and one-sided Lipschitz multifunctions, Nonlinear Anal., 60 (2005), 849-862.
doi: 10.1016/j.na.2004.09.050. |
[7] |
A. F. Filippov, "Differential Equations with Discontinuous Righthand Sides," Translated from the Russian, Mathematics and its Applications (Soviet Series), 18, Kluwer Academic Publishers Group, Dordrecht, 1988. |
[8] |
P. Giesl, "Construction of Global Lyapunov Functions Using Radial Basis Functions," Lecture Notes in Math., 1904, Springer, Berlin, 2007. |
[9] |
P. Giesl and S. F. 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. |
[10] |
L. Grüne and O. Junge, "Gewöhnliche Differentialgleichungen. Eine Einführung aus der Perspektive der dynamischen Systeme. Bachelorkurs Mathematik," Vieweg Studium, Vieweg+Teubner, Wiesbaden, 2009. |
[11] |
S. F. Hafstein, "An Algorithm for Constructing Lyapunov Functions,", Electron. J. Differential Equ. Monogr., 8, Texas State Univ., Dep. of Mathematics, San Marcos, TX, 2007., Available from: \url{http://ejde.math.txstate.edu}., ().
|
[12] |
D. Hinrichsen and A. J. Pritchard, "Mathematical Systems Theory I. Modelling, State Space Analysis, Stability and Robustness," Texts in Applied Mathematics, 48, Springer-Verlag, Berlin, 2005. |
[13] |
T. A. 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. |
[14] |
M. Johansson, "Piecewise Linear Control Systems. A Computational Approach," Lecture Notes in Control and Inform. Sci., 284, Springer-Verlag, Berlin, 2003. |
[15] |
P. Julián, J. Guivant and A. Desages, A parametrization of piecewise linear Lyapunov functions via linear programming. Multiple model approaches to modelling and control, Internat. J. Control, 72 (1999), 702-715. |
[16] |
B. Kummer, Newton's method for nondifferentiable functions, in "Advances in Mathematical Optimization," Math. Res., 45, Akademie-Verlag, Berlin, (1988), 114-125. |
[17] |
G. Leoni, "A First Course in Sobolev Spaces," Graduate Studies in Mathematics, 105, American Mathematical Society, Providence, RI, 2009. |
[18] |
D. Liberzon, "Switching in Systems and Control," Systems & Control: Foundations & Applications, Birkhäuser Boston, Inc., Boston, MA, 2003. |
[19] |
S. Marinósson, Lyapunov function construction for ordinary differential equations with linear programming, Dyn. Syst., 17 (2002), 137-150.
doi: 10.1080/0268111011011847. |
[20] |
I. P. Natanson, "Theory of Functions of a Real Variable," Translated by L. F. Boron with the collaboration of E. Hewitt, Frederick Ungar Publishing Co., New York, 1955. |
[21] |
E. P. Ryan, An integral invariance principle for differential inclusions with applications in adaptive control, SIAM J. Control Optim., 36 (1998), 960-980 (electronic).
doi: 10.1137/S0363012996301701. |
[22] |
S. Scholtes, "Introduction to Piecewise Differentiable Equations," habilitation thesis, Universität Karlsruhe, Institut für Statistik und Mathematische Wirtschaftstheorie, Karlsruhe, Germany, May, 1994., Preprint no. 53/1994., ().
|
[23] |
D. Stewart, A high accuracy method for solving ODEs with discontinuous right-hand side, Numer. Math., 58 (1990), 299-328.
doi: 10.1007/BF01385627. |
[24] |
A. R. Teel and L. Praly, A smooth Lyapunov function from a class-$\mathcal{KL}$ estimate involving two positive semidefinite functions, ESAIM Control Optim. Calc. Var., 5 (2000), 313-367 (electronic).
doi: 10.1051/cocv:2000113. |
[25] |
H. Whitney, Analytic extensions of differentiable functions defined in closed sets, Trans. Amer. Math. Soc., 36 (1934), 63-89.
doi: 10.1090/S0002-9947-1934-1501735-3. |
show all references
References:
[1] |
A. Bacciotti and F. Ceragioli, Stability and stabilization of discontinuous systems and nonsmooth Lyapunov functions, ESAIM Control Optim. Calc. Var., 4 (1999), 361-376 (electronic).
doi: 10.1051/cocv:1999113. |
[2] |
F. Camilli, L. Grüne and F. Wirth, A regularization of Zubov's equation for robust domains of attraction, in "Nonlinear Control in the Year 2000, Vol. 1," Lecture Notes in Control and Inform. Sci., 258, Springer, London, (2001), 277-289. |
[3] |
G. Chesi, Estimating the domain of attraction for uncertain polynomial systems, Automatica J. IFAC, 40 (2004), 1981-1986.
doi: 10.1016/j.automatica.2004.06.014. |
[4] |
F. H. Clarke, "Optimization and Nonsmooth Analysis," Second edition, Classics in Applied Mathematics, 5, SIAM, Philadelphia, PA, 1990, First edition published in John Wiley & Sons, Inc., New York, 1983. |
[5] |
F. H. Clarke, Yu. S. Ledyaev and R. J. Stern, Asymptotic stability and smooth Lyapunov functions, J. Differential Equations, 149 (1998), 69-114. |
[6] |
T. Donchev, V. Ríos and P. Wolenski, Strong invariance and one-sided Lipschitz multifunctions, Nonlinear Anal., 60 (2005), 849-862.
doi: 10.1016/j.na.2004.09.050. |
[7] |
A. F. Filippov, "Differential Equations with Discontinuous Righthand Sides," Translated from the Russian, Mathematics and its Applications (Soviet Series), 18, Kluwer Academic Publishers Group, Dordrecht, 1988. |
[8] |
P. Giesl, "Construction of Global Lyapunov Functions Using Radial Basis Functions," Lecture Notes in Math., 1904, Springer, Berlin, 2007. |
[9] |
P. Giesl and S. F. 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. |
[10] |
L. Grüne and O. Junge, "Gewöhnliche Differentialgleichungen. Eine Einführung aus der Perspektive der dynamischen Systeme. Bachelorkurs Mathematik," Vieweg Studium, Vieweg+Teubner, Wiesbaden, 2009. |
[11] |
S. F. Hafstein, "An Algorithm for Constructing Lyapunov Functions,", Electron. J. Differential Equ. Monogr., 8, Texas State Univ., Dep. of Mathematics, San Marcos, TX, 2007., Available from: \url{http://ejde.math.txstate.edu}., ().
|
[12] |
D. Hinrichsen and A. J. Pritchard, "Mathematical Systems Theory I. Modelling, State Space Analysis, Stability and Robustness," Texts in Applied Mathematics, 48, Springer-Verlag, Berlin, 2005. |
[13] |
T. A. 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. |
[14] |
M. Johansson, "Piecewise Linear Control Systems. A Computational Approach," Lecture Notes in Control and Inform. Sci., 284, Springer-Verlag, Berlin, 2003. |
[15] |
P. Julián, J. Guivant and A. Desages, A parametrization of piecewise linear Lyapunov functions via linear programming. Multiple model approaches to modelling and control, Internat. J. Control, 72 (1999), 702-715. |
[16] |
B. Kummer, Newton's method for nondifferentiable functions, in "Advances in Mathematical Optimization," Math. Res., 45, Akademie-Verlag, Berlin, (1988), 114-125. |
[17] |
G. Leoni, "A First Course in Sobolev Spaces," Graduate Studies in Mathematics, 105, American Mathematical Society, Providence, RI, 2009. |
[18] |
D. Liberzon, "Switching in Systems and Control," Systems & Control: Foundations & Applications, Birkhäuser Boston, Inc., Boston, MA, 2003. |
[19] |
S. Marinósson, Lyapunov function construction for ordinary differential equations with linear programming, Dyn. Syst., 17 (2002), 137-150.
doi: 10.1080/0268111011011847. |
[20] |
I. P. Natanson, "Theory of Functions of a Real Variable," Translated by L. F. Boron with the collaboration of E. Hewitt, Frederick Ungar Publishing Co., New York, 1955. |
[21] |
E. P. Ryan, An integral invariance principle for differential inclusions with applications in adaptive control, SIAM J. Control Optim., 36 (1998), 960-980 (electronic).
doi: 10.1137/S0363012996301701. |
[22] |
S. Scholtes, "Introduction to Piecewise Differentiable Equations," habilitation thesis, Universität Karlsruhe, Institut für Statistik und Mathematische Wirtschaftstheorie, Karlsruhe, Germany, May, 1994., Preprint no. 53/1994., ().
|
[23] |
D. Stewart, A high accuracy method for solving ODEs with discontinuous right-hand side, Numer. Math., 58 (1990), 299-328.
doi: 10.1007/BF01385627. |
[24] |
A. R. Teel and L. Praly, A smooth Lyapunov function from a class-$\mathcal{KL}$ estimate involving two positive semidefinite functions, ESAIM Control Optim. Calc. Var., 5 (2000), 313-367 (electronic).
doi: 10.1051/cocv:2000113. |
[25] |
H. Whitney, Analytic extensions of differentiable functions defined in closed sets, Trans. Amer. Math. Soc., 36 (1934), 63-89.
doi: 10.1090/S0002-9947-1934-1501735-3. |
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