CONSTRUCTION OF A FINITE-TIME LYAPUNOV FUNCTION BY MESHLESS COLLOCATION

被引：6

作者：

Giesl, Peter ^{[1
]}

机构：

[1] Univ Sussex, Dept Math, Falmer BN1 9QH, England

来源：

DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS-SERIES B | 2012年 / 17卷 / 07期

关键词：

Nonautonomous ordinary differential equation; finite-time Lyapunov function; basin of attraction; meshless collocation; radial basis functions; error estimates; LAGRANGIAN COHERENT STRUCTURES; DIFFERENTIAL-EQUATIONS; ATTRACTION;

D O I：

10.3934/dcdsb.2012.17.2387

中图分类号：

O29 [应用数学];

学科分类号：

070104 ;

摘要：

We consider a nonautonomous ordinary differential equation of the form (x) over dot = f (t; x), x is an element of R-n over a finite-time interval t is an element of [T-1; T-2]. The basin of attraction of an attracting solution can be determined using a finite-time Lyapunov function. In this paper, such a finite-time Lyapunov function is constructed by Meshless Collocation, in particular Radial Basis Functions. Thereto, a finite-time Lyapunov function is characterised as the solution of a second-order linear partial differential equation with boundary values. This problem is approximately solved using Meshless Collocation, and it is shown that the approximate solution can be used to determine the basin of attraction. Error estimates are obtained and verified in examples.

引用

页码：2387 / 2412

页数：26

共 31 条

[1] [Anonymous], 2005, Cambridge Monograph, Applied Comput. Math.
[2] Berger A, 2008, DISCRETE CONT DYN-B, V9, P463
[3] ON FINITE-TIME HYPERBOLICITY
Berger, Arno
[J]. COMMUNICATIONS ON PURE AND APPLIED ANALYSIS, 2011, 10 (03) : 963 - 981
[4] On the approximation of complicated dynamical behavior
Dellnitz, M
Junge, O
[J]. SIAM JOURNAL ON NUMERICAL ANALYSIS, 1999, 36 (02) : 491 - 515
[5] Dellnitz M., 2002, Handbook of Dynamical Systems, P221, DOI [DOI 10.1016/S1874-575X(02)80026-1, 10.1016/s1874-575x(02)80026-1]
[6] Transient spectral theory, stable and unstable cones and Gershgorin's theorem for finite-time differential equations
Doan, T. S.
Palmer, K.
Siegmund, S.
[J]. JOURNAL OF DIFFERENTIAL EQUATIONS, 2011, 250 (11) : 4177 - 4199
[7] Doan T. S., UNIFIED APPROACH FIN
[8] Solving partial differential equations by collocation using radial basis functions
Franke, C
Schaback, R
[J]. APPLIED MATHEMATICS AND COMPUTATION, 1998, 93 (01) : 73 - 82
[9] Giesl P, 2007, LECT NOTES MATH, V1904, P1, DOI 10.1007/978-3-540-69909-5
[10] Giesl P., 2009, Series, P259

← 1 2 3 4 →