# American Institute of Mathematical Sciences

October  2012, 17(7): 2387-2412. doi: 10.3934/dcdsb.2012.17.2387

## Construction of a finite-time Lyapunov function by meshless collocation

 1 Department of Mathematics, University of Sussex, Falmer BN1 9QH

Received  July 2011 Revised  February 2012 Published  July 2012

We consider a nonautonomous ordinary differential equation of the form $\dot{x}=f(t,x)$, $x\in \mathbb{R}^n$ over a finite-time interval $t\in [T_1,T_2]$. 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.
Citation: Peter Giesl. Construction of a finite-time Lyapunov function by meshless collocation. Discrete & Continuous Dynamical Systems - B, 2012, 17 (7) : 2387-2412. doi: 10.3934/dcdsb.2012.17.2387
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##### References:
 [1] Peter Giesl, Holger Wendland. Approximating the basin of attraction of time-periodic ODEs by meshless collocation. Discrete & Continuous Dynamical Systems - A, 2009, 25 (4) : 1249-1274. doi: 10.3934/dcds.2009.25.1249 [2] Peter Giesl, James McMichen. Determination of the area of exponential attraction in one-dimensional finite-time systems using meshless collocation. Discrete & Continuous Dynamical Systems - B, 2018, 23 (4) : 1835-1850. doi: 10.3934/dcdsb.2018094 [3] Peter Giesl, Holger Wendland. Approximating the basin of attraction of time-periodic ODEs by meshless collocation of a Cauchy problem. Conference Publications, 2009, 2009 (Special) : 259-268. doi: 10.3934/proc.2009.2009.259 [4] Peter Giesl, James McMichen. Determination of the basin of attraction of a periodic orbit in two dimensions using meshless collocation. Journal of Computational Dynamics, 2016, 3 (2) : 191-210. doi: 10.3934/jcd.2016010 [5] Peter Giesl. Construction of a global Lyapunov function using radial basis functions with a single operator. Discrete & Continuous Dynamical Systems - B, 2007, 7 (1) : 101-124. doi: 10.3934/dcdsb.2007.7.101 [6] Hjörtur Björnsson, Sigurdur Hafstein, Peter Giesl, Enrico Scalas, Skuli Gudmundsson. Computation of the stochastic basin of attraction by rigorous construction of a Lyapunov function. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 4247-4269. doi: 10.3934/dcdsb.2019080 [7] Arno Berger, Doan Thai Son, Stefan Siegmund. Nonautonomous finite-time dynamics. Discrete & Continuous Dynamical Systems - B, 2008, 9 (3&4, May) : 463-492. doi: 10.3934/dcdsb.2008.9.463 [8] Najla Mohammed, Peter Giesl. Grid refinement in the construction of Lyapunov functions using radial basis functions. Discrete & Continuous Dynamical Systems - B, 2015, 20 (8) : 2453-2476. doi: 10.3934/dcdsb.2015.20.2453 [9] Imtiaz Ahmad, Siraj-ul-Islam, Mehnaz, Sakhi Zaman. Local meshless differential quadrature collocation method for time-fractional PDEs. Discrete & Continuous Dynamical Systems - S, 2020, 13 (10) : 2641-2654. doi: 10.3934/dcdss.2020223 [10] Huijuan Li, Junxia Wang. Input-to-state stability of continuous-time systems via finite-time Lyapunov functions. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 841-857. doi: 10.3934/dcdsb.2019192 [11] Sanjeeva Balasuriya. Uncertainty in finite-time Lyapunov exponent computations. Journal of Computational Dynamics, 2020, 7 (2) : 313-337. doi: 10.3934/jcd.2020013 [12] Peter Giesl, Najla Mohammed. Verification estimates for the construction of Lyapunov functions using meshfree collocation. Discrete & Continuous Dynamical Systems - B, 2019, 24 (9) : 4955-4981. doi: 10.3934/dcdsb.2019040 [13] Martin D. Buhmann, Slawomir Dinew. Limits of radial basis function interpolants. Communications on Pure & Applied Analysis, 2007, 6 (3) : 569-585. doi: 10.3934/cpaa.2007.6.569 [14] Khalid Addi, Samir Adly, Hassan Saoud. Finite-time Lyapunov stability analysis of evolution variational inequalities. Discrete & Continuous Dynamical Systems - A, 2011, 31 (4) : 1023-1038. doi: 10.3934/dcds.2011.31.1023 [15] Gerard Gómez, Josep–Maria Mondelo, Carles Simó. A collocation method for the numerical Fourier analysis of quasi-periodic functions. II: Analytical error estimates. Discrete & Continuous Dynamical Systems - B, 2010, 14 (1) : 75-109. doi: 10.3934/dcdsb.2010.14.75 [16] Oliver Junge, Alex Schreiber. Dynamic programming using radial basis functions. Discrete & Continuous Dynamical Systems - A, 2015, 35 (9) : 4439-4453. doi: 10.3934/dcds.2015.35.4439 [17] Sohana Jahan, Hou-Duo Qi. Regularized multidimensional scaling with radial basis functions. Journal of Industrial & Management Optimization, 2016, 12 (2) : 543-563. doi: 10.3934/jimo.2016.12.543 [18] Arno Berger. On finite-time hyperbolicity. Communications on Pure & Applied Analysis, 2011, 10 (3) : 963-981. doi: 10.3934/cpaa.2011.10.963 [19] Thierry Cazenave, Yvan Martel, Lifeng Zhao. Finite-time blowup for a Schrödinger equation with nonlinear source term. Discrete & Continuous Dynamical Systems - A, 2019, 39 (2) : 1171-1183. doi: 10.3934/dcds.2019050 [20] Luis Barreira, Claudia Valls. Stability of nonautonomous equations and Lyapunov functions. Discrete & Continuous Dynamical Systems - A, 2013, 33 (7) : 2631-2650. doi: 10.3934/dcds.2013.33.2631

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