Construction of Lyapunov functions for nonlinear planar systems by linear programming

Giesl, Peter and Hafstein, Sigurdur (2012) Construction of Lyapunov functions for nonlinear planar systems by linear programming. Journal of Mathematical Analysis and Applications, 388 (1). pp. 463-479. ISSN 0022-247X

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Recently the authors proved the existence of piecewise affine Lyapunov functions for dynamical systems with an exponentially stable equilibrium in two dimensions (Giesl and Hafstein, 2010 [7]). Here, we extend these results by designing an algorithm to explicitly construct such a Lyapunov function. We do this by modifying and extending an algorithm to construct Lyapunov functions first presented in Marinosson (2002) [17] and further improved in Hafstein (2007) [10]. The algorithm constructs a linear programming problem for the system at hand, and any feasible solution to this problem parameterizes a Lyapunov function for the system. We prove that the algorithm always succeeds in constructing a Lyapunov function if the system possesses an exponentially stable equilibrium. The size of the region of the Lyapunov function is only limited by the region of attraction of the equilibrium and it includes the equilibrium.

Item Type: Article
Keywords: Exponentially stable equilibrium; Piecewise affine Lyapunov function; Triangulation; Linear programming
Schools and Departments: School of Mathematical and Physical Sciences > Mathematics
Subjects: Q Science > QA Mathematics
Depositing User: Peter Giesl
Date Deposited: 23 Jul 2012 14:38
Last Modified: 23 Jul 2012 14:38
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