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16th International Conference on Dynamical Systems - Theory and Applications (DSTA)

作者： Giesl, Peter Hafstein, Sigurdur Univ Sussex Dept Math Falmer England Univ Iceland Sci Inst Reykjavik Iceland

ISBN: (纸本)9783031564987;9783031564963;9783031564956

Lyapunov functions for nonlinear systems, whose dynamics are defined by ordinary differential equations, are computed by solving linear programming feasibility problems in the cpa method. Further, the cpa method is constructive and can generate a Lyapunov function on any compact subset of the basin of attraction of an asymptotically stable equilibrium. Instead of solving the linear programming feasibility problem, one can use converse theorems to determine a candidate solution and then verify the constraints of the feasibility problem. This procedure has the advantage of being usually much faster. Further, a partial solution to the feasibility problem that violates the constraints in some areas can be analyzed, whereas a solver either generates a feasible solution or assures that a feasible solution does not exist. In this paper we prove that the numerical quadrature of numerically integrated solutions will deliver a feasible solution to the linear programming problem, given that the time horizon is large enough and the time steps are small enough in the numerical integration and quadrature. Further, the relevant theorems are general enough to allow for considerable flexibility in the particular implementation as they cover a wider range of numerical methods both for integration and quadrature.

关键词： Lyapunov function cpa algorithm Numerical integration Numerical quadrature

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Triangulations and Error Estimates for Interpolating Lyapunov Functions

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SN Computer Science 2023年第4期4卷 1-15页

作者： Giesl, Peter Hafstein, Sigurdur Department of Mathematics University of Sussex Falmer BN1 9QH United Kingdom Science Institute University of Iceland Dunhagi 3 Reykjavík 107 Iceland

The cpa method to compute Lyapunov functions depends on a triangulation of the relevant part of the state space. In more detail, a cpa (Continuous and Piecewise Affine) function is affine on each simplex of a given triangulation and is determined by the values at the vertices of the triangulation. Two important aspects in the proof that the cpa method is always able to generate a cpa Lyapunov function if the triangulation is sufficiently fine, are (a) the geometry of the simplices of the triangulation and (b) error estimates of cpa interpolations of functions. In this paper the aspect (a) is tackled by extending the notion of (h, d)-boundedness, which so far has depended on the order of the vertices in each simplex, and it is shown that it is essentially independent of the order and can be expressed in terms of the condition number of the shape matrix. Concerning (b), existing error estimates are generalised to other norms to increase the flexibility of the cpa method. In particular, when the cpa method is used to verify Lyapunov function candidates generated by other methods. Parts of the results were presented in Giesl and Hafstein (Uniformly regular triangulations for parameterizing Lyapunov functions. In: Proceedings of the 18th International Conference on Informatics in Control, Automation and Robotics (ICINCO), 549–557, 2021). © 2023, The Author(s).

关键词： cpa algorithm Error estimates Linear programming Lyapunov function Triangulation

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Uniformly Regular Triangulations for Parameterizing Lyapunov Functions 18

Uniformly Regular Triangulations for Parameterizing Lyapunov...

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18th International Conference on Informatics in Control, Automation and Robotics (ICINCO)

作者： Giesl, Peter Hafstein, Sigurdur Univ Sussex Dept Math Falmer BN1 9QH England Univ Iceland Sci Inst Dunhagi 3 IS-107 Reykjavik Iceland

ISBN: (纸本)9789897585227

The computation of Lyapunov functions to determine the basins of attraction of equilibria in dynamical systems can be achieved using linear programming. In particular, we consider a cpa (continuous piecewise affine) Lyapunov function, which can be fully described by its values at the vertices of a given triangulation. The method is guaranteed to find a cpa Lyapunov function, if a sequence of finer and finer triangulations with a bound on their degeneracy is considered. Hence, the notion of (h;d)-bounded triangulations was introduced, where h is a bound on the diameter of each simplex and d a bound on the degeneracy, expressed by the so-called shape-matrices of the simplices. However, the shape-matrix, and thus the degeneracy, depends on the ordering of the vertices in each simplex. In this paper, we first remove the rather unnatural dependency of the degeneracy on the ordering of the vertices and show that an (h;d)-bounded triangulation, of which the ordering of the vertices is changed, is still (h;d*)-bounded, where d* is a function of d, h, and the dimension of the system. Furthermore, we express the degeneracy in terms of the condition number, which is a well-studied quantity.

关键词： Triangulation Lyapunov Function cpa algorithm Linear Programming

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