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| Lyapunov functions are an important tool to determine the basin of attraction of equilibria in Dynamical Systems. A Lyapunov function is a function which is decreasing along trajectories; sublevel sets of the Lyapunov function are subsets of the basin of attraction. The explicit construction of Lyapunov functions for a given system is a difficult problem. Recently, several numerical construction methods have been proposed, among them the RBF and CPA methods. In this talk, we will combine the advantages of both.
The RBF method formulates the decreasing property as a linear PDE and solves it approximately using meshless collocation, in particular Radial Basis Functions (RBF). Error estimates show that the method always constructs a (smooth) Lyapunov function if the collocation points are dense enough and placed in the appropriate area. So far, however, the method lacks a verification of whether a given approximation is decreasing along trajectories.
The CPA method triangulates the phase space and constructs a continuous Lyapunov function, which is piece-wise affine (CPA) on each simplex of the triangulation, using linear optimization. The method includes error estimates, which guarantee that the CPA function is indeed decreasing along trajectories.
In this talk, we propose a combination of these two methods: we use the RBF method to construct a Lyapunov function. Then we interpolate this function and thus construct a CPA Lyapunov function. Checking a finite number of inequalities, we are able to verify that this interpolation is indeed decreasing along trajectories. Moreover, sublevel sets get arbitrarily close to the basin of attraction. |
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