Dynamic programming using radial basis functions
Oliver Junge Alex Schreiber
Discrete & Continuous Dynamical Systems - A 2015, 35(9): 4439-4453 doi: 10.3934/dcds.2015.35.4439
We propose a discretization of the optimality principle in dynamic programming based on radial basis functions and Shepard's moving least squares approximation method. We prove convergence of the value iteration scheme, derive a statement about the stability region of the closed loop system using the corresponding approximate optimal feedback law and present several numerical experiments.
keywords: optimal feedback. radial basis function Dynamic programming moving least squares Shepard's method
Topological method for rigorously computing periodic orbits using Fourier modes
Anthony W. Baker Michael Dellnitz Oliver Junge
Discrete & Continuous Dynamical Systems - A 2005, 13(4): 901-920 doi: 10.3934/dcds.2005.13.901
We present a technique for the rigorous computation of periodic orbits in certain ordinary differential equations. The method combines set oriented numerical techniques for the computation of invariant sets in dynamical systems with topological index arguments. It not only allows for the proof of existence of periodic orbits but also for a precise (and rigorous) approximation of these. As an example we compute a periodic orbit for a differential equation introduced in [2].
keywords: periodic orbit set oriented methods Dynamical system computer assisted proof.
Computing coherent sets using the Fokker-Planck equation
Andreas Denner Oliver Junge Daniel Matthes
Journal of Computational Dynamics 2016, 3(2): 163-177 doi: 10.3934/jcd.2016008
We perform a numerical approximation of coherent sets in finite-dimensional smooth dynamical systems by computing singular vectors of the transfer operator for a stochastically perturbed flow. This operator is obtained by solution of a discretized Fokker-Planck equation. For numerical implementation, we employ spectral collocation methods and an exponential time differentiation scheme. We experimentally compare our approach with the more classical method by Ulam that is based on discretization of the transfer operator of the unperturbed flow.
keywords: transfer operator Coherent set Fokker-Planck equation.
Global optimal feedbacks for stochastic quantized nonlinear event systems
Stefan Jerg Oliver Junge Marcus Post
Journal of Computational Dynamics 2014, 1(1): 163-176 doi: 10.3934/jcd.2014.1.163
We consider nonlinear control systems for which only quantized and event-triggered state information is available and which are subject to random delays and losses in the transmission of the state to the controller. We present an optimization based approach for computing globally stabilizing controllers for such systems. Our method is based on recently developed set oriented techniques for transforming the problem into a shortest path problem on a weighted hypergraph. We show how to extend this approach to a system subject to a stochastic parameter and propose a corresponding model for dealing with transmission delays.
keywords: global feedback event system. quantized system Dynamic programming set oriented numerics

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