The Algorithms Behind GAIO — Set Oriented Numerical Methods for Dynamical Systems

Dellnitz, Michael; Froyland, Gary; Junge, Oliver

doi:10.1007/978-3-642-56589-2_7

Cited by 155 publications

(141 citation statements)

References 21 publications

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“…The algorithms for constructing these approximations and their numerical convergence properties appear in [50]. As one takes finer partitions, the invariant measure of Markov matrix P converges to a weak limit ν * that approximates the invariant measure of the P-F operator P; see Theorem 3.1 in [42].…”

Section: Stochastic Model and Bifurcation Analysismentioning

confidence: 99%

A non-equilibrium analysis and control framework for active queue management

et al. 2008

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Section: Stochastic Model and Bifurcation Analysismentioning

confidence: 99%

A non-equilibrium analysis and control framework for active queue management

et al. 2008

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“…(For the unstable manifold W u , the same holds in backward time (t ≤ 0)). To approximate the global (un)stable manifolds in our numerical example, we use the method GAIO (Global Analysis of Invariant Objects, [4]). …”

Section: Introductionmentioning

confidence: 99%

Optimal Control on Stable Manifolds for a Double Pendulum

Flaßkamp

Timmermann

Ober‐Blöbaum

et al. 2012

Proc Appl Math and Mech

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Optimal control problems for mechanical systems often arise in technical applications, e.g. in the computation of energy efficient or time optimal steering maneuvers between operation points. In this contribution, we propose a control strategy for mechanical systems that includes uncontrolled trajectories on (un)stable manifolds in a sequence together with short control maneuvers to design a feedforward control. In particular, we present optimal swing-up solutions for a double pendulum that are based on trajectories on the stable manifold of the pendulum's upper equilibrium.

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“…Computation of reachable sets can also be performed by the general-purpose package GAIO [9] for set-based computations.…”

Section: Introductionmentioning

confidence: 99%

Optimal Semicomputable Approximations to Reachable and Invariant Sets

Collins

2007

Theory Comput Syst

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Abstract. In this paper we consider the computation of reachable, viable and invariant sets for discrete-time systems. We use the framework of type-two effectivity, in which computations are performed by Turing machines with infinite input and output tapes, with the representations of computable topology. We see that the reachable set is lower-semicomputable, and the viability and invariance kernels are uppersemicomputable. We then define an upper-semicomputable over-approximation to the reachable set, and lower-semicomputable under-approximations to the viability and invariance kernels, and show that these approximations are optimal.

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The Algorithms Behind GAIO — Set Oriented Numerical Methods for Dynamical Systems

Cited by 155 publications

References 21 publications

A non-equilibrium analysis and control framework for active queue management

A non-equilibrium analysis and control framework for active queue management

Optimal Control on Stable Manifolds for a Double Pendulum

Optimal Semicomputable Approximations to Reachable and Invariant Sets

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