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A deep learning algorithm for the stable manifolds of the Hamilton-Jacobi equations

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Document pages: 19 pages

Abstract: In this paper, we propose a deep learning method to approximate the stablemanifolds of the Hamilton-Jacobi (HJ) equations from nonlinear control systemsbased on some mathematically rigorous asymptotic analysis, and then numericallycompute optimal feedback controls. The algorithm is devised from geometricfeatures of the stable manifolds, and relies on adaptive data generation byfinding trajectories of two-point boundary value problems (BVP) for theassociated Hamiltonian systems of the HJ equations. A number of samples arechosen on each trajectory according to exponential distribution with respect tothe time. Some adaptive samples are selected near the the points with largeerrors after the previous round of training. These may make the training of theneural network (NN) approximations more efficient. Our algorithm iscausality-free basically, hence it has a potential to apply to varioushigh-dimensional nonlinear systems. We illustrate the effectiveness of ourmethod by swinging up and stabilizing the Reaction Wheel Pendulums.

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