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Manifold-Following Approximate Solution of Completely Hypersensitive Optimal Control Problems

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Abstract

The solution to a completely hypersensitive optimal control problem may shadow trajectories on the stable and unstable manifolds of an equilibrium point in the state–costate phase space. If such shadowing occurs, the solution of the Hamiltonian boundary value problem, that constitutes the first-order necessary conditions for optimality, can be approximated as a composite of an initial segment on the stable manifold leading to the equilibrium point and a final segment on the unstable manifold departing from the equilibrium point. Using a dichotomic basis, the Hamiltonian vector field can be decomposed into stable and unstable components, and the unspecified boundary conditions for the initial and terminal segments can be determined such that the initial and final conditions are, respectively, on the stable and unstable manifolds of the equilibrium point. In this paper, we propose and justify the use of finite-time Lyapunov vectors to construct an approximate dichotomic basis and develop a corresponding manifold-following solution approximation method. The method is illustrated on two examples and shown to be more accurate than a similar method that uses eigenvectors of the frozen-time linearized dynamics.

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Acknowledgments

This work was supported by the National Science Foundation under Grants CMMI-0010085 and CMMI-1069331.

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  1. Ufuk Topcu

    Present address: Department of Aerospace Engineering and Engineering Mechanics, The University of Texas at Austin, 210 E 24th St, Austin, TX, 78712, USA

Authors and Affiliations

  1. Department of Mechanical and Aerospace Engineering, Henry Samueli School of Engineering, 4200 Engineering Gateway Building, University of California, Irvine, Irvine, CA, 92697-3975, USA

    Erkut Aykutlug, Ufuk Topcu & Kenneth D. Mease

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  1. Erkut Aykutlug
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  2. Ufuk Topcu
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  3. Kenneth D. Mease
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Correspondence to Kenneth D. Mease.

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Aykutlug, E., Topcu, U. & Mease, K.D. Manifold-Following Approximate Solution of Completely Hypersensitive Optimal Control Problems. J Optim Theory Appl 170, 220–242 (2016). https://doi.org/10.1007/s10957-016-0868-3

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