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Controlled dynamic model with boundary-value problem of minimizing a sensitivity function

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Abstract

We formulate a controlled dynamic model with a boundary-value problem of minimizing a sensitivity function under constraints. Solution of boundary-value problem implicitly defines a terminal condition for the dynamic model. In the model, a unique trajectory corresponds to each control taken from a bounded set. The problem is to select the control such that the corresponding trajectory takes an object from an arbitrary initial state to the terminal state. In this paper, the dynamic model is treated as a problem of stabilization, and the terminal state of the object is interpreted as a state of equilibrium. If under the influence of external disturbances the object loses equilibrium then this object is returned back by selecting the appropriate control. A saddle-point method for solving problem is proposed. We prove its convergence to solution of the problem in all the variables.

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Notes

  1. Hereinafter, the index “1” next to a variable means belonging to the finite-dimensional space of the right-hand end of a time interval \([t_0,t_1]\).

  2. The method is called “dual”, since iterations run only in dual variables \(p_1\) and \(\psi (\cdot )\), while the optimization is done by primal variables \(y_1,x_1,x(\cdot )\).

  3. \(|u(t)|=\sqrt{\sum \limits _{i=1}^r u_i^2(t)}= \sqrt{\langle u(t),u(t)\rangle },\; t \in [t_0,t_1], \quad \Vert u(\cdot )\Vert = \sqrt{\int _{t_0}^{t_1}|u(t)|^2dt}\).

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Acknowledgements

The reported study was supported by the Russian Science Foundation (Research Project 17-11-01353).

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Authors and Affiliations

  1. Dorodnicyn Computing Centre, Federal Research Center “Computer Science and Control”, Russian Academy of Sciences, Vavilova str., 40, Moscow, Russia, 119333

    Anatoly Antipin

  2. Faculty of Computational Mathematics and Cybernetics, Lomonosov Moscow State University, Moscow, Russia, 119991

    Elena Khoroshilova

Authors
  1. Anatoly Antipin
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  2. Elena Khoroshilova
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Correspondence to Anatoly Antipin.

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Antipin, A., Khoroshilova, E. Controlled dynamic model with boundary-value problem of minimizing a sensitivity function. Optim Lett 13, 451–473 (2019). https://doi.org/10.1007/s11590-017-1216-8

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  • DOI: https://doi.org/10.1007/s11590-017-1216-8

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