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On Methods of Terminal Control with Boundary-Value Problems: Lagrange Approach

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Optimization and Its Applications in Control and Data Sciences

Part of the book series: Springer Optimization and Its Applications ((SOIA,volume 115))

Abstract

A dynamic model of terminal control with boundary value problems in the form of convex programming is considered. The solutions to these finite-dimensional problems define implicitly initial and terminal conditions at the ends of time interval at which the controlled dynamics develops. The model describes a real situation when an object needs to be transferred from one state to another. Based on the Lagrange formalism, the model is considered as a saddle-point controlled dynamical problem formulated in a Hilbert space. Iterative saddle-point method has been proposed for solving it. We prove the convergence of the method to saddle-point solution in all its components: weak convergence—in controls, strong convergence—in phase and conjugate trajectories, and terminal variables.

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Notes

  1. 1.

    Scalar products and norms are defined, respectively, as

    $$\displaystyle\begin{array}{rcl} & \langle x(\cdot ),y(\cdot )\rangle =\int _{ t_{0}}^{t_{1}}\langle x(t),y(t)\rangle dt,\;\|x(\cdot )\|^{2} =\int _{ t_{ 0}}^{t_{1}}\vert x(t)\vert ^{2}dt, & {}\\ & \text{where }\;\langle x(t),y(t)\rangle =\sum \limits _{ 1}^{n}x_{i}(t)y_{i}(t),\;\vert x(t)\vert ^{2} =\sum \limits _{ 1}^{n}x_{i}^{2}(t),\quad t_{0} \leq t \leq t_{1},& {}\\ & x(t) = (x_{1}(t),\ldots,x_{n}(t))^{\mathrm{T}},\;y(t) = (y_{1}(t),\ldots,y_{n}(t))^{\mathrm{T}}. & {}\\ \end{array}$$
  2. 2.

    For simplicity, the positive orthant R+ m hereinafter will also be referred to p ≥ 0.

  3. 3.

    See below the proof of the theorem.

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Acknowledgements

The work was carried out with financial support from RFBR (project No. 15-01-06045-a), and the Ministry of Education and Science of the Russian Federation in the framework of Increase Competitiveness Program of NUST “MISiS” (agreement Ń02.A03.21.0004).

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Authors and Affiliations

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

    Anatoly Antipin

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

    Elena Khoroshilova

Authors
  1. Anatoly Antipin
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  2. Elena Khoroshilova
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Corresponding author

Correspondence to Anatoly Antipin .

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Editors and Affiliations

  1. Department of Industrial and Systems Engineering, Ohio University, Athens, Ohio, USA

    Boris Goldengorin

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Antipin, A., Khoroshilova, E. (2016). On Methods of Terminal Control with Boundary-Value Problems: Lagrange Approach. In: Goldengorin, B. (eds) Optimization and Its Applications in Control and Data Sciences. Springer Optimization and Its Applications, vol 115. Springer, Cham. https://doi.org/10.1007/978-3-319-42056-1_2

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