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The quasidifferential descent method in a control problem with nonsmooth objective functional

Abstract

The paper is devoted to the problem of optimal control of an object described by a system with a continuously differentiable right-hand side and a nondifferentiable (but only quasidifferentiable) quality functional. We consider a problem in the form of Mayer with both a free and a fixed right end. Admissible controls are piecewise continuous (with a finite number of discontinuity points) and bounded vector-functions, which belong to certain polyhedron at each moment of time. Standard discretization of the initial system and control parameterization are performed, and theorems on the convergence of the discrete system solution obtained to the desired solution of the continuous problem are presented. Further, for the discrete system obtained, the necessary and, in some cases, sufficient minimum conditions are written in terms of quasidifferential. The quasidifferential descent method is applied to this problem. The algorithm developed is demonstrated by examples.

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Acknowledgements

The author is sincerely grateful to his colleague M. Dolgopolik for numerous fruitful discussions on the problem considered and to the anonymous referees, whose comments helped to improve the paper.

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Correspondence to A. V. Fominyh.

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Fominyh, A.V. The quasidifferential descent method in a control problem with nonsmooth objective functional. Optim Lett 15, 2773–2792 (2021). https://doi.org/10.1007/s11590-021-01710-7

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Keywords

  • Nonsmooth optimal control problem
  • Quasidifferential
  • Parameterization of control
  • Method of quasidifferential descent