Differential Stability of Convex Discrete Optimal Control Problems

Abstract

Differential stability of convex discrete optimal control problems in Banach spaces is studied in this paper. By using some recent results of An and Yen (Appl. Anal. 94, 108–128, 2015) on differential stability of parametric convex optimization problems under inclusion constraints, we obtain an upper estimate for the subdifferential of the optimal value function of a parametric convex discrete optimal control problem, where the objective function may be nondifferentiable. If the objective function is differentiable, the obtained upper estimate becomes an equality. It is shown that the singular subdifferential of the just mentioned optimal value function always consists of the origin of the dual space.

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Acknowledgements

The research of Duong Thi Viet An was supported by the College of Sciences, Thai Nguyen University, Vietnam. The research of Nguyen Thi Toan was supported by the National Foundation for Science and Technology Development (Vietnam) under grant number 101.01-2015.04. The authors thank Professor Nguyen Dong Yen for useful discussions and the anonymous referees for valuable remarks.

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Correspondence to Duong Thi Viet An.

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An, D.T.V., Toan, N. Differential Stability of Convex Discrete Optimal Control Problems. Acta Math Vietnam 43, 201–217 (2018). https://doi.org/10.1007/s40306-017-0227-y

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Keywords

  • Parametric convex discrete optimal control problem
  • Optimal value function
  • Subdifferentials
  • Linear operator with closed range
  • Adjoint operator

Mathematics Subject Classification (2010)

  • 93C55
  • 93C73
  • 49K40
  • 49J53
  • 90C31
  • 90C25