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Set-Valued and Variational Analysis

, Volume 26, Issue 4, pp 843–866

# An Approximation Scheme for Uncertain Minimax Optimal Control Problems

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## Abstract

In this work, we address an uncertain minimax optimal control problem with linear dynamics where the objective functional is the expected value of the supremum of the running cost over a time interval. By taking an independently drawn random sample, the expected value function is approximated by the corresponding sample average function. We study the epi-convergence of the approximated objective functionals as well as the convergence of their global minimizers. Then we define an Euler discretization in time of the sample average problem and prove that the value of the discrete time problem converges to the value of the sample average approximation. In addition, we show that there exists a sequence of discrete problems such that the accumulation points of their minimizers are optimal solutions of the original problem. Finally, we propose a convergent descent method to solve the discrete time problem, and show some preliminary numerical results for two simple examples.

## Keywords

Minimax control problems Uncertain control problems Sample average approximation Epi-convergence Numerical solutions

## Mathematics Subject Classification (2010)

49K35 49M25 49M37

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## Copyright information

© Springer Science+Business Media B.V. 2017

## Authors and Affiliations

• Laura S. Aragone
• 1
• Justina Gianatti
• 1
• Pablo A. Lotito
• 2
• Lisandro A. Parente
• 1
1. 1.CIFASIS-CONICET-UNR, Ocampo y EsmeraldaRosarioArgentina
2. 2.PLADEMA-UNCPBA-CONICET, Paraje Arroyo SecoTandilArgentina