An Approximation Scheme for Uncertain Minimax Optimal Control Problems

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...

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Bibliographic Details
Published in:Set-valued and variational analysis 2018-12, Vol.26 (4), p.843-866
Main Authors: Aragone, Laura S., Gianatti, Justina, Lotito, Pablo A., Parente, Lisandro A.
Format: Article
Language:English
Subjects:
Analysis
Approximation
Convergence
Expected values
Mathematics
Mathematics and Statistics
Minimax technique
Optimal control
Optimization
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Summary: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.
ISSN:1877-0533
1877-0541
DOI:10.1007/s11228-017-0450-7