Minimax problems for ensembles of control-affine systems
<|reference_start|>Minimax problems for ensembles of control-affine systems: In this paper, we consider ensembles of control-affine systems in $\mathbb{R}^d$, and we study simultaneous optimal control problems related to the worst-case minimization. After proving that such problems admit solutions, denoting with $(\Theta^N)_N$ a sequence of compact sets that parametrize the ensembles of systems, we first show that the corresponding minimax optimal control problems are $\Gamma$-convergent whenever $(\Theta^N)_N$ has a limit with respect to the Hausdorff distance. Besides its independent interest, the previous result plays a crucial role for establishing the Pontryagin Maximum Principle (PMP) when the ensemble is parametrized by a set $\Theta$ consisting of infinitely many points. Namely, we first approximate $\Theta$ by finite and increasing-in-size sets $(\Theta^N)_N$ for which the PMP is known, and then we derive the PMP for the $\Gamma$-limiting problem. The same strategy can be pursued in applications, where we can reduce infinite ensembles to finite ones to compute the minimizers numerically. We bring as a numerical example the Schr\"odinger equation for a qubit with uncertain resonance frequency.<|reference_end|>
@article{scagliotti2024minimax,
    title={Minimax problems for ensembles of control-affine systems},
    author={Alessandro Scagliotti},
    journal={arXiv preprint arXiv:2405.05782},
    year={2024},
    archivePrefix={arXiv},
    eprint={2405.05782},
    primaryClass={math.OC cs.SY eess.SY}
}