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.