MINIMAX PROBLEMS FOR ENSEMBLES OF CONTROL-AFFINE SYSTEMS

In this paper, we consider ensembles of control-affine systems in \BbbR^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.