Minimax problems for ensembles of control-affine systems

In this paper, we consider ensembles of control-affine systems in ℝd, and we study simultaneous optimal control problems related to the worst-case minimization. After proving that such problems admit solutions, denoting with (ThetaN)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 (ThetaN)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 (ThetaN)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.