Minimax problems for ensembles of affine-control systems

In this paper, we consider ensembles of affine-control systems in Rⁿ, and we study simultaneous optimal control problems related to the worst-case minimization. After proving that such problems admit solutions, denoting with (ΘN)N a sequence of compact sets that parametrize the ensembles of systems, we first show that the corresponding minimax optimal control problems are Γ-convergent whenever (ΘN)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 Θ consisting of infinitely many points. Namely, we first approximate Θ by finite and increasing-in-size sets (ΘN)N for which the PMP is known, and then we derive the PMP for the Γ-limiting problem. The same strategy can be pursued in applications where we can reduce infinite ensembles to finite ones to compute the minimizers numerically.