Minimax problems for ensembles of control-affine systems

Abstract

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 $(\hat{I}{˜}^N{)}_N$ a sequence of compact sets that parametrize the ensembles of systems, we first show that the corresponding minimax optimal control problems are $\hat{I}“$-convergent whenever $(\hat{I}{˜}^N{)}_N$ 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 $\hat{I}˜$ consisting of infinitely many points. Namely, we first approximate $\hat{I}˜$ by finite and increasing-in-size sets $(\hat{I}{˜}^N{)}_N$ for which the PMP is known, and then we derive the PMP for the $\hat{I}“$-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 Schrödinger equation for a qubit with uncertain resonance frequency.

24 pages, 1 Figure, 2 Tables. Correction of typos. Accepted in SIAM J Control Optim