On the Pareto control and no-regret control for distributed systems with incomplete data

We discuss the control of distributed systems with incomplete data following the notion of no-regret control (or, equivalently, Pareto control) used by Lions in [C.R. Acad. Sci. Paris Ser. I Math., 302 (1986), pp. 223-227] and [C.R. Acad. Sci. Paris Ser. I Math., 302 (1992), pp. 1253-1257]. We associate with the no-regret control a sequence of low-regret controls defined by a quadratic perturbation previously used by Nakoulima, Omrane, and Velin in [C.R. Acad. Sci. Paris Ser. I Math., 330 (2000), pp. 801-806]. We prove that the perturbed system corresponds to a sequence of standard control problems and converges to the no-regret (or Pareto) control for which we obtain a singular optimality system.