Robust Stackelberg controllability for the Navier–Stokes equations

Abstract

In this paper we deal with a robust Stackelberg strategy for the Navier-Stokes system. The scheme is based in considering a robust control problem for the "follower control" and its associated disturbance function. Afterwards, we consider the notion of Stackelberg optimization (which is associated to the "leader control") in order to deduce a local null controllability result for the Navier-Stokes system.

Robust Stackelberg Controllability for the Kuramoto-Sivashinsky Equation

“…It is worth mentioning again that the theoretical results known up to now on robust Stackelberg controllability (Problem 3) are [20], [32] and [21], and there is no evidence on both numerical algorithms and a controllability to trajectories constrain for the leader control for nonlinear systems. Therefore, this paper we pretend to show theoretical results and carry out numerical schemes jointly with its implementation to Problems 1,3 for the KS equation (1.6).…”

“…From a theoretical perspective, recent works have mixed the concept of robust control with a Stackelberg strategy, and applied it to semilinear and linear heat equations [20,21], and to the Navier-Stokes system [32]. This new idea in control theory is being abridged and called "Robust Stackelberg controllability" (RSC), see Problem 3 below.…”

“…In [21] the leader control satisfies the null controllability property. In the RSC problem for the Navier-Stokes system [32] all controls are external forces acting on the systems, the leader control has a local null controllability objective, while the perturbation and the follower control solve a robust control problem. However, these three works have three things in common: 1) they deal with systems whose main operator is a secondorder operator (Laplace operator, Stokes operator), 2) independent of the configuration or localization of forces (either interior or bounded), the property of the exact controllability to the trajectories for the leader control remains open for nonlinear systems, and 3) as it can see, they do not present any numerical framework.…”

Controllability results for cascade systems of $m$ coupled $N$-dimensional Stokes and Navier-Stokes systems by $N-1$ scalar controls

“…In the case of scalar (heat) coupled equations an important number of challenging problems has been solved (see [1] for a survey of results until 2011) and sometimes the results have been surprising [2][3][4]. In the case of coupled Stokes or Navier-Stokes systems, to our knowledge, only some cases of two coupled systems have been treated [5,6,18,29]. Here our aim is to generalize results for a m scalar cascade system [17] to a m N -dimensional Stokes or Navier-Stokes cascade system but including an extra deal: to eliminate one component on the N -dimensional control.…”

Some remarks on the Robust Stackelberg controllability for the heat equation with controls on the boundary

“…In the recent past, several authors have applied successfully the hierarchic control method for a wide variety of equations and solving different kind of objectives, see, among others, [1,2,3,11,22,24,28,33]. In particular, in [24], the authors proposed to combine the notion of hierarchic control for the heat equation introduced in [31] with the concept of robust control (see e.g.…”