Remarks on Hierarchic Control for a Linearized Micropolar Fluids System in Moving Domains
Isaías Pereira de Jesus
TL;DR
This work analyzes a hierarchical Stackelberg–Nash control problem for a linearized micropolar fluid in moving domains, mapping the moving region to a cylinder via a diffeomorphism $K(t)$ and studying a Nash equilibrium for followers and an approximate controllability objective for the leader. The authors prove existence and uniqueness of the follower Nash equilibrium under explicit smallness/coercivity conditions, and they characterize it through an adjoint system with explicit follower controls in terms of the adjoints. They establish approximate controllability of the state at final time and then use convex duality to derive the leader’s optimality system, showing the leader controls are given by dual variables supported on the observation domain. The results provide a rigorous foundation for hierarchical control of micropolar fluids in moving domains and suggest avenues for extending to nonlinear models and related PDEs.
Abstract
We study a Stackelberg strategy subject to the evolutionary linearized micropolar fluids equations in domains with moving boundaries, considering a Nash multi-objective equilibrium (non necessarily cooperative) for the "follower players" (as is called in the economy field) and an optimal problem for the leader player with approximate controllability objective. We will obtain the following main results : the existence and uniqueness of Nash equilibrium and its characterization, the approximate controllability of the linearized micropolar system with respect to the leader control and the existence and uniqueness of the Stackelberg-Nash problem, where the optimality system for the leader is given.