Hierarchical Control for the Oldroyd Equation in Memoriam to Professor Luiz Adauto Medeiros
Isaías Pereira de Jesus, Marcondes Rodrigues Clark, Alexandro Marinho Oliveira, Aldo Trajano Louredo
TL;DR
The Oldroyd fluid model with memory is given by $\left(1 + \lambda \frac{\partial}{\partial t}\right)\tau = 2\nu \left(1 + k\nu^{-1} \frac{\partial}{\partial t}\right)D$ and the motion by $\frac{\partial u}{\partial t} + (u \cdot \nabla)u - \mu \Delta u - \int_0^t g(t-\sigma)\Delta u(\sigma)\,d\sigma + \nabla p = F, \quad \nabla \cdot u = 0$ in the domain, with boundary conditions. The paper studies a linearized Oldroyd-type system (nonlinearity removed) under a Stackelberg-Nash hierarchy: a leader control $v$ acting on $\mathcal{O}$ and follower controls $w_i$ acting on $\mathcal{O}_i$, and seeks approximate controllability of the state at time $T$ via a Nash equilibrium of the follower problems and leader optimization. Existence and uniqueness of a Nash equilibrium are established via the Lax–Milgram lemma under bounded weight functions $\rho_i \in L^{\infty}(\Omega)$ and sufficiently small costs $\alpha_i$, and the reachable set $R(T) = \{u(\cdot,T,v,\mathbf{w}(v))\}$ is shown to be dense in the Hilbert space $H$, yielding approximate controllability. An optimality system coupling the leader and followers is derived: the follower controls are given by $w_i = -\alpha_i \psi_i$, where $\psi_i$ solves a backward adjoint-type system; the leader optimization is treated via Fenchel–Rockafellar duality, producing a variational inequality that yields the optimal leader control $v$ as $v = \varphi \chi_{\mathcal{O}}$ with $\{u,\varphi,\psi_i,\xi_i,\widehat{p},p\}$ solving a coupled forward–backward system. The results suggest extensions to nonlinear Oldroyd systems and Navier–Stokes-type models, and provide a rigorous framework for hierarchical control of viscoelastic fluids with memory.
Abstract
This manuscript deals with a hierarchical control problem for Oldroyd equation under the Stackelberg-Nash strategy. The Oldroyd equation model is defined by non-regular coefficients, that is, they are bounded measurable functions. We assume that we can act in the dynamic of the system by a hierarchy of controls, where one main control (the leader) and several additional secondary control (the followers) act in order to accomplish their given tasks: controllability for the leader and optimization for followers. We obtain the existence and uniqueness of Nash equilibrium and its characterization, the approximate controllability with respect to the leader control, and the optimality system for leader control.