Approximate velocity control of a viscoelastic fluid with prescribed transport
Let $\Omega \subset \mathbb{R}^2$ be a bounded connected smooth domain, $\Gamma \subset \partial\Omega$ a nonempty relatively open set, $T > 0$ and $a, b, \nu > 0$. Let $z \in C^\infty(\overline\Omega \times [0,T]; \mathbb{R}^2)$ satisfy $\nabla \cdot z = 0$, with $z = 0$ on $\partial\Omega$, and consider $$u_t + (z \cdot \nabla) u - \nu \Delta u + \nabla p = \nabla \cdot \tau, \qquad \nabla \cdot u = 0, \qquad \tau_t + a \tau = b D(u)$$ in $\Omega \times (0,T)$, where $D(u) = (\nabla u + \nabla u^T)/2$, the extra-stress tensor $\tau$ is symmetric, $u = f \mathbf{1}_\Gamma$ on $\partial\Omega \times (0,T)$, and $u(0) = u_0$, $\tau(0) = \tau_0$. The control $f \in L^2(\Gamma \times (0,T); \mathbb{R}^2)$ is required to satisfy the zero-total-flux compatibility condition. Let $H_\sigma$ be the closure in $L^2(\Omega; \mathbb{R}^2)$ of the divergence-free fields in $C_c^\infty(\Omega; \mathbb{R}^2)$.
With $z = 0$ the system is the linearisation at zero of the Oldroyd model of a viscoelastic fluid. In that case it is approximately controllable in $u$ (Doubova and Fernández-Cara): for all $u_0, u_T \in H_\sigma$, all $\tau_0$ and all $\varepsilon > 0$ some boundary control brings $u(T)$ within $\varepsilon$ of $u_T$ in $L^2$. It is not null controllable in $u$, even with controls on the whole boundary (Fernández-Cara, Machado and Souza).
Problem. Is it true that for every $u_0, u_T \in H_\sigma$, every $\tau_0 \in L^2(\Omega; \mathbb{R}^{2 \times 2}_{\mathrm{sym}})$ and every $\varepsilon > 0$ there are such a control $f$ and an associated weak solution, continuous in $L^2$ at $t = 0$ and $t = T$, with $$\|u(T) - u_T\|_{L^2(\Omega)} < \varepsilon?$$ No terminal condition is imposed on $\tau$.
Source: Fernández-Cara (2025), §7, Problem 14, for a given transport field z in a bounded Lipschitz domain of R^N; stated in this form, in the plane with a smooth divergence-free field vanishing on the boundary, by AIM — Open Applied Problems.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- OC — Optimization and Control
- Secondary
- AP
- Keywords
- AIM Open Applied Problems, approximate controllability, boundary control, viscoelastic fluids
References
@article{fernandezcara2025,
author = {Fern{\'a}ndez-Cara, Enrique},
title = {Remarks on control and inverse problems for {PDE}s},
journal = {SeMA J.},
volume = {82},
number = {3},
year = {2025},
pages = {267--288},
doi = {10.1007/s40324-024-00363-7},
}
@article{doubova2012,
author = {Doubova, Anna and Fern{\'a}ndez-Cara, Enrique},
title = {On the control of viscoelastic {J}effreys fluids},
journal = {Systems Control Lett.},
volume = {61},
number = {4},
year = {2012},
pages = {573--579},
doi = {10.1016/j.sysconle.2012.02.003},
}
@article{fernandezcara2020,
author = {Fern{\'a}ndez-Cara, Enrique and Machado, Jos{\'e} Lucas F. and Souza, Diego A.},
title = {Non null controllability of {S}tokes equations with memory},
journal = {ESAIM Control Optim. Calc. Var.},
volume = {26},
year = {2020},
pages = {72},
doi = {10.1051/cocv/2019067},
}
@misc{aim-oseen-oldroyd-approximate-boundary-control,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Approximate velocity control of a viscoelastic fluid with prescribed transport}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/397-oseen-oldroyd-approximate-boundary-control.md},
note = {Entry 397 at commit a153f46, 2026-10-13},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00542, "Approximate velocity control of a viscoelastic fluid with prescribed transport", clopen.solutions.