Higher-dimensional optimal control theory for quadratic-gradient heat equations

Extend the well-posedness and optimal-control framework for the quasilinear nonlinear heat-conduction equation with quadratic gradient term from one spatial dimension to spatial dimensions two and three.

Background

The paper develops well-posedness, differentiability of the control-to-state map, existence and optimality conditions for a one-dimensional quasilinear parabolic heat-conduction model containing the nonlinear term b(u)u_x2. The one-dimensional analysis relies essentially on embeddings such as H1(I) into L-infinity(I), which provide the regularity needed to control the quadratic gradient term.

The authors explicitly identify extension to two and three spatial dimensions as unresolved because the corresponding H1 embeddings into L-infinity fail and the multidimensional quadratic gradient term b(u)|nabla u|2 reaches the critical regularity threshold for their energy method. They mention gradient-constrained admissible frameworks as a possible starting point, but do not resolve the higher-dimensional problem.

References

The extension of the present framework to $d=2$ and $d=3$ stands out as the main open problem we leave for future work. The two- and three-dimensional cases are genuinely open and, in our view, the most interesting direction in which this work can be continued.

— Optimal control of a class of nonlinear heat conduction models  (2608.18896 - Menezes et al., 19 Aug 2026) in Section 1, Introduction; Section Conclusions and future perspectives

A side question --- of independent interest --- is the behaviour of the smallness threshold $\delta_0$ and of the regularization threshold $k_0$ as one passes from $d=1$ to $d\geq 2$: one would like to understand whether these constants degenerate in a controlled way and, if so, what the resulting effective dimension-dependent theory looks like.

— Optimal control of a class of nonlinear heat conduction models  (2608.18896 - Menezes et al., 19 Aug 2026) in Section Conclusions and future perspectives