Uniform null controllability for linearizations about arbitrary states

Establish null controllability, with control and state estimates uniform in the reference function \(\bar y\), for the linear parabolic systems obtained by linearizing the quasi-linear diffusion equation \(y_t-\nabla\cdot[a(\nabla y)\nabla y]=\chi_\omega v\) about \(\bar y\), in order to support a global inversion argument for global null controllability.

Background

The paper proves only local null controllability for the quasi-linear parabolic equation whose diffusion coefficient depends on the state gradient. The authors explain that a global inversion approach would require establishing null controllability for the displayed linearized system around every reference state yˉ\bar y, together with estimates for the control vv and state yy that are uniform with respect to yˉ\bar y. They explicitly identify this required result as open and probably difficult.

References

However, this requires in practice the resolution of null controllability problems for systems of the form

\begin{cases} y_t - \nabla \cdot \left[ a(\nabla \bar{y}) \nabla y + (a'(\nabla \bar{y})\cdot\nabla y) \nabla\bar{y} \right] = \chi_\omega v \ \text{ in } Q, \ y = 0 \ \text{ on } \Sigma, \ y|_{t=0} = y_0 \ \text{ in } \Omega . \end{cases}

with good estimates of~$v$ and~$y$, uniform with respect to~$\bar y$. And this is an open and probably difficult problem.

— Local null controllability of a quasi-linear system and related numerical experiments  (2608.19023 - Fernández-Cara et al., 19 Aug 2026) in Section 1, immediately before Section 1 discussion of the numerical algorithm

Finally, recall that there are many open problems concerning the null control of semilinear and nonlinear systems. In particular, as already mentioned at the beginning of the paper, it is unknown whether the global controllability property holds for~sistema1fase.

— Local null controllability of a quasi-linear system and related numerical experiments  (2608.19023 - Fernández-Cara et al., 19 Aug 2026) in Section 5, Conclusions, further remarks and open questions