Higher-dimensional null-controllability with distributed and boundary controls

Establish the analogue, in spatial dimensions N>1, of the one-dimensional null-controllability and Kalman rank condition for linear parabolic systems with constant coefficients and controls acting on multiple distributed subdomains and boundary portions.

Background

The paper proves a necessary and sufficient Kalman rank condition for null-controllability of one-dimensional linear parabolic systems with several distributed and boundary controls. It also extends the distributed-control result to arbitrary spatial dimensions when no boundary controls are present.

The authors explain that their proof relies on a change of variables and on the commutation of the Laplacian with the change-of-basis matrix. They state that this strategy does not extend directly to space-varying coefficients and explicitly identify the multidimensional case with N>1 as unresolved. The open problem therefore concerns extending the one-dimensional result to genuinely higher-dimensional systems involving the combined distributed and boundary control configuration.

References

In this paper we used a particular strategy which does not apply to many other problems. For instance the case of space varying coefficients can not be analyzed the same way. Indeed we did a change of variable so we used the fact that $\Delta$ commutes with $P_1$.

As our result is based on the one of , the $N$-dimensional case with $N>1$ is still open.

— Null-controllability for some linear parabolic systems with controls acting on different parts of the domain and its boundary  (2609.09959 - Olive, 9 Sep 2026) in Section “Further results and comments,” item 3

We insist on the fact that this is a result in 1D; except in the framework of , the problem is open in higher space dimension.

— Boundary approximate controllability of some linear parabolic systems  (2609.09982 - Olive, 9 Sep 2026) in Section 'Only one control: m=1' (Section 2.4), immediately following Theorem 2.8