Null-controllable data for cascade heat equations: almost sharp spectral spaces and modal critical times

Abstract

We study one-dimensional cascades of heat equations controlled through their last component. For a fixed time $T>0$, we consider the Hilbert space ${\mathcal{C}}_T$ of all initial data that can be steered exactly to zero at time $T$, endowed with the norm of the minimal $L^2$ control. This space can be strongly anisotropic: in explicit examples, it is incomparable with every isotropic spectral Sobolev or sub-Gaussian Gevrey scale associated with an adapted basis. For two equations, under the block Fattorini--Hautus condition, we give an almost sharp spectral description of ${\mathcal{C}}_T$. More precisely, in an adapted Riesz basis of generalized eigenvectors we define an explicit increasing scale $({\mathcal{V}}_{\tau }{)}_{\tau \in \mathbb{R}}$ of Hilbert spaces such that ${\mathcal{V}}_{T-\epsilon }\hookrightarrow {\mathcal{C}}_T\hookrightarrow {\mathcal{V}}_T$ for every $\epsilon >0$. The weights of ${\mathcal{V}}_T$ combine heat dissipation with the loss of observation inside each double generalized eigenspace. When the coupling and control regions are disjoint, this recovers the known minimal null-control time $T^{\star }$ and identifies it as a transition time for controllable data: for $T{T}^{\star }$, the natural state space is properly contained in ${\mathcal{C}}_T$. We extend the analysis to arbitrary finite upper triangular heat cascades. The observation losses of the different directions in each spectral block may have distinct exponential orders, leading to a hierarchy of critical times rather than a single threshold. This hierarchy describes successive gains of null controllability on adapted modal subspaces and accounts for the interactions and cancellations between different coupling paths.