Controllability of wave-heat and heat-wave cascades
Abstract
We study boundary controllability of one-dimensional coupled hyperbolic-parabolic cascades, focusing on the fine structure of reachable sets. The main model is a wave-heat cascade in which a boundary control acts on the wave equation and drives the heat equation through an internal coupling. We provide a sharp minimal time for the hyperbolic part (T > 2L) and a complete spectral characterization of exact controllability in weighted Hilbert spaces, whose definition depends explicitly on the coupling profile through a sequence of modal coefficients. In particular, internal couplings may generate nonstandard highly irregular controllability spaces and yield a generic (full measure) but non-robust controllability property. The analysis relies on Riesz basis decompositions and on an Ingham-M{\"u}ntz inequality. We also prove that the exact controllability space is not invariant along Hilbert Uniqueness Method trajectories: even if both endpoints belong to the controllability space, the associated minimal-energy trajectory may leave it at intermediate times. Finally, we compare with the reversed (heat-wave) cascade and discuss how reversing the direction of the coupling transfers the loss of regularity between the parabolic and hyperbolic components.
Keywords
Cite
@article{arxiv.2601.18212,
title = {Controllability of wave-heat and heat-wave cascades},
author = {Hugo Lhachemi and Christophe Prieur and Emmanuel Trélat},
journal= {arXiv preprint arXiv:2601.18212},
year = {2026}
}