Controllability of wave-heat and heat-wave cascades

TL;DR

This paper investigates the boundary controllability of coupled wave-heat systems, providing precise controllability times, spectral characterizations, and analyzing how internal couplings affect controllability spaces and regularity properties.

Contribution

It offers a sharp minimal controllability time, a spectral framework for controllability, and insights into how internal couplings influence controllability spaces and regularity.

Findings

Minimal controllability time T > 2L for hyperbolic part

Spectral characterization of controllability in weighted spaces

Couplings can create irregular controllability spaces and affect regularity

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.