Exact controllability to eigensolutions of the bilinear heat equation on compact networks

TL;DR

This paper establishes the exact controllability to eigensolutions for the heat equation on compact networks using bilinear control, adapting recent methods and constructing biorthogonal families under non-uniform eigenvalue gaps.

Contribution

It introduces a novel approach to control the heat equation on networks, extending controllability results to diffusive models with bilinear control.

Findings

Achieved exact controllability to eigensolutions on networks.

Constructed biorthogonal families under non-uniform eigenvalue gaps.

Applied results to star and tadpole graph structures.

Abstract

Partial differential equation on networks have been widely investigated in the last decades in view of their application to quantum mechanics (Schr\"odinger type equations) or to the analysis of flexible structures (wave type equations). Nevertheless, very few results are available for diffusive models despite an increasing demand arising from life sciences such as neurobiology. This paper analyzes the controllability properties of the heat equation on a compact network under the action of a single input bilinear control. By adapting a recent method due to [F.~Alabau-Boussouira, P.~Cannarsa and C.~Urbani, {\em Exact controllability to eigensolutions for evolution equations of parabolic type via bilinear control}, arXiv:1811.08806], an exact controllability result to the eigensolutions of the uncontrolled problem is obtained in this work. A crucial step has been the construction of a suitable biorthogonal family under a non-uniform gap condition of the eigenvalues of the Laplacian on a graph. Application to star graphs and tadpole graphs are included.