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

<p style='text-indent:20px;'>Partial differential equations on networks have been widely investigated in the last decades view of their application to quantum mechanics (Schrödinger type equations) or analysis flexible structures (wave equations). Nevertheless, very few results are available for diffusive models despite an increasing demand arising from life sciences such as neurobiology. This paper analyzes controllability properties heat equation a compact network under action single input bilinear control.</p><p style='text-indent:20px;'>By adapting recent method due [F. Alabau-Boussouira, P. Cannarsa, C. Urbani, <i>Exact eigensolutions evolution parabolic via control</i>, arXiv: 1811.08806], exact result uncontrolled problem is obtained this work. A crucial step has construction suitable biorthogonal family non-uniform gap condition eigenvalues Laplacian graph. Application star graphs and tadpole included.</p>