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

  • *Corresponding author: Piermarco Cannarsa

    *Corresponding author: Piermarco Cannarsa 
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  • Partial differential equations on networks have been widely investigated in the last decades in view of their application to quantum mechanics (Schrödinger 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, C. Urbani, 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.

    Mathematics Subject Classification: Primary: 93B05, 35R02.


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  • Figure 1.  Internal and external vertices in a compact graph

    Figure 2.  Figure (A) illustrates the result of Thereom 4.2: the solution of problem (22) with initial condition lying in the colored region can be driven to the first eigensolution $ \varphi_1 $ in time $ T_R $ (which is uniform for any $ y_0 $ in the strip). In figure (B) we highlighted the cone of amplitude $ 2\arctan(R) $ of initial conditions which can be steered to the trajectory $ \zeta_1 $ in time $ T_R $. Since $ R $ is arbitrary, we are able to apply Theorem 4.3 for any $ y_0\in X\setminus \phi_1^\perp $

    Figure 3.  The figure shows the parametrization of a star graph with $ 3 $ edges

    Figure 4.  The parametrization of the tadpole graph and its symmetry axis $ r $

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