Partial differential equations on networks have been widely investigated in the last decades in view of their application to quantum mechanics (Schrodinger 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.

Cannarsa, P., Duca, A., Urbani, C. (2022). Exact controllability to eigensolutions of the bilinear heat equation on compact networks. DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS. SERIES S, 15(6), 1377-1401 [10.3934/dcdss.2022011].

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

Cannarsa P.
;
Urbani C.
2022-01-01

Abstract

Partial differential equations on networks have been widely investigated in the last decades in view of their application to quantum mechanics (Schrodinger 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.
2022
Pubblicato
Rilevanza internazionale
Articolo
Esperti anonimi
Settore MAT/05
Settore MATH-03/A - Analisi matematica
English
Con Impact Factor ISI
Key words and phrases
Bilinear control
heat equations
compact graph
biorthogonal family
Cannarsa, P., Duca, A., Urbani, C. (2022). Exact controllability to eigensolutions of the bilinear heat equation on compact networks. DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS. SERIES S, 15(6), 1377-1401 [10.3934/dcdss.2022011].
Cannarsa, P; Duca, A; Urbani, C
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/2108/389744
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