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June  2022, 15(6): 1377-1401. doi: 10.3934/dcdss.2022011

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

1. 

Department of Mathematics, University of Rome Tor Vergata, Via della Ricerca Scientifica 1, 00133 Rome, Italy

2. 

Université Paris-Saclay, UVSQ, Laboratoire de Mathématiques de Versailles, 45 avenue des États-Unis 78035 Versailles cedex, France

*Corresponding author: Piermarco Cannarsa

Received  October 2021 Published  June 2022 Early access  January 2022

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.

Citation: Piermarco Cannarsa, Alessandro Duca, Cristina Urbani. Exact controllability to eigensolutions of the bilinear heat equation on compact networks. Discrete and Continuous Dynamical Systems - S, 2022, 15 (6) : 1377-1401. doi: 10.3934/dcdss.2022011
References:
[1]

L. F. AbbottE. Farhi and S. Gutmann, The path integral for dendritic trees, Biological Cybernetics, 66 (1991), 49-60. 

[2]

F. Alabau-Boussouira, P. Cannarsa and C. Urbani, Exact controllability to eigensolutions for evolution equations of parabolic type via bilinear control, preprint, arXiv: 1811.08806.

[3]

F. Alabau-BoussouiraP. Cannarsa and C. Urbani, Superexponential stabilizability of evolution equations of parabolic type via bilinear control, J. Evol. Equ., 21 (2021), 941-967.  doi: 10.1007/s00028-020-00611-z.

[4]

S. Alexander, Superconductivity of networks. A percolation approach to the effects of disorder, Phys. Rev. B, 27 (1983), 1541-1557.  doi: 10.1103/physrevb.27.1541.

[5]

K. Ammari and A. Duca, Controllability of localized quantum states on infinite graphs through bilinear control fields, Internat. J. Control, 94 (2021), 1824-1837.  doi: 10.1080/00207179.2019.1680868.

[6]

K. Ammari and A. Duca, Controllability of periodic bilinear quantum systems on infinite graphs, J. Math. Phys., 61 (2020), 101507, 15 pp. doi: 10.1063/5.0010579.

[7]

S. A. Avdonin and W. Moran, Simultaneous controllability of elastic systems and beams, Systems Control Lett., 44 (2001), 147-155.  doi: 10.1016/S0167-6911(01)00137-2.

[8]

S. A. Avdonin and M. Tucsnark, Simultaneous controllability in sharp time for two elastic strings, ESAIM Control Optim. Calc. Var., 6 (2001), 259-273.  doi: 10.1051/cocv:2001110.

[9]

J. M. BallJ. E. Marsden and M. Slemrod, Controllability for distributed bilinear systems, SIAM J. Control Optim., 20 (1982), 575-597.  doi: 10.1137/0320042.

[10]

J. A. Bárcena-Petisco, M. Cavalcante, G. Coclite, N. de Nitti and E. Zuazua, Control of hyperbolic and parabolic equations on networks and singular limits, preprint, hal-03233211.

[11]

K. Beauchard, Local controllability and non-controllability for a 1D wave equation with bilinear control, J. Differential Equations, 250 (2011), 2064-2098.  doi: 10.1016/j.jde.2010.10.008.

[12]

K. Beauchard and C. Laurent, Local controllability of 1D linear and nonlinear Schrödinger equations with bilinear control, J. Math. Pures Appl., 94 (2010), 520-554.  doi: 10.1016/j.matpur.2010.04.001.

[13]

A. BenabdallahF. Boyer and M. Morancey, A block moment method to handle spectral condensation phenomenon in parabolic control problems, Ann. H. Lebesgue, 3 (2020), 717-793.  doi: 10.5802/ahl.45.

[14]

G. Berkolaiko and P. Kuchment, Introduction to Quantum Graphs, Mathematical Surveys and Monographs, 186. American Mathematical Society, Providence, RI, 2013. doi: 10.1090/surv/186.

[15]

P. CannarsaP. Martinez and J. Vancostenoble, Sharp estimate of the cost of controllability for a degenerate parabolic equation with interior degeneracy, Minimax Theory Appl., 6 (2021), 251-280. 

[16]

P. CannarsaP. Martinez and J. Vancostenoble, The cost of controlling weakly degenerate parabolic equations by boundary controls, Math. Control Relat. Fields, 7 (2017), 171-211.  doi: 10.3934/mcrf.2017006.

[17]

P. Cannarsa and C. Urbani, Superexponential stabilizability of degenerate parabolic equations via bilinear control, In Inverse Problems and Related Topics, Springer Singapore, 310 (2020), 31–45. doi: 10.1007/978-981-15-1592-7_2.

[18]

C. Castro and E. Zuazua, A hybrid system consisting of two flexible beams connected by a point mass: Spectral analysis and well-posedness in asymmetric spaces, In Elasticité, Viscoélasticité et Contrôle Optimal, (ESAIM Proc.), 2 (1997), 17–54. doi: 10.1051/proc:1997022.

[19]

R. Dáger, Observation and control of vibrations in tree-shaped networks of strings, SIAM J. Control Optim., 43 (2004), 590-623.  doi: 10.1137/S0363012903421844.

[20]

R. Dáger and E. Zuazua, Wave Propagation, Observation and Control in $1\text{-}d$ Flexible Multi-Structures, Springer-Verlag, Berlin, 2006. doi: 10.1007/3-540-37726-3.

[21]

R. Dáger and E. Zuazua, Spectral boundary controllability of networks of strings, C. R. Math., 334 (2002), 545-550.  doi: 10.1016/S1631-073X(02)02314-2.

[22]

B. Dekoninck and S. Nicaise, Control of networks of Euler-Bernoulli beams, ESAIM Control Optim. Calc. Var., 4 (1999), 57-81.  doi: 10.1051/cocv:1999103.

[23]

B. Dekoninck and S. Nicaise, The eigenvalue problem for networks of beams, Linear Algebra Appl., 314 (2000), 165-189.  doi: 10.1016/S0024-3795(00)00118-X.

[24]

A. Duca, Bilinear quantum systems on compact graphs: Well-posedness and global exact controllability, Automatica J. IFAC, 123 (2021), Paper No. 109324, 13 pp. doi: 10.1016/j.automatica.2020.109324.

[25]

A. Duca, Global exact controllability of bilinear quantum systems on compact graphs and energetic controllability, SIAM J. Control Optim., 58 (2020), 3092-3129.  doi: 10.1137/18M1212768.

[26]

C. Flesia, R. Johnston and H. Kunz, Strong localization of classical waves: A numerical study, Europhysics Letters (EPL), 3 (1987), 497–502.

[27]

A. L. Hodgkin and A. F. Huxley, A quantitative description of membrane current and its application to conduction and excitation in nerve, J. Physic, (1952), 500–544.

[28]

A. Y. Khapalov, Controllability of Partial Differential Equations Governed by Multiplicative Controls, Lecture Notes in Mathematics, 1995. Springer-Verlag, Berlin, 2010. doi: 10.1007/978-3-642-12413-6.

[29]

P. Kuchment, Quantum graphs. I. Some basic structures, Waves Random Media, 14 (2004), 107-128. 

[30]

R. Mittra and S. W. Lee, Analytical Techniques in the Theory of Guided Waves, New York: Macmillan, 1971.

[31]

D. Mugnolo, Parabolic Systems and Evolution Equations on Networks, Universität Ulm, Zusammenfassung der Habilitation, 2010.

[32]

L. Pauling, The diamagnetic anisotropy of aromatic molecules, The Journal of Chemical Physics, 4 (1936), 673-677.  doi: 10.1063/1.1749766.

[33]

W. Rall, Branching dendritic trees and motoneuron membrane resistivity, Experimental Neurology, 1 (1959), 491-527.  doi: 10.1016/0014-4886(59)90046-9.

[34]

M. J. Richardson and N. L. Balazs, On the network model of molecules and solids, Annals of Physics, 73 (1972), 308-325.  doi: 10.1016/0003-4916(72)90285-0.

[35]

K. Ruedenberg and C. W. Scherr, Free-electron network model for conjugated systems. I. Theory, The Journal of Chemical Physics, 21 (1953), 1565-1581.  doi: 10.1063/1.1699299.

show all references

References:
[1]

L. F. AbbottE. Farhi and S. Gutmann, The path integral for dendritic trees, Biological Cybernetics, 66 (1991), 49-60. 

[2]

F. Alabau-Boussouira, P. Cannarsa and C. Urbani, Exact controllability to eigensolutions for evolution equations of parabolic type via bilinear control, preprint, arXiv: 1811.08806.

[3]

F. Alabau-BoussouiraP. Cannarsa and C. Urbani, Superexponential stabilizability of evolution equations of parabolic type via bilinear control, J. Evol. Equ., 21 (2021), 941-967.  doi: 10.1007/s00028-020-00611-z.

[4]

S. Alexander, Superconductivity of networks. A percolation approach to the effects of disorder, Phys. Rev. B, 27 (1983), 1541-1557.  doi: 10.1103/physrevb.27.1541.

[5]

K. Ammari and A. Duca, Controllability of localized quantum states on infinite graphs through bilinear control fields, Internat. J. Control, 94 (2021), 1824-1837.  doi: 10.1080/00207179.2019.1680868.

[6]

K. Ammari and A. Duca, Controllability of periodic bilinear quantum systems on infinite graphs, J. Math. Phys., 61 (2020), 101507, 15 pp. doi: 10.1063/5.0010579.

[7]

S. A. Avdonin and W. Moran, Simultaneous controllability of elastic systems and beams, Systems Control Lett., 44 (2001), 147-155.  doi: 10.1016/S0167-6911(01)00137-2.

[8]

S. A. Avdonin and M. Tucsnark, Simultaneous controllability in sharp time for two elastic strings, ESAIM Control Optim. Calc. Var., 6 (2001), 259-273.  doi: 10.1051/cocv:2001110.

[9]

J. M. BallJ. E. Marsden and M. Slemrod, Controllability for distributed bilinear systems, SIAM J. Control Optim., 20 (1982), 575-597.  doi: 10.1137/0320042.

[10]

J. A. Bárcena-Petisco, M. Cavalcante, G. Coclite, N. de Nitti and E. Zuazua, Control of hyperbolic and parabolic equations on networks and singular limits, preprint, hal-03233211.

[11]

K. Beauchard, Local controllability and non-controllability for a 1D wave equation with bilinear control, J. Differential Equations, 250 (2011), 2064-2098.  doi: 10.1016/j.jde.2010.10.008.

[12]

K. Beauchard and C. Laurent, Local controllability of 1D linear and nonlinear Schrödinger equations with bilinear control, J. Math. Pures Appl., 94 (2010), 520-554.  doi: 10.1016/j.matpur.2010.04.001.

[13]

A. BenabdallahF. Boyer and M. Morancey, A block moment method to handle spectral condensation phenomenon in parabolic control problems, Ann. H. Lebesgue, 3 (2020), 717-793.  doi: 10.5802/ahl.45.

[14]

G. Berkolaiko and P. Kuchment, Introduction to Quantum Graphs, Mathematical Surveys and Monographs, 186. American Mathematical Society, Providence, RI, 2013. doi: 10.1090/surv/186.

[15]

P. CannarsaP. Martinez and J. Vancostenoble, Sharp estimate of the cost of controllability for a degenerate parabolic equation with interior degeneracy, Minimax Theory Appl., 6 (2021), 251-280. 

[16]

P. CannarsaP. Martinez and J. Vancostenoble, The cost of controlling weakly degenerate parabolic equations by boundary controls, Math. Control Relat. Fields, 7 (2017), 171-211.  doi: 10.3934/mcrf.2017006.

[17]

P. Cannarsa and C. Urbani, Superexponential stabilizability of degenerate parabolic equations via bilinear control, In Inverse Problems and Related Topics, Springer Singapore, 310 (2020), 31–45. doi: 10.1007/978-981-15-1592-7_2.

[18]

C. Castro and E. Zuazua, A hybrid system consisting of two flexible beams connected by a point mass: Spectral analysis and well-posedness in asymmetric spaces, In Elasticité, Viscoélasticité et Contrôle Optimal, (ESAIM Proc.), 2 (1997), 17–54. doi: 10.1051/proc:1997022.

[19]

R. Dáger, Observation and control of vibrations in tree-shaped networks of strings, SIAM J. Control Optim., 43 (2004), 590-623.  doi: 10.1137/S0363012903421844.

[20]

R. Dáger and E. Zuazua, Wave Propagation, Observation and Control in $1\text{-}d$ Flexible Multi-Structures, Springer-Verlag, Berlin, 2006. doi: 10.1007/3-540-37726-3.

[21]

R. Dáger and E. Zuazua, Spectral boundary controllability of networks of strings, C. R. Math., 334 (2002), 545-550.  doi: 10.1016/S1631-073X(02)02314-2.

[22]

B. Dekoninck and S. Nicaise, Control of networks of Euler-Bernoulli beams, ESAIM Control Optim. Calc. Var., 4 (1999), 57-81.  doi: 10.1051/cocv:1999103.

[23]

B. Dekoninck and S. Nicaise, The eigenvalue problem for networks of beams, Linear Algebra Appl., 314 (2000), 165-189.  doi: 10.1016/S0024-3795(00)00118-X.

[24]

A. Duca, Bilinear quantum systems on compact graphs: Well-posedness and global exact controllability, Automatica J. IFAC, 123 (2021), Paper No. 109324, 13 pp. doi: 10.1016/j.automatica.2020.109324.

[25]

A. Duca, Global exact controllability of bilinear quantum systems on compact graphs and energetic controllability, SIAM J. Control Optim., 58 (2020), 3092-3129.  doi: 10.1137/18M1212768.

[26]

C. Flesia, R. Johnston and H. Kunz, Strong localization of classical waves: A numerical study, Europhysics Letters (EPL), 3 (1987), 497–502.

[27]

A. L. Hodgkin and A. F. Huxley, A quantitative description of membrane current and its application to conduction and excitation in nerve, J. Physic, (1952), 500–544.

[28]

A. Y. Khapalov, Controllability of Partial Differential Equations Governed by Multiplicative Controls, Lecture Notes in Mathematics, 1995. Springer-Verlag, Berlin, 2010. doi: 10.1007/978-3-642-12413-6.

[29]

P. Kuchment, Quantum graphs. I. Some basic structures, Waves Random Media, 14 (2004), 107-128. 

[30]

R. Mittra and S. W. Lee, Analytical Techniques in the Theory of Guided Waves, New York: Macmillan, 1971.

[31]

D. Mugnolo, Parabolic Systems and Evolution Equations on Networks, Universität Ulm, Zusammenfassung der Habilitation, 2010.

[32]

L. Pauling, The diamagnetic anisotropy of aromatic molecules, The Journal of Chemical Physics, 4 (1936), 673-677.  doi: 10.1063/1.1749766.

[33]

W. Rall, Branching dendritic trees and motoneuron membrane resistivity, Experimental Neurology, 1 (1959), 491-527.  doi: 10.1016/0014-4886(59)90046-9.

[34]

M. J. Richardson and N. L. Balazs, On the network model of molecules and solids, Annals of Physics, 73 (1972), 308-325.  doi: 10.1016/0003-4916(72)90285-0.

[35]

K. Ruedenberg and C. W. Scherr, Free-electron network model for conjugated systems. I. Theory, The Journal of Chemical Physics, 21 (1953), 1565-1581.  doi: 10.1063/1.1699299.

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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